of a towing carriage and a parallel linkage mechanism capable of translation motion in both IL and CF directions. The motion was calculated by measuring the spring force via calibrated load cells. Apparently, this device is based on a mass-spring-damper model.
The damping ratioζ is around 0.6%. The mechanism employed ensured identical mass ratios and natural frequencies in both directions. The mass ratio range covered was 2.36 to 12.96, achieved by the addition of lumped masses to the plate attaching the cylinder test section to the parallel linkage. Part of the natural frequencies fn (angular natural frequencyωn) in still water are given in Table 4.3.
Table 4.3: Natural frequencies in still water at different mass ratios m∗ m∗ζ fn(Hz) ωn(rad/s)
2.36 0.014 1.711 10.75 3.68 0.022 1.502 9.44 5.19 0.031 1.359 8.54 ... ... ... ...
when theAz/Do reaches 1.5 approximately without considering the vortex mode refer-ring to [97]), and lower branch. The jump phenomenon is apparent when the mode is transferred to the lower branch. With the increase of the static drag coefficient, the CF amplitudes decrease while the lock-in ranges are almost the same. When v = 0.4 m/s anddt= 0.005 s atCDs= 1, numerical iterations fromt= 500dttot= 6000dtcan yield steadyy/Do and z/Do values, where t is the time anddt is the adopted time step-size.
As expected, the trajectories of the bottom end exhibited the so-called crescent-shaped figure of eight as clearly shown in Figure 4.3.
Referring to [24], it is necessary to look at the amplitudes in the results and see if they converge to a particular value as the number of modes is increased or not. The convergence and stability of numerical solutions can be ascertained by increased number of eigenfunctions N and decreased time step-size dt. Notes are made beforehand: (1) 4 beam modes are enough for 2-D nonlinear dynamic study of a cantilevered pipe with a bottom end-mass discharging fluid in the air [21]. For 3-D nonlinear dynamic study of a flexible cantilevered pipe system attached with an end-mass discharging fluid in the air, 6 beam modes are sufficient to find the quasiperiodic planar motion and 8 modes are enough to find higher-frequency periodic planar motion [24]. (2) 5 modes are sufficient for the dynamic analysis of a marine riser in [46]. (3) It should be emphasized that the external fluid only excite the first natural modes of the cantilever in Exp. 1 [141].
Wake oscillator model is based on phenomenological method, and it is required to per-form dynamical analysis to confirm that the simulations can capture the main vibration characteristics of VIVs. Dynamical analysis can be conducted by comparing simulation predictions with former experimental observations. In this study, 4 case studies (i.e. be-fore the onset of lock-in; at the onset of lock-in; in the lock-in region; departing lock-in) atCDs= 1 are performed and examined.
Before the onset of lock-in
At v = 0.18 m/s (Vr ≈ 2.997), the time traces of the bottom end adopting increased numbers of eigenfunctions and decreased time step-sizes are shown in Figure 4.4 (IL direction) and Figure 4.5 (CF direction).
(1) Convergent solution can be achieved adopting sufficient eigenfucntions (N = 5) at appropriate time step-size (dt= 0.01 s) in both IL and CF directions.
(2) In the IL direction, the static displacement is small due to the small current ve-locity. The oscillation amplitudes display modulations obviously.
(3) In the CF direction, the amplitude is small. The time trace is irregular and un-steady. This is in consistent with the experimental observation in [108]. The modulation phenomenon is not so obvious as that in the IL direction.
At the onset of lock-in
At v = 0.20 m/s (Vr ≈ 3.33, at the onset of lock-in), the time traces of the bottom end with increased number of eigenfunctions and decreased time step-sizes are shown in Figure 4.6 (IL direction) and Figure 4.7 (CF direction).
(1) Convergent solution can be achieved adopting sufficient eigenfucntions (N = 5) at appropriate time step-size (dt= 0.01 s) in both IL and CF directions.
(2) In the IL direction, the modulations in the time traces become obscure.
(3) In the CF direction, the amplitude reaches 0.6Do approximately from small ampli-tude in the initial branch (Az/Do <0.1). The time histories are more or less sinusoidal.
It indicates the vortex shedding frequency synchronize with the structure vibration fre-quency and lock-in occurs at a single frefre-quency. Similarly, at the onset of lock-in, the amplitude reached 0.7Do from small amplitudes (<0.1Do−0.2Do) in the experimental study in [108].
In the lock-in region
At v = 0.40 m/s (Vr ≈ 6.661, in the lock-in region), the time traces of the bottom end with increased number of eigenfunctions and decreased time step-sizes are shown in Figure 4.8 (IL direction) and Figure 4.9 (CF direction).
(1) Convergent solution can be achieved adopting sufficient eigenfucntions (N = 5) at appropriate time step-size (dt= 0.01 s) in both IL and CF directions.
(2) In the IL direction, the modulations disappear in the lock-in band.
(3) In the CF direction, it vibrates at a single frequency and the amplitude reaches 1.3Do approximately at Vr≈6.661.
Departing lock-in
When the cylinder exits lock-in at v = 0.70 m/s (Vr ≈11.656), the time traces of the bottom end with increased number of eigenfunctions and decreased time step-sizes are shown in Figure 4.12 (IL direction) and Figure 4.13 (CF direction).
(1) The jump phenomenon is observed from the supper-upper branch to the lower branch (Az≈0.25Do). Vibration amplitude modulation reappears in both IL and CF directions as shown in Figure 4.10 and Figure 4.11 respectively.
(2) Convergent solution can be achieved adopting sufficient eigenfucntions (N = 10) and appropriate time step-size (dt = 0.003 s) in both IL and CF directions. When de-parting lock-in, fs = Sv/Do is applicable again and thus convergent solutions requires more eigenfunctions and smaller time step-size as shown in Figure 4.12 and Figure 4.13.
Due to the relationshipfys= 2fs (fys is the IL vortex shedding frequency), we enlarge IL time histories at different time step-sizes for investigations (see Figure 4.14). At suf-ficiently small time step-size, the curve becomes smooth. The phase shift phenomenon is observed which is acceptable and can be attributed to the Houbolt’s finite difference method referring to [34].
In the second step, it is going to test the newly calibrated wake coefficients proposed by Srinil and Zanganeh based on 2-DOF VIV study in [115]. It conducts simulations us-ing the data from Table 4.2 and employus-ingy = 0.3,z=f(m∗), and makes comparisons with the CF measurements in Exp. 1. The curves describing the CF dimensionless am-plitudeAz/Do of the bottom end versus the reduced velocityVr at different static drag coefficients are plotted in Figure 4.15. The amplitude decreases with the increase of the mean drag coefficient while keeping the same lock-in region. Therefore, the simulation can predict the amplitudes if a proper mean drag coefficient is assumed. The simulation can predict the size of lock-in region. However, the onset of lock-in region is predicted earlier (at smaller reduced velocity compared with experiment observation). In order to explain this phenomenon, we findy = 0.3,z =f(m∗) were calibrated base on Exp. 2, in which the natural frequency is 1.711 Hz at mass ratio m∗ = 23.6 and damping ratio ζ = 0.57% (see Table 4.3). However, the natural frequency is 2.313 Hz at m∗ = 23.6 andζ = 0.6% in Exp.1. That would be the reason for the shift of lock-in region between the simulation and experiment, because the reduced velocity is calculated by equation (4.11) and this new set of wake coefficients (i.e. y = 0.3 and z =f(m∗)) is calibrated based on Exp. 2.
2 4 6 8 10 12 14 16 18 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Vr A z/Do
Exp. 1 data Simulation at C
Ds=0.8 Simulation at C
Ds=1.0 Simulation at C
Ds=1.2
Figure 4.2: Comparisons between simulations employingy =z= 0.3 and Exp. 1 in the CF direction
0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26
−1.5
−1
−0.5 0 0.5 1 1.5
y/Do
z/Do
Simulation data
Figure 4.3: Trajectory of the bottom end fromt= 500dttot= 6000dt
10 20 30 40 0.03
0.04 0.05 0.06 0.07 0.08 0.09
t (s)
y/Do
(a) N=5; dt=0.01 s
10 20 30 40
0.03 0.04 0.05 0.06 0.07 0.08 0.09
t (s)
y/Do
(b) N=10; dt=0.01 s
10 20 30 40
0.03 0.04 0.05 0.06 0.07 0.08 0.09
t (s)
y/Do
(c) N=10; dt=0.005 s
Figure 4.4: Convergence tests atv= 0.18 m/s (Vr≈2.997) in the IL direction
10 20 30 40
−0.08
−0.06
−0.04
−0.02 0 0.02 0.04 0.06 0.08
t (s)
z/Do
(a) N=5; dt=0.01 s
10 20 30 40
−0.08
−0.06
−0.04
−0.02 0 0.02 0.04 0.06 0.08
t (s)
z/Do
(b) N=10; dt=0.01 s
10 20 30 40
−0.08
−0.06
−0.04
−0.02 0 0.02 0.04 0.06 0.08
t (s)
z/Do
(c) N=10; dt=0.005 s
Figure 4.5: Convergence tests atv= 0.18 m/s (Vr≈2.997) in the CF direction
10 20 30 40 0.045
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085
t (s)
y/Do
(a) N=5; dt=0.01 s
10 20 30 40
0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085
t (s)
y/Do
(b) N=10; dt=0.01 s
10 20 30 40
0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085
t (s)
y/Do
(c) N=10; dt=0.005 s
Figure 4.6: Convergence tests atv= 0.20 m/s (Vr≈3.33) in the IL direction
0 10 20 30
−0.4
−0.2 0 0.2 0.4 0.6
t (s)
z/Do
(a) N=5; dt=0.01 s
0 10 20 30
−0.4
−0.2 0 0.2 0.4 0.6
t (s)
z/Do
(b) N=10; dt=0.01 s
0 10 20 30
−0.4
−0.2 0 0.2 0.4 0.6
t (s)
z/Do
(c) N=10; dt=0.005 s
Figure 4.7: Convergence tests atv= 0.20 m/s (Vr≈3.33) in the CF direction
0 10 20 30 0.16
0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25
t (s)
y/Do
(a) N=5; dt=0.01 s
0 10 20 30
0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25
t (s)
y/Do
(b) N=10; dt=0.01 s
0 10 20 30
0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25
t (s)
y/Do
(c) N=10; dt=0.005 s
Figure 4.8: Convergence tests atv= 0.40 m/s (Vr≈6.661) in the IL direction
0 10 20 30
−1.5
−1
−0.5 0 0.5 1 1.5
t (s)
z/Do
(a) N=5; dt=0.01 s
0 10 20 30
−1.5
−1
−0.5 0 0.5 1 1.5
t (s)
z/Do
(b) N=10; dt=0.01 s
0 10 20 30
−1.5
−1
−0.5 0 0.5 1 1.5
t (s)
z/Do
(c) N=10; dt=0.005 s
Figure 4.9: Convergence tests atv= 0.40 m/s (Vr≈6.661) in the CF direction
0 5 10 15 20 25 30 0.93
0.935 0.94 0.945 0.95 0.955 0.96 0.965 0.97
Time s
y/Do
N=10, dt=0.03 s
Figure 4.10: Time trace of the bottom end atVr≈11.656 in the IL direction
0 5 10 15 20 25 30
−0.2
−0.1 0 0.1 0.2 0.3
t (s)
z/Do
N=10, dt=0.003 s
Figure 4.11: Time trace of the bottom end at Vr≈11.656 in the CF direction
0 10 20 0.93
0.935 0.94 0.945 0.95 0.955 0.96 0.965 0.97
t (s)
y/Do
(a) N=10; dt=0.005 s
0 10 20
0.93 0.935 0.94 0.945 0.95 0.955 0.96 0.965 0.97
t (s)
y/Do
(b) N=10; dt=0.003 s
0 10 20
0.93 0.935 0.94 0.945 0.95 0.955 0.96 0.965 0.97
t (s)
y/Do
(c) N=10; dt=0.002 s
Figure 4.12: Convergence tests atv= 0.70 m/s (Vr≈11.656) in the IL direction
0 10 20
−0.3
−0.2
−0.1 0 0.1 0.2 0.3
t (s)
z/Do
(a) N=10; dt=0.005 s
0 10 20
−0.2
−0.1 0 0.1 0.2 0.3
t (s) z/D e
(b) N=10; dt=0.003 s
0 10 20
−0.2
−0.1 0 0.1 0.2 0.3
t (s) z/D e
(c) N=10; dt=0.002 s
Figure 4.13: Convergence tests at v= 0.70 m/s (Vr≈11.656) in the CF direction
1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 0.935
0.94 0.945 0.95 0.955 0.96 0.965
t (s)
y/Do
N=10, dt=0.010 s
N=10, dt=0.005 s N=10, dt=0.002 s N=10, dt=0.003 s
Figure 4.14: Time traces of the bottom end at different time step-sizes in the IL direction when departing lock-in
2 4 6 8 10 12 14 16 18
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Vr
A Z/Do
Exp. 1 data Simulation at C
Ds=2.0 Simulation at C
Ds=2.5
Figure 4.15: Comparisons between simulations employingy = 0.3,z =f(m∗) and experiments in the CF direction
Driven by curiosity and Griffin plot, we compared the two sets of experiment data (i.e.
Exp. 1 and Exp. 2) in the CF direction at m∗ = 23.6 andζ ≈0.6% as shown in Figure 4.16. We can find they compare very well in the CF direction although this may be just
2 4 6 8 10 12 14 16 18
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Vr
Az/Do
Exp. 1 data Exp. 2 data
Figure 4.16: Comparisons of Exp. 1 and Exp. 2 atm∗ = 23.6 and ζ≈0.6% in the CF direction
a coincidence. Note should be made here that the two curves in Figure 4.16 are plotted against their own reduced velocities. That means the reduced velocities are calculated by their own natural frequencies. According to the Griffin plot, the coincidence seems to be reasonable and acceptable. We can also find the similarities after compare the time histories at Vr = 7 in 2-DOF VIV study in Exp. 2 in [130] and the time traces in our simulations atv= 0.4m/s(Vr ≈6.661) (see Figure 4.8 and Figure 4.9). Therefore, it seems reasonable to suppose that the IL displacements in Exp. 1 and Exp. 2 may compare very well.
In the third step, because there is no IL measurements in Exp. 1 and our predictions are 2-DOF VIV study, we compare the our IL simulations with the IL experimental measurements in Exp. 2.
Based on the simulations imputing the data from Table 4.2 and employingy =z = 0.3, the IL predictions and IL measurements in Exp. 2 are compared in Figure 4.17. The simulation can predict the lock-in region very well. But it fails to predict the amplitudes.
That is acceptable becausey =z = 0.3 was calibrated based on 1-DOF study. The IL amplitude decreases with the increase of mean drag coefficient. That means this set of coefficient can be used for parametric study in the IL direction.
Based on the simulations imputing the data from Table 4.2 and employing y = 0.3, z=f(m∗) , the IL predictions and IL measurements in Exp. 2 are compared in figure 4.18. The IL amplitude decreases with the increase of mean drag coefficient. The simu-lation can predict the amplitudes at proper mean drag coefficients and the size of lock-in region. But the onset of lock-in region was predicted earlier at smaller reduced velocity.
These are the same with the CF predictions when compared with the CF measurements.