Journal of Coastal Research
31
4
879–891
Coconut Creek, Florida
July 2015
Numerical Simulation of In-Line Response of a Vertical Cylinder in Regular Waves Yi Han†, Jie-Min Zhan†*, Wei Su†, Y.S. Li‡, and Quan Zhou† † Department of Applied Mechanics and Engineering SunYat-sen University Guangzhou, 510275, P.R. China
‡ Department of Civil and Environmental Engineering The Hong Kong Polytechnic University Kowloon, Hong Kong
ABSTRACT Han, Y.; Zhan, J-M.; Su, W.; Li, Y.S., and Zhou, Q. 2015. Numerical simulation of in-line response of a vertical cylinder in regular waves. Journal of Coastal Research, 31(4), 879–891. Coconut Creek (Florida), ISSN 0749-0208. In-line response of a flexibly mounted, vertical cylinder in a series of regular waves was studied with numerical simulation. A dynamic mesh scheme and a laminar flow model were adopted. The experimental and simulated time histories of five wave gages and the response of the cylinder were compared. At low-incident wave frequencies, although the ratios of the diameters of the cylinders to the wavelength were small, the oscillating cylinder had little influence on the flow field. The numerical responses of the cylinder were in very good agreement with the experimental data. As the frequency of the incoming wave increased, the diffraction effect caused by the cylinder became significant, and the simulated responses of the cylinder were slightly weaker than those of experiment. When the incident wave frequency approached the natural frequency of the cylinder, the cylinder oscillated in resonance, and the flow field was strongly influenced. The wave height behind the cylinder was reduced. At high-incident wave frequencies, the Reynolds number increased, and the numerical response of the cylinder became stronger than those of experiment. In conclusion, the numerical results of the cylinder response agreed well with the results of the experimental model. To improve the predictive accuracy of the numerical model, further study, such as with the use of a turbulence model, should be carried out in the future.
ADDITIONAL INDEX WORDS: Oscillation, wave-structure interaction, dynamic mesh, Stokes wave.
INTRODUCTION Cylindrical structures are basic elements in coastal and ocean engineering works. The study of many coastal and offshore elements can be simplified as circular cylinders, such as bridge piers, cables, tension leg platforms, and aquatic plants. Therefore, there is a need to study the flow-induced vibration of cylindrical structures. In the past decades, many studies have been conducted on the performance of oscillatory cylinders in a fluid. Comprehensive reviews on this topic can be found in various publications (Gabbai and Benaroya, 2005; Sarpkaya and Isaacson, 1981; Sumer and Fredse, 2006). Besides theoretical analysis and experimental measurement, numerical methods are an alternative method for investigating the coupled problem of the interaction between structures and fluids. For the flow-induced vibration problem, four basic issues should be considered in any numerical simulation: the modeling of the flow field, the modeling of the structural vibration, the modeling of the fluid–structure interaction, and the data analysis. The coupling aspect can be addressed within the modeling of the fluid–structure interaction (So and Wang, 2003). In numerical simulations, one of the difficulties is the varying position of the structures during each cycle of oscillation. Therefore, updating the meshes around the moving boundary of the structure should be carefully considered. So far, two ideas have been used to try to address this problem. DOI: 10.2112/JCOASTRES-D-13-00052.1; accepted in revision 3 March 2013; accepted in revision 4 June 2013; corrected proofs received 16 September 2013; published pre-print online 6 November 2013. *Corresponding author:
[email protected] Ó Coastal Education and Research Foundation, Inc. 2015
One approach has been to allow the meshes to deform following the movement of the structure (Bai and Eatock Taylor, 2006; Placzek, Sigrist, and Hamdouni, 2009; Wu and Hu, 2004; Zhao, Cheng, and An, 2012). This method is straightforward, but the challenge is in how to maintain a fine grid and avoid excessive distortion of the meshes in the boundary layer, especially when the magnitude of oscillation of the structure is large. The other method is to use a structure-fixed coordinate system (Dutsch ¨ et al., 1998; Zhou, So, and Lam, 1999). The advantage of this method is that, no matter how the structure moves, the shape of the grid around it does not be changed. However, additional computational time has to be spent on retransforming the vector fields. If the computational domain is large, the time consumption can be considerable. Mittal and Kumar (2001) proposed a compromise scheme, which combined the cores of the two ideas mentioned above, to deal with the remesh problem. In their model, the structure was located in a square box, in which, the mesh moved together with the structure. On the other hand, the computational domain outside the box was fixed. Therefore, the mesh close to the structure did not undergo any deformation and contributed to the accuracy of the resulting solution by preserving the gradients in the boundary layers. The motion of the cylinder was addressed by adjusting the nodes at the boundaries of the box. General speaking, cylinder oscillation in flows are due to currents and waves. Nevertheless, the basic study is the interaction between a moving cylinder and the still water. Dutsch ¨ et al. (1998) investigated the laminar flow induced by the harmonic inline oscillation of a circular cylinder in water
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at rest, with time-averaged laser Doppler anemometry (LDA) measurements and time-resolved numerical flow predictions. Hu, Wu, and Ma (2002) used a fully nonlinear potential model based on a finite-element method to investigate the nonlinear wave motion around a moving circular cylinder, with both transient and sinusoidal motions. They found that fully nonlinear results agreed better than did those of linear theory. As a main aspect of current-induced motion, vortexinduced vibration (VIV) of an elastically mounted cylinder has been widely investigated in recent years. The responses of the cylinder, sorted by the relative direction of motion with that of the incident flow, are of three types: inline, transverse or cross-flow, and bidirectional. Early works have been primarily focused on the one-degree-of-freedom (1DOF) vibration in the cross-flow direction. For instance, Shiels, Leonard, and Roshko (2001) used a high-resolution viscousvortex method to simulate the oscillation of a dynamically supported circular cylinder in a flow for a range of dynamic parameters. Guilmineau and Queutey (2004) and Pan, Cui, and Miao (2007) studied the dynamics of an elastically mounted, rigid cylinder by using Reynolds-averaged Navier Stokes (RANS) codes with a turbulence model. Zhao, Cheng, and An (2012) investigated the 1DOF VIV of a circular cylinder in an oscillatory flow with a numerical method. Recently, studies have begun to analyze the 2DOF responses of a flexibly mounted cylinder in fluid. For example, Jauvtis and Williamson (2004) studied 2DOF VIV numerically and found a new response branch, the ‘‘super upper’’ branch. Lee, Lee, and Chang (2013) investigated the vibration characteristics of two-dimensional motions of cylinders by using the immersed finite element method (IFEM). Zhao, Tong, and Cheng (2012) simulated the 2DOF VIV of a circular cylinder between two lateral plane walls in steady currents. Comparatively, few numerical investigations have focused on the current-induced inline response of an elastically mounted cylinder. Anagnostopoulos and Iliadis (1998) studied numerically the inline response of a flexible cylinder in an oscillating stream with a Reynolds number equal to 200 and Keulegan-Carpenter numbers ranging between 2 and 20. The current-induced vibration of an elastically mounted cylinder is usually simulated by assuming two-dimensional flows because the vertical motion of fluid is usually considered small. On the other hand, characteristics of the excited oscillation of a flexibly mounted cylinder in a wavy flow field have been well studied experimentally (Downes, 1999; Downes and Rockwell, 2003; Li et al., 2007; Li, Zhan, and Lau, 1997; Ozgoren and Rockwell, 2007; Verley, 1980). However, there are few numerical investigations on this topic. Although some similar studies, such as the response of floating cylinders in waves (Liu, Xue, and Yue, 2001; Wu and Hu, 2004), are available for reference, further study on this topic should be carried out to fill the gap. In this study, the inline response of a flexibly mounted, vertical cylinder in a series of regular waves was simulated. The flow field was solved by a three-dimensional, unsteady, incompressible numerical model. The free surface was tracked by the volume of fluid (VOF) model. A dynamic mesh scheme
was used to update the grid system in accordance with the motion of the cylinder. The equation for the oscillation of the cylinder was solved by the fourth-order Runge-Kutta algorithm.
GOVERNING EQUATIONS A three-dimensional, unsteady, incompressible numerical model was employed for solving the flow field. The conservation equations are expressed as follows: ]q þ Ñ ðq~ vÞ ¼ 0 ]t ] ~ ðq~ vÞ þ Ñ ðq~ v~ vÞ ¼ Ñp þ q~ g þ lÑ2~ vþS ]t
ð1Þ
ð2Þ
where q is density, ~ v is velocity vector, p is the static pressure, l ~ is is dynamic viscosity, ~ g is gravitational force vector, and S external body force vector. The tracking of the free surface was accomplished by solving a continuity equation for the volume fraction. For the qth phase, the equation was as follows: ] ðaq qq Þ þ Ñ ðaq qq~ vq Þ ¼ 0 ]t
ð3Þ
where aq is the volume fraction of the qth phase. In the present study, only two phases, water and air, are considered: aw þ aa ¼ 1
ð4Þ
Second-order Stokes wave theory is usually applied to intermediate and deep water waves with a small but finite wave height and is used as the incident wave: g¼
uin ¼
H cosðkx xtÞ 2 pH 2 3 þ 1þ cot kd cos 2ðkx xtÞ 4L 2 sinh2 kd pH cosh kðz þ dÞ cosðkx xtÞ T sinh kd þ
win ¼
ð5Þ
3p2 H 2 cosh 2kðz þ dÞ cos 2ðkx xtÞ 4TL sinh4 kd
ð6Þ
pH sinh kðz þ dÞ sinðkx xtÞ T sinh kd þ
3p2 H 2 sinh 2kðz þ dÞ sin 2ðkx xtÞ 4TL sinh4 kd
ð7Þ
where g is the free-surface elevation, H is the wave height, k is the angular wave number, x is the angular frequency, t is time, L is wave length, d is water depth, T is wave period, and T ¼ 1/f, where f is wave frequency. uin and win are the velocities at x and z directions of the inlet boundary. Porous structures have been widely used as an energydissipation mechanism to eliminate wave reflection by adding a porous zone near the outlet boundary of the computational domain. Zhan et al. (2010) developed a wave-absorbing method, based on the porous media model, by adding a dissipation term in the Navier-Stokes equations. The additional term can be
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written as follows: Si ¼
l 1 vi þ Cr qjvjvi a 2
ð8Þ
where Si is the source term for the ith momentum equation as shown in Equation (2), jvj is the magnitude of the velocity, 1/a is the viscous resistance coefficient, and Cr is the inertial resistance factor. In the wave-absorbing zone, 1/a is given a linear form as follows: 1 x x0 ¼j a xe x0
ð9Þ
x0 , x , xe
where j is a fixed empirical coefficient, and x0 and xe are the xcoordinates of the start and end boundaries of the waveabsorbing zone, respectively. Simulation in this article skips the inertial resistance and Cr ¼ 0. Results in the following parts show that the wave-absorbing zone can work effectively with just the viscous-dissipation term. The cylinder is taken as a rigid body. and only its inline motion is considered. The 1DOF equation to be solved is the following: mX¨ þ cs X˙ þ ks X ¼ Fx
ð10Þ
where X is the displacement of the cylinder in the inline direction with respect to the static balance position, m is the mass of the cylinder, and cs and ks are the structural damping and spring stiffness, respectively. Fx is the fluid force in the inline direction. The dynamic mesh technique was used in the simulation. The integral form of the conservation equation for a general scalar / on a control volume V, whose boundary is moving, is written as follows: Z Z Z Z d ~ ¼ CÑ/ dA ~ þ S/ dV q/dV þ q/ð~ v~ vg Þ dA dt V
]V
V
]V
ð11Þ where ~ vg is the velocity of the moving mesh, C is the diffusion coefficient, S/ is the source term of /, and ]V presents the boundary of the control volume V. The first term in Equation (11) can be written in the following first-order backward-difference form: d dt
Z
q/dV ¼
ðq/VÞnþ1 ðq/VÞn Dt
ð12Þ
V
where n and n þ 1 denote the current and the next time level, respectively. At the (n þ 1)th time level, volume Vnþ1 can be written as follows: V nþ1 ¼ V n þ
dV Dt dt
ð13Þ
To satisfy the mesh conservation law, the volume–time derivative of the control volume, dV/dt, is computed as follows: dV ¼ dt
Z ]V
~¼ ~ vg dA
nf X j
~j ¼ ~ vg;j A
nf X dVj j
Dt
ð14Þ
~j is the j where nf is the number of faces of the control volume, A
Figure 1. Sketch of the computational model.
face area vector, and dVj is the swept volume by the control volume face j over the time step Dt.
NUMERICAL PROCEDURE The inline response of a flexibly mounted vertical cylinder in regular waves was investigated. Figure 1 shows a sketch of the computational model, which was set up according to the experiment conducted by Li et al. (2007). The numerical wave flume was 1.5 m wide, with d water depth of 0.6 m. The total length of the flume was 9L, where L is the wavelength of the incident wave. The diameter of the circular cylinder, D was 0.215 m. The cylinder was located at 5L from the wave inlet boundary. The wave-absorbing zone started at 7L and went to the end of the flume. Five wave gages were installed near the cylinder to capture the variations of the wave height for comparison with the experimental results. To simplify the model, the steel plates under the cylinder, which were used to constrain its movement, were not considered in this study. In other words, the cylinder was governed by Equation (10) in the computational model, with the values of the structural damping and stiffness coefficients, cs and ks, respectively, assigned according to the experimental data. The force term F x was calculated by integrating pressure and shear stress over the cylinder surface in every time step. For each layer of fluid, there were 80 nodes around the perimeter of the circular cylinder. The minimum mesh size in the radial direction (adjacent to the cylinder surface) was Dr ¼ 0.01D. To capture the free surface accurately, the grid system in the vertical direction was nonuniform. Meshes near the still water level were refined to Dz ¼ H/30 and became coarser near the bottom. In the wave-propagation direction, the mesh in front of the cylinder was set to Dx ¼ L/80, and the mesh behind the cylinder began at Dx ¼ L/60 and gradually became coarser toward the end of the flume. In summary, the total number of grids in the moving zone is 350,000, whereas that of the whole computational domain was about 2.2 millions. The size of the time step was set to Dt ¼ T/640. In every time step, the flow field was solved first. Then, the force term Fx in Equation (10) was updated by integrating the pressure and shear stress over the cylinder surface. The Fx
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Table 1. Parameters of the simulation.
Mass of cylinder (kg) Structural damping coefficient (N s/m) Spring stiffness (N/m) Wave-absorbing coefficient Mesh split coefficient Mesh merge coefficient
Parameter
Value
m cs ks j aspl amer
14.7 1.6 2711 7.5e6 0.5 0.5
VALIDATION Figure 2. Schematic sketch of the dynamic mesh scheme.
term was considered a constant within a small enough time step (Guilmineau and Queutey, 2004). After that, Equation (10) was solved with a fourth-order Runge-Kutta algorithm. The result of X˙ was used to update the velocities of the cylindrical boundary and the moving zone. A dynamic mesh scheme, similar to the ideas of Mittal and Kumar (2001), was used in the present study. Figure 2 displays the computational mesh near the cylinder. The grey zone is the moving zone, whereas the remaining colors are stationary zones. The two vertical planes separating the moving and stationary zones were fixed. All meshes inside the moving zone moved with the same velocity as the oscillatory cylinder at every time step. At the same time, the meshes inside the moving zone, adjacent to its boundaries, were updated at every time step in accordance with the movement of the moving zone. The detail is shown in Figure 2. These meshes were split or merged according to the following conditions: hj . ð1 þ aspl Þh0
ð15Þ
hj , amer h0
ð16Þ
where hj is the cell width of segment j, h0 is the cell width before oscillation, and aspl and amer are split and merge factors, respectively. Once Equation (15) was satisfied, cells in segment j were split. On the other hand, if Equation (16) was met, cells in segment j were merged with those of segment i. This scheme provided a simple rule to remesh the moving zone and to preserve the mesh quality around the cylinder to eliminate any distortion. The details of parameters used in this study are listed in Table 1. The Commercial Computation Fluid Dynamics package, FLUENT, was used to solve the three-dimensional flow field, similar to the method of other researchers (Maghsoodi et al., 2012; Salaheldin, Imran, and Chaudhry, 2004; Yazdi et al., 2010). The Pressure Staggering Option (PRESTO) discretization scheme was used for pressure. The Quadratic Upwind Interpolation of Convective Kinematics (QUICK) algorithm was used for the momentum, and the Pressure Implicit with Splitting of Operators (PISO) algorithm was used for the pressure–velocity coupling. A geometric reconstruction scheme was used to track the free surface. The code for solving the oscillation equation was programmed and incorporated into the solver.
Li et al. (2007) allowed the cylinder to oscillate freely in the air and measured the structural parameters through the oscillation time history. According to their records, the structural damping was (1.6 N s)/m, whereas the spring stiffness was 2711 N/m. Then, the free oscillation of the circular cylinder with an initial displacement in still water was simulated. Figure 3 compares the time series of the acceleration of the cylinder in the numerical simulation and in the experiment. The figure shows that the numerical result agrees well with that of the experiment, which proves that the chosen numerical method and dynamic mesh system are effective for simulation of cylinder oscillation. The natural frequency of the cylinder in water fcw was determined as 1.62 Hz.
RESULTS The inline response of a flexibly mounted vertical cylinder in a series of regular waves was simulated. The numerical experiments consisted of 15 cases, whose parameters are listed in Table 2. Reynolds number (Re) and Keulegan–Carpenter number (KC) were defined as follows: Re ¼
DUm m
ð17Þ
KC ¼
Um Tw D
ð18Þ
where Um is maximum velocity, m is the kinematic viscosity, and Tw is the period of the oscillatory flow. These two dimensionless numbers can be used to judge the regimes of flow around a smooth, circular cylinder in oscillatory flow. By summarizing results from different studies, Sumer and
Figure 3. Experimental and numerical records of free vibration in still water.
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Table 2. Parameters of incident regular waves. Case No.
f (Hz)
T (s)
H (m)
d (m)
uin_max (m/s)
KC
Re
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.50 0.67 0.84 1.00 1.15 1.28 1.39 1.49 1.56 1.61 1.67 1.72 1.82 1.92 2.00
2.00 1.49 1.19 1.00 0.87 0.78 0.72 0.67 0.64 0.62 0.60 0.58 0.55 0.52 0.50
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60
0.0235 0.0253 0.0284 0.0326 0.0373 0.0417 0.0455 0.0490 0.0515 0.0533 0.0556 0.0574 0.0613 0.0652 0.0683
0.218 0.175 0.157 0.152 0.151 0.151 0.152 0.153 0.154 0.154 0.155 0.155 0.157 0.158 0.159
5.05e3 5.43e3 6.11e3 7.01e3 8.02e3 8.96e3 9.78e3 1.05e4 1.11e4 1.15e4 1.20e4 1.23e4 1.32e4 1.40e4 1.47e4
Fredse (2006) presented several figures to clarify the relationships among different factors. In this research, we were able to predict Re and KC through the maximum wave velocity in the propagation direction uin_max, as shown in Table 2. Under this condition, KC ranges from 0.159 to 0.218, whereas Re ranges from 5000 to 15,000. Within these ranges, according to Sumer and Fredse (2006), separation in the flow field may not occur. Therefore, the laminar flow model was used in this study. In the following sections, detailed analyses of four typical cases are presented. Then, the response of the cylinder for all cases is summarized. Finally, possible errors are discussed. Figure 4a presents the time history of the cylinder acceleration at f ¼ 0.67 Hz. Here, the nondimensionalized parameters 2 ¨ were defined as s ¼ (t/T) and a* ¼ ½X=ðDf cw Þ. The numerical result showed that the acceleration record had sharp crests and
Figure 5. (a) Cylinder velocity and the free surface record of wg4 within a wave period at f ¼ 0.67 Hz. (b) Profile (left column) and vector field (right column) of the free surface near the cylinder at f ¼ 0.67 Hz.
Figure 4. (a) Time history of the cylinder acceleration at f¼0.67 Hz. (b) Time records of the wave gages at f ¼ 0.67 Hz.
flat troughs, which revealed that the cylinder oscillated nonlinearly in this wave frequency. Moreover, the experimental time history was not as stable and regular as the numerical one, although both the period and the amplitude of the two curves are close to each other before s ¼ 20. The reason can be found by examining Figure 4b, which hwg/d is the record of the nondimensionalized free-surface elevation of the wave gage. As shown in Figure 4b, wave gage records in experiment and simulation are identical before s¼20, but afterward, the shapes of experimental records gradually change, e.g., amplitude increases at wg1 and wg4, and the phase changes at wg2, wg3, and wg5, because of secondary reflection within the experimental flume. The wave tank used by Li et al. (2007) was 25 m long, whereas the wavelength for f ¼ 0.67 Hz was 2.97 m.
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Table 3. Average nondimensionalized wave amplitude (wg) of the wave gages vs. incident wave. Percentages are shown in parentheses. f (Hz)
H/2d
0.67 1.00 1.61 2.00
0.00835 0.00835 0.00835 0.00835
wg1 0.00880 0.00760 0.00810 0.00785
(5.4) (8.9) (2.7) (6.1)
wg2 0.00905 0.00680 0.00940 0.00960
(8.3) (18.3) (12.5) (15.1)
Although wave absorbing was adopted in the experiment, s¼20 is a long enough time for the initial, small secondary reflections to grow and affect the measurement. In comparison, numerical results are stable within a longer time span, and the reflection effects are not apparent. This shows the effectiveness of the wave-absorbing zone, and the contamination in the simulation results is avoided. Features of the interaction between the oscillating cylinder and the flow field should be captured. Figure 5a illustrates the cylinder motion and the free-surface record of wg4 within a ˙ wave period. In the figure, U*, defined as U* ¼ [X/(Df cw)], is the nondimensionalized velocity of the cylinder. The starting point s0, is selected when U* ¼ 0. Four characteristic points of time, s ¼ s0, s0 þ 1/4, s0 þ 1/2, and s0 þ 3/4, were selected for analysis. The graph shows there is a phase difference of about p between the cylinder velocity and the incident wave. Near the s0 þ 1/4 time, the cylinder reaches maximum positive velocity when the wave trough passes it. Nevertheless, even though the wave crest passes the cylinder near s0 þ 3/4, the maximum negative velocity appears after a time lag. This phenomenon indicates that the cylinder encounters more resistance when it moves opposite to the incident wave direction. Figure 5b displays the
wg3 0.00910 0.00755 0.01075 0.00625
(8.7) (9.9) (28.5) (25.3)
wg4 0.00885 0.00745 0.01010 0.00580
(5.7) (10.9) (20.7) (30.7)
wg5 0.00885 0.00855 0.00380 0.00545
(6.1) (2.2) (54.5) (35.0)
profile and vector field of the free surface near the cylinder. The nondimensional velocity of the flow field was defined as pffiffiffiffiffiffi ~ v¼~ v= gd. By comparing the subfigures, the vector fields around the cylinder show little difference, even though the cylinder is oscillating. The reason is that, with f ¼ 0.67 Hz and D/L ’ 0.07, the motion of the cylinder is small and has little influence on the flow field (Sumer and Fredse, 2006). Table 3 lists the numerical results of average nondimensional wave amplitudes measured by the wave gages compared with the incident wave amplitude. The percentages are the differences. Shown in Table 3, the wave amplitudes are close to each other. Therefore, along with the supplementary evidence shown in Figure 4b, the distortions of the experimental wave records after s ¼ 20 are not caused by the oscillating cylinder but are caused primarily by flume reflections. Figure 6a presents the time history of the cylinder acceleration at f ¼ 1.00 Hz. After the cylinder oscillation becomes stable at s . 12, the experimental records and numerical results coincide with each other. Figure 6b compares the time series of the wave gage records from experimental and simulation approaches. The agreement of results is within the error range. As Table 3 shows, although the differences
Figure 6. (a) Time history of the cylinder acceleration at f ¼ 1.00 Hz. (b) Time records of the wave gages at f ¼ 1.00 Hz.
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Figure 7. (a) Cylinder velocity and the free surface record of wg4 within a wave period at f ¼ 1.00 Hz. (b) Profile (left column) and vector field (right column) of the free surface near the cylinder at f ¼ 1.00 Hz.
between some wave gages (e.g., wg1, wg3, and wg4) and the incident wave are still around 10, that of wg2 already reaches 18.3. Therefore, as the value of D/L increases to 0.14, the influence of the cylinder on the flow field becomes significant. Figure 7a depicts the cylinder motion and the free surface record of wg4 within a wave period. The phase difference between the cylinder velocity and the wave record is 0.9p, slightly smaller than that under f ¼ 0.67 Hz. The profile of cylinder velocity has a similar shape at the crest and at the trough, which indicates that the motion of the cylinder has less nonlinearity than it does at f ¼ 0.67 Hz. Figure 7b displays the flow fields near the cylinder at four selected time points. At s ¼ s0, the cylinder is stationary and begins to move in the positive
885
x-direction. Water converges immediately in front of the cylinder after passing around it, and the water level in front of the cylinder is higher than that behind it, where a surface depression exists. At s ¼ s0 þ 1/4, the cylinder reaches maximum positive velocity when the wave trough just passes it. Water in front of the cylinder is pushed and part of it moves in the negative x-direction on both sides of the cylinder. At the same time, water behind the cylinder flows to fill the space left by the cylinder as it moves forward. As Figure 7b shows, these interactions result in a higher water level around the cylinder than in other parts of the wave trough. At s ¼ s0 þ 1/2, the motion of the cylinder stops. A low-velocity zone thus appears in front of the cylinder, within the range of the wave trough, which just leaves the cylinder. At the same time, a bulge caused by the interaction of wave crest and the cylinder emerges behind the cylinder. At s ¼ s0 þ 3/4, the cylinder moves at maximum speed in the negative x-direction when the wave crest has just passed. Similar to the phenomenon described for s ¼ s0 þ 1/4, the water level around the cylinder is lower than in other parts within the wave crest that are away from the cylinder. Under this condition, the frequency of the incident wave (1.61 Hz) is close to the natural frequency of the cylinder (1.62 Hz). The oscillation of the cylinder is near the resonance point. From Figure 8a, the amplitude of the oscillating cylinder is shown to be larger than in the previous cases. The results of experimental and simulation approaches are close to each other, except for having an amplitude difference of around 10 when s . 20. This discrepancy will be discussed further in a later section. Figure 8b compares the experimental and numerical time histories of the wave gages. Except for wg1, the matching of results from all other wave gages is acceptable during the recording time. For wg1, the amplitude of the experimental results becomes larger than the numerical ones after s ¼ 17. This phenomenon is due to the interaction of the reflected wave caused by the cylinder resonance and the rereflected wave from the wave paddle. The value of D/L is 0.36 in the present case. The diffraction of the wave field by the cylinder cannot be ignored (Isaacson, 1979; Sarpkaya and Isaacson, 1981; Sumer and Fredse, 2006). Results listed in Table 3 demonstrate the evident variations of the wave gage records. Compared with the incident wave, the amplitudes measured by wg3 and wg4 increased by 28.5 and 20.7, respectively, whereas the amplitude of wg5 decreased by 54.5. That reveals that the oscillating cylinder, at f ¼ 1.61 Hz, could reflect part of the wave energy and reduce the free surface elevation behind it. Figure 9a shows the cylinder motion and the free surface record of wg4 within a wave period. The phase difference of the curves is 0.43p. In contrast to the case of f ¼ 0.67 Hz, the cylinder is almost stationary when the wave crest or trough passes it. Figure 9b displays the free-surface elevation and the vector field near the cylinder at four time points. At s ¼ s0, the cylinder was ready to move in the positive x-direction when the wave crest is passing it. Crests and troughs, caused by the superposition of the incoming and reflected waves, could be seen behind the cylinder. Flows in front of the cylinder tended to converge at low speed. At s ¼ s0 þ 1/4, the cylinder reached the maximum positive velocity. Water in front of the cylinder was pushed sideways. Two low-speed
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Figure 8. (a) Time history of the cylinder acceleration at f ¼ 1.61 Hz. (b) Time records of the wave gages at f ¼ 1.61 Hz.
zones were generated symmetrically. At the same time, two high-speed zones appeared behind the cylinder. The wave trough at s ¼ s0 is split into two surface depressions, leaving the cylinder separate. The free surface behind the cylinder becomes moderately high under the influence of the incoming and reflected waves. At s ¼ s0 þ 1/2, the wave crest passed the cylinder, which just finished its motion in the positive xdirection when the water in front of it was still moving away. Obvious crests and troughs again emerge from behind the cylinder. At s ¼ s0 þ 3/4, the cylinder reached the maximum velocity in the negative x-direction. Flows behind the cylinder converged and moved in the same direction to fill the space left by the cylinder, which generated a low-speed zone. At the same time, water in front of the cylinder was pushed against
the incoming wave. That action lead to the generation of a narrow, low-velocity zone in front of the cylinder. Similar to the situation at s ¼ s0 þ 1/4, the wave crest was split into two separate bulges, resulting from the redistribution of energy induced by the oscillating cylinder. Figure 10a presents the time history for the cylinder acceleration at f ¼ 2.00 Hz. The phases of the experimental and simulated curves are the same. However, the amplitude of the simulation is larger than that of the experimental one. Figure 10b shows the records of the wave gage time series in both the experimental and simulation approaches. For all wave gages, the two records match satisfactorily, although they have some differences in both amplitude and phase. In this case, D/L ¼ 0.66, which means that the diameter of the
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Figure 9. (a) Cylinder velocity and the free surface record of wg4 within a wave period at f ¼ 1.61 Hz. (b) Profile (left column) and vector field (right column) of the free surface near the cylinder at f ¼ 1.61 Hz.
cylinder was more than one-half of the wavelength. The
According to Figure 11a, the phase difference between the
diffraction effect was more evident. From the results shown
cylinder velocity and the wave gage record is about 0.2p. The
in Table 3, it can be seen that the flow field is totally changed by the cylinder.
shape of the wave recorded by wg4 is distorted, which is another proof of the significant influence of the cylinder on
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Figure 10. (a) Time history of the cylinder acceleration at f ¼ 2.00 Hz. (b) Time records of the wave gages at f ¼ 2.00 Hz.
the flow field. Figure 11b displays the flow fields near the cylinder at the same four time points. At s ¼ s0, the cylinder is ready to move in the positive x-direction when the wave crest is at its back, which generates a runup. At the same time, the previous wave trough is split into two smaller troughs, generating a low-velocity zone. At s ¼ s0 þ 1/4, the cylinder reached its maximum positive velocity. The interaction of the reflected and incoming wave gives rise to a zone of negative velocity behind the cylinder. Water in front the cylinder was pushed to move into the two small troughs. At s ¼ s0 þ 1/2, the wave trough is at the back of the cylinder and causes a surface depression. Meanwhile, two low-speed zones emerge in front of and behind the cylinder. At s ¼ s0 þ 3/4, the cylinder reaches the maximum velocity in the negative x-direction and splits the surface depression in front of it. Flows behind the cylinder converge to the centerline and push the low-speed zone away.
Response of the Cylinder Both the experimental and numerical results for the amplitude of the response of the circular cylinder at the excitation frequency in regular waves at a wave height of 0.01 m are shown in Figure 12. All values in the figure are nondimensionalized. The data were analyzed using a standard Fast Fourier Transform package. The numerical results agree well with the experimental ones. The cylinder primarily oscillates at the same frequency as the incident waves. The resonance occurs at f/fcw ¼ 0.99 (e.g., f ¼ 1.61 Hz), near the natural frequency of the cylinder.
DISCUSSION The simulation results have been discussed in the above paragraphs, however, there are several aspects that warrant more-detailed discussion. From Figure 12, the simulated and experimental responses of the cylinder at f/fcw 0.62 (i.e. D/L 0.14) are almost the same. In the range 0.71 f/fcw 1.03 (i.e. 0.18 D/L 0.38), the simulated responses are underestimated when compared with the experimental ones. For larger ratios of frequencies, f/ fcw 1.06 (i.e. D/L 0.41), the simulated responses are overestimated. The difference in cylindrical motion reveals the errors in the flow field. Therefore, an accurate calculation of flow field is the prerequisite for an accurate prediction of the response of the cylinder. However, the behavior of the coupled oscillating cylinder and flow field system is complicated. A small change in the system could induce significant effects on the results. In the present study, the errors might come from several sources (e.g., the scheme of dynamic mesh update, the experimental data acquisition, and the numerical solution of the oscillation equation, etc.) or a combination of them. Thus, to improve prediction accuracy, further investigation will be needed in the future. Moreover, in the last four cases (f/fcw 1.06 ), while Re . 1.2e4, the numerical responses of the cylinder is stronger than the experimental ones. It suggests that there was more power for driving the cylinder in the numerical model than there was in the experiment. Because of the use of a laminar model, energy dissipation due to turbulence in the flow field was not considered. There is doubt about whether the flow is left unseparated in such a high Reynolds number range. Unfortu-
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Figure 11. (a) Cylinder velocity and the free surface record of wg4 within a wave period at f¼2.00 Hz. (b) Profile (left column) and vector field (right column) of the free surface near the cylinder at f ¼ 2.00 Hz.
nately, Li et al. (2007) does not provide detailed information about the flow regimes in their article. As a reference, Sumer and Fredse (2006) (fig. 3.15 in their book) provides a diagram about the effect of Reynolds number on flow regimes. It shows that there is no flow separation at KC , 0.16. However, the upper limit of the Reynolds number in the diagram is 8000. To our knowledge, there is no information in the literature on flow at such a low KC number with such a large Reynolds number. Therefore, a turbulence model should be used in the future to clarify the flow regime.
CONCLUSION The in-line response of a flexibly mounted vertical cylinder in regular waves was investigated numerically. A dynamic mesh
scheme and a laminar flow model were used. Four typical cases are selected and analyzed in detail. Experimental and simulated time records of five wave gages and the responses of the cylinder were compared. With a low-incident wave frequency and values for D/L and Reynolds number that are small as well, the oscillating cylinder has little influence on the flow field. Numerical responses of the cylinder are in good agreement with the experimental ones. As the frequency of the incoming wave increases and the value of D/L exceed 0.2, the diffraction effect caused by the cylinder becomes significant. At this stage, responses of the cylinder in simulations are slightly weaker than are those of the experiment.
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Figure 12. Comparison between cylindrical response in simulation and experiment.
When the incident wave frequency approaches the natural frequency of the cylinder, the cylinder oscillates in resonance. Because of the resonant motion of the cylinder, the flow field is strongly influenced. Part of the wave energy is reflected, and the wave height behind the cylinder is reduced. At high incoming-wave frequencies, the Reynolds number increases and the numerical responses of the cylinder become stronger than those of the experiment. In general, the responses of the cylinder in simulation and in the experiment agree well. To improve the accuracy of the numerical prediction, further study, along with the use of a suitable turbulence model, should be conducted.
ACKNOWLEDGMENT The work described in this paper was supported by an important project of National Marine Public Welfare Research Projects of China (Project No. 201005002).
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