3.1. Cylinder displacement

Figure 3 shows the variation of the cylinder vibration amplitude in non-dimensionalized form, ${A^\ast } = A/D$, with the reduced velocity, ${U^\ast }$, for the cylinder end conditions of no endplate, attached endplate and unattached endplate with varying gap ratios. Therein, the case without any endplate is marked by a gap ratio of g* = 280 % since the distance from the free-ended cylinder to the channel floor was 2.8 times the cylinder diameter, and the case with the attached endplate is marked by g* = 0 %. Also, note that the vibration amplitude was virtually the same for ${g^\ast } \ge 24\,\%$. Therefore, for ${g^\ast } \ge 24\,\%$, the variation of the amplitude with reduced velocity is shown with a single line in figure 3. For the attached endplate and the unattached endplates with gap ratios of up to 4 %, the amplitude variation shows the familiar three-branch behaviour, with a jump between the upper and lower branches and a relatively constant amplitude throughout the lower branch before reaching desynchronization. On the other hand, for larger gap ratios (g* > 4 %), while the initial and upper branches maintain nearly the same amplitude, the upper-to-lower branch transition shows a significant distinction from smaller gap ratios, such that the amplitude response of g* > 4 % gradually decreases with increasing reduced velocity from its peak value with no discrete transition to the lower branch. These results agree well with the literature presented in the introduction section and prove that the two different characteristics in amplitude response are indeed caused by the cylinder end conditions, given that all other parameters were kept the same in the current experiments.

Figure 3.

Response amplitude, ${A^\ast }$, against reduced velocity, ${U^\ast }$, for various end conditions. Here, g* = 0 % corresponds to the case with the attached endplate, and g* = 280 % depicts the gap ratio between the cylinder and the channel floor for the free-ended cylinder. The gap ratio $2\,\%\le {g^\ast } \le 40\,\%$ involves an unattached endplate. As the response lines coincided for ${g^\ast } \ge 24\,\%$, these cases are shown with a single line in the plot for clarity.

Another significant understanding comes from the inspection of the response data at g* = 8 % in figure 3. This gap ratio is well below the g* = 15 % threshold, which was thought previously to be the critical point where the free-vibration response characteristics in the lower branch would change from depicting a sudden jump at the upper-to-lower branch transition, similar to the case with an attached endplate, to a gradual variation, identical to the case with no endplate. This critical gap ratio value was deduced based on the sudden increase of fluid forces measured on a forced-oscillating cylinder, which was prescribed to vibrate sinusoidally with a fixed amplitude and fixed frequency corresponding to the oscillation amplitude and frequency of a comparable free-vibrating cylinder fitted with an attached endplate (Morse et al. Reference Morse, Govardhan and Williamson2008); however, this critical value was never validated for cylinders undergoing free vibrations. Considering that the oscillation amplitude varies with the value of g* in the lower branch, as seen in figure 3, as well as the oscillation frequency (as will be seen later in this section), this critical g* value, identified by prescribing a fixed amplitude and a fixed frequency to the cylinder, may not represent a critical gap ratio for free-vibrating cylinders. It can be seen from figure 3 that even a gap ratio as small as g* = 8 % can significantly alter the response amplitude of the free-vibrating cylinder. In fact, the transition between the two distinct response characteristics in the lower branch appears gradual with the gap ratio as opposed to an abrupt change believed to occur at one critical gap ratio.

To further examine the effects of the gap ratio on the vibration response of the cylinder, figure 4 is given, where the response amplitude, ${A^\ast }$, is plotted as a function of the gap ratio, g*, at different constant values of reduced velocity, ${U^\ast }$, in the lower branch. Note that, in this figure, error bars are depicted on the data pertaining to the ${U^\ast } = 10.8$ case to show the uncertainty in oscillation amplitude measurements. This uncertainty value (around 0.025D) was the same for all test cases; therefore, it is not shown in other figures to improve readability. Several important observations can be derived from this plot. For a given ${U^\ast }$, when the gap between the cylinder and the endplate is small, the response amplitude of the cylinder is around that of the cylinder having an attached endplate (i.e. g* = 0 %) at the corresponding ${U^\ast }$. At the other end of the spectrum, once the gap is large enough, the oscillation amplitude is unaffected by the gap ratio, and the cylinder essentially behaves like a free-ended cylinder. Based on the forced-oscillation data of Morse et al. (Reference Morse, Govardhan and Williamson2008), as indicated above, the gap ratio of 15 % was previously thought to be a threshold as a sudden increase was observed in the excitation force of the forced-oscillating cylinder at that g*. Here, it can be seen that the transition for a free-oscillating cylinder is more gradual than expected. Another salient point is that the end effects are more pronounced at the beginning of the lower branch (i.e. at lower U*). As the reduced velocity increases, the gap ratio effects slowly fade away. By the end of the lower branch (${U^\ast } \approx 10$ to 11), the amplitude remains roughly the same regardless of the gap ratio. The reasons for this behaviour will be investigated in the following sections.

Figure 4.

Response amplitude, ${A^\ast }$, for various end conditions as a function of gap ratio, g*. Each curve shows a constant reduced velocity, ${U^\ast }$, in the lower branch, as indicated in the legend. In the plot, g* = 0 % corresponds to the case with the attached endplate, and g* = 280 % depicts the gap ratio between the cylinder and the channel floor for the free-ended cylinder. The gap ratios of $2\,\%\le {g^\ast } \le 40\,\%$ involve an unattached endplate. The uncertainty in amplitude measurements is the same for all test cases and is only shown for the case with U* = 10.8.

The end conditions also affect the frequency of oscillations, as can be seen in figure 5, where the oscillation frequency of the cylinder, ${f^\ast } = f/{f_n}$, is plotted as a function of reduced fluid velocity, ${U^\ast }$, for different gap ratios, g*. The oscillation frequency generally decreases with an increase in gap ratio, and, similar to the trend observed for the response amplitude, the difference in frequency between the different gap ratios fades away at higher reduced velocities in the lower synchronization branch. It must be noted that only a subset of the gap ratios investigated in this study is shown in figure 5 for better clarity; however, all other gap ratios investigated in the present work follow the same trends.

Figure 5.

Variation of the cylinder's oscillation frequency, ${f^\ast } = f/{f_n}$, against reduced velocity, ${U^\ast }$, for various gap ratios. In the lower branch, larger gaps lead to a lower oscillation frequency.

Other than the vibration amplitude and frequency, the response of the cylinder can be further characterized by observing the time traces of its displacement, which are shown in figure 6 for three reduced velocities representing the three response branches. In this figure, the plots on the left have an unattached endplate with a very small gap ratio of g* = 2 %, while those on the right have the unattached endplate placed with a large gap ratio of g* = 40 %. Here, the former represents the typical oscillation patterns observed when an attached endplate or an unattached endplate with a small gap ratio is used, and the latter is representative of the cylinder displacements encountered when no endplate or an unattached endplate with a large gap is used. There is no discernible difference in the time traces of cylinder displacements between the two end conditions in the initial or upper branches. For both gap ratios, the initial branch is characterized by the beating motion, which is expected to be the result of the 2S vortex shedding mode, and the upper branch shows large modulations in response amplitude due to the intermittency of the 2P and 2P_O vortex shedding modes, which is known to be encountered in this range of reduced velocities. Hence, it can be observed that the end conditions have limited influence on the initial and upper branches, but they notably alter the cylinder's movement in the lower branch. The markedly larger-amplitude and undulated cylinder oscillations observed in the lower branch in the case of a very large gap ratio (g* = 40 % in figure 6) are replaced by relatively lower-amplitude and steadier oscillations in the presence of an endplate placed very close to the cylinder end (the case with g* = 2 % in figure 6). It has been apparent from figures 3 and 4 that the changes in cylinder response occur gradually rather than abruptly as the gap ratio varies. Therefore, in the upcoming sections, the focus will be placed on the gap ratios of g* = 2 % and 40 % as the two end points of the existing spectrum.

Figure 6.

Comparison of the typical time traces of cylinder displacement between unattached endplates with small and large gap ratios (g* = 2 % on the left and g* = 40 % on the right) in different response branches. In the lower branch, the higher gap ratio results in larger oscillation amplitudes with more modulation. (a,c,e) Unattached endplate (g* = 2 %) and (b,d,f) unattached endplate (g* = 40 %).

3.2. Force analysis

To gain physical insight into the cylinder's movement, it is essential to investigate the lift force and its relationship with the cylinder displacement. While there was no direct measurement of force data in this investigation, the following equation of motion for a body undergoing transverse oscillations can be leveraged to give an accurate estimation of the transverse force component acting on the cylinder:

(3.1)\begin{equation}m\ddot{y} + c\dot{y} + ky = F(t),\end{equation}

where F(t) is the instantaneous force imparted to the body by the fluid, y(t) is the instantaneous cylinder displacement and m, c and k are the system's mass, structural damping and spring stiffness, respectively, which are known properties of the experimental set-up. Measuring the instantaneous displacement, y(t), one can evaluate the left side of the equation and find the instantaneous fluid force, $F(t)$, acting on the cylinder. Here, a sixth-order Butterworth filter with a cutoff frequency of 4 Hz was applied after each differentiation to eliminate the measurement noise enhanced by the numerical differentiation of a digitized signal. This cutoff frequency was carefully selected to be small enough to filter out most of the noise while still being sufficiently far from the frequencies of interest. The body oscillation is synchronized with the periodic vortex shedding in the synchronization range, where a reasonable approximation to the motion and force are often provided by the following sinusoidal functions:

(3.2)\begin{gather}y(t) = A\sin (2{\rm \pi} ft),\end{gather}

(3.3)\begin{gather}F(t) = {F_0}\sin (2{\rm \pi} ft + \; \varphi ).\end{gather}

Here, f is the body oscillation frequency, and $\varphi $ is the phase angle between the fluid force and the body displacement. This phase difference, $\varphi $, is a key factor in determining the energy transfer between the fluid and the cylinder. Therefore, the precise value of the phase difference significantly affects the amplitude of oscillations. We note that, for free oscillations, while the cylinder displacement is close to a sinusoidal motion, the forcing on the cylinder may differ slightly from a sinusoidal signal. If the fluid force is decomposed into its Fourier components as

(3.4)\begin{equation}F(t) = \{ {F_1}\sin (2{\rm \pi} {f_1}t + {\varphi _1}) + {F_2}\sin (2{\rm \pi} {f_2}t + {\varphi _2}) + \cdots \} ,\end{equation}

then the force component of interest is the one closest to the frequency of oscillation f as this is the only component making a lasting contribution to the energy transfer between the fluid and the cylinder. In this investigation, the short-time Fourier transformation method is utilized to find the instantaneous phase difference between the fluid force and the body displacement corresponding to the oscillation frequency.

For stable oscillations of a free-oscillating body, the energy transfer from the fluid to the oscillating system $({E_{in}})$ in each cycle is balanced by the dissipation of energy through structural damping $({E_{out}})$

(3.5)\begin{equation}{E_{in}} = {E_{out}}.\end{equation}

Integrating over each cycle of oscillation with non-dimensionalization of parameters gives the following amplitude equation, provided by Khalak & Williamson (Reference Khalak and Williamson1999), which is used in various studies to successfully estimate the amplitude of free oscillations using force measurements from forced-oscillation experiments:

(3.6)\begin{equation}{A^\ast } = \frac{1}{{4{{\rm \pi} ^3}}}\frac{{{C_y}\sin \varphi }}{{({m^\ast } + {C_A})\zeta }}{\left( {\frac{{{U^\ast }}}{{{f^\ast }}}} \right)^2}{f^\ast }.\end{equation}

Here, ${C_A}$ is the potential added-mass coefficient (equal to 1 for circular cylinders) and ${C_y}$ is the transverse force coefficient found by normalizing the amplitude of the force component corresponding to the frequency of body oscillation by $({\textstyle{1 \over 2}})\rho {U^2}DL$, where $\rho $ is the fluid density, L is the cylinder length and D is the cylinder diameter. As only the end conditions are altered between the different test cases studied in this work, parameters related to the experimental set-up in (3.6) (mass ratio, ${m^\ast }$, damping coefficient, $\zeta $, reduced velocity, ${U^\ast }$, and potential added-mass coefficient, ${C_A}$) remain constant. Hence, the following relation is valid for the present study:

(3.7)\begin{equation}{A^\ast } \propto \frac{{{C_y}\sin \varphi }}{{{f^\ast }}}.\end{equation}

The term ${C_y}\sin (\varphi )$, which is known as fluid excitation, combines the effects of the magnitude of the transverse force component corresponding to the frequency of body oscillation with its phase difference, $\varphi $, relative to body motion, and directly influences the oscillation amplitude. It should also be noted that the frequency ratio, ${f^\ast }$, has a reverse relation with amplitude. When considering the effects of a single change in the experimental conditions, such as the change in the end condition, all of the parameters on the right side of (3.7) must be examined.

To better understand the force exerted on the cylinder, it is also useful to examine the concept of the vortex force. Following the analysis of Lighthill (Reference Lighthill1986), the total transverse force $({F_{tot}})$ acting on the cylinder can be decomposed into a potential force $({F_{pot}})$ due to the inertia of the fluid being accelerated by the cylinder's motion and a vortex force component $({F_{vor}})$ due to vortex shedding

(3.8)\begin{equation}{F_{tot}} = {F_{pot}} + {F_{vor}}.\end{equation}

The instantaneous potential added-mass force, ${F_{pot}}(t)$, acting on the cylinder is given by the inertial resistance of the mass being displaced by the cylinder's movement

(3.9)\begin{equation}{F_{pot}}(t) ={-} {C_A}{m_d}\ddot{y}(t),\end{equation}

where ${m_d} = \rho ({\rm \pi} {D^2}L/4)$ is the displaced fluid mass. Assuming a sinusoidal motion $(y = A\sin (2{\rm \pi} ft))$ and normalizing this force by $({\textstyle{1 \over 2}})\rho {U^2}DL$ gives the instantaneous potential force coefficient

(3.10)\begin{equation}{C_{pot}}(t) = 2{{\rm \pi} ^3}\frac{{y(t)/D}}{{{{({U^\ast }/{f^\ast })}^2}}}.\end{equation}

It can be seen from the equation above that the instantaneous potential force remains always in phase with the cylinder's motion. It is only due to the vortex force component that there can be a phase difference between the total transverse force and the body motion. The vortex force coefficient can be found by subtracting the potential force coefficient from the total force coefficient:

(3.11)\begin{equation}{C_{vor}}(t) = {C_{tot}}(t) - {C_{pot}}(t).\end{equation}

Typical time traces of cylinder displacement, y/D, in the initial and upper branches are given in figure 7, along with the total transverse force coefficient, ${C_{tot}}$, its breakdown into the potential added-mass force and the vortex force coefficients, ${C_{pot}}$ and ${C_{vor}}$, respectively, and the phase difference, $\varphi $, between the total transverse force and the body motion. In the initial and upper branches, there were no observed differences in results among the different end conditions considered; therefore, only the results for the unattached endplate at a gap ratio of g* = 2 % are shown in figure 7. In the initial branch, which is known to have two single vortices shed in each cycle (i.e. the 2S mode), both the potential and vortex force components are virtually in phase with the body motion. As a result, the phase difference, $\varphi $, between the total force component and the body motion fluctuates around a small value, alternating between positive and negative values. As this phase difference becomes positive or negative, the flux of energy into the body also changes sign (from (3.5) and (3.6)). This results in the repeated increase and decrease of the oscillation amplitude and the beating motion observed in the initial branch (figure 6). In the upper branch, as the vortex shedding mode is 2P/2P_O mode, the vortex force goes against the potential force component that remains in phase with the body motion. Ultimately, the phase of the total force with respect to the body motion is determined by the relative strength of these two force components. In the upper branch, as mentioned earlier, the vortex shedding alternates between the 2P mode, during which two pairs of similar-strength vortices are shed in each cycle, and the 2P_O mode, during which the second vortex in each pair is weaker than the first. These vortex modes will be discussed in detail in the next section, where vorticity contours in the wake are examined, but the effects of these modes are also apparent from the force diagram in figure 7. At times when the vortex shedding is in 2P mode where strong vortex pairs shed (for example, during t* = 60–70), the vortex force dominates the potential force, and the phase difference between the total force and the cylinder's motion becomes close to 180°. In contrast, when the vortex shedding is in 2P_O mode and hence the second vortex of each vortex pair is weaker (for example, during t* = 40–60), the effects of the vortex and potential forces balance each other, resulting in a much lower phase difference between the total force and the cylinder's motion.

Figure 7.

(a,b) Typical time traces of body motion, y/D, and transverse force coefficient, ${C_{tot}}$, (c,d) comparison of the total transverse force coefficient, ${C_{tot}}$, potential added-mass force coefficient, ${C_{pot}}$, and vortex force coefficient, ${C_{vor}}$ and, (e,f) the phase difference, $\varphi $, between the total transverse force and the body motion for the initial and upper branches shown in the first and second columns, respectively. The diagrams show examples from the unattached endplate at g* = 2 %, but all other gap ratios yield similar results.

As discussed in the previous section, the oscillation response in the lower synchronization branch shows a clear dependency on the gap ratio, and the reason starts to reveal itself when one looks at the force data provided in figure 8. The case with a small gap ratio (g* = 2 %) is shown in the left column of figure 8. In this case, the total force follows the dominant vortex force component resulting from a strong 2P vortex shedding, which is shown to persist for this case in the next section. Therefore, the total force maintains a constant phase difference of around $\varphi = 180^\circ $ relative to the body motion. In contrast, the case with a gap ratio of g* = 40 %, shown in the right column, demonstrates a vortex force that becomes intermittently dominant, similar to what was observed for the upper branch, because sometimes the vortex force is larger than the potential force and dominates the movement, the phase difference is around 180°, but other times, the vortex force gets weaker, and as a result, the phase difference reduces significantly. This can be further corroborated by observing the phase angle variations between the total force and the oscillatory motion at various reduced velocities, given in figure 9. In the upper branch, shown in the first row of figure 9, a similar alternating phase difference between the total force and the cylinder's motion exists for both the small and large gap ratios. In this branch, the vortex shedding alternates between the 2P and 2P_O modes, as will be shown in the next section. As a result of this, as discussed above, at times of 2P vortex shedding, with the second vortex of each vortex pair being as strong as the first vortex, the vortex force dominates, resulting in a phase difference of around 180° between the total force and the cylinder's motion. However, at times of 2P_O, as the vortex shedding has a weaker second vortex in each pair, the vortex and potential forces approach each other in strength, resulting in a much lower phase difference between the total force and the body motion. When the reduced velocity is increased and the lower branch is entered (U* > 7), the case with the small gap ratio (g* = 2 %), which is shown in the left column of figure 9, starts to show a phase difference of around 180°, and this same phase difference persists throughout the entire lower branch (i.e. for all U* in the lower branch). As the flow patterns given in the next section will show, this is related to the immediate switch of the wake mode to the steady 2P mode. In contrast, for the case with the larger gap ratio (g* = 40 %), shown in the right column of figure 9, the transition to the lower branch is pushed to higher reduced velocities, and this transition occurs gradually rather than with an immediate shift. These observations explain the gradual decrease of amplitude with increasing reduced velocity, seen in the lower branch at large gap ratios, rather than an abrupt change (in figure 3).

Figure 8.

(a,b) Typical time traces of body motion, y/D, and transverse force coefficient, ${C_{tot}}$, (c,d) comparison of the total transverse force coefficient, ${C_{tot}}$, potential added-mass force coefficient, ${C_{pot}}$, and vortex force coefficient, ${C_{vor}}$, (e,f) and the phase difference, $\varphi $, between the total transverse force and the body motion for the lower branch. Results for the case with the unattached endplate at g* = 2 % and 40 % are shown in the first and second columns, respectively.

Figure 9.

Typical time traces of the phase difference $\varphi $, between the total transverse force and the body motion for various reduced velocities, values of which are given on the left. The first and second columns show the results for the unattached endplate at g* = 2 % and 40 %, respectively.

By returning to (3.7) $({A^\ast } \propto {C_y}\sin \varphi /{f^\ast })$, the increased amplitudes of oscillation for large gap ratios in the early parts of the lower branch compared with those of lower gap ratios, and then the gradual decrease of the amplitude with increasing reduced velocity toward the amplitudes of lower gap ratios (all of which were observed in figure 3) can be explained. First, as detected in figure 5, the vortex shedding frequency, thereby ${f^\ast }$ in (3.7), decreases at a given reduced velocity as the gap gets larger in the initial parts of the lower branch. Second, as seen in figure 9, the average phase difference at a given reduced velocity also decreases from being around 180° at the lower gap ratios to a much lower value with increasing gap in the initial parts of the lower branch (such as at ${U^\ast } = 7.01$ or 7.44 in figure 9), which leads to an increase in the excitation force, ${C_y}\sin (\varphi )$, for larger gaps compared with smaller gaps. In accordance with (3.7), these changes in ${f^\ast }$ and ${C_y}\sin (\varphi )$ both tend to increase the oscillation amplitude with increasing gap ratio compared with that of a smaller gap in the early parts of the lower branch. However, as the reduced velocity is increased, and thereby the later parts of the lower branch are entered, these effects of the gap ratio on ${f^\ast }$ and ${C_y}\sin (\varphi )$ start to fade (as seen in figures 5 and 9), and as a result, the oscillation amplitudes of larger gap ratios get closer to that of the lower gap ratio.

4.1. Wake modes

First, the case of an unattached endplate with a small gap ratio of g* = 2 % is assessed here for wake modes in each response branch. As observed from the amplitude response data in figure 3, there is no significant difference in cylinder oscillations between this end condition (i.e. g* = 2 %) and the end condition with an attached end plate, which is the most studied case in the literature. Thus, the unattached endplate with g* = 2 % will serve as a baseline scenario with which the higher gap ratios can be compared. Figures 11, 12 and 14 show contours of phased-averaged vorticity, ${\langle \omega \rangle _p}D/U$, in the wake of the free-oscillating cylinder for the unattached endplate condition with the small gap ratio of g* = 2 % in the initial, upper and lower branches, respectively. In each of these figures, (a) depicts the reference time, ${t_0}$, which corresponds to the time when the cylinder passes the centreline (i.e. y/D = 0 location), moving in the positive y direction, while (b–d) correspond to the subsequent quarter periods in the oscillation cycle, where the symbol T denotes the full period of the body oscillation. The extent of the displacement of the cylinder's centre is marked by a thick dashed line on the vertical axis in each plot, and the direction of the cylinder's movement is illustrated by an arrow placed on the image of the cylinder when appropriate.

Figure 11.

Phase-averaged contours of vorticity, ${\langle \omega \rangle _p}D/U$, in the initial branch $({U^\ast } = 4.05,\; Re = 4000)$ for the cylinder end condition of the unattached endplate placed at the small gap ratio of g* = 2 %. The patterns indicate the 2S vortex shedding mode. Here, ${t_0}$ is the time when the cylinder passes the centreline (i.e. y/D = 0 location) in the positive y direction, and each consecutive plot is separated by a quarter period in the oscillation cycle; (a) t = t ₀, (b) t = t ₀ + T/4, (c) t = t ₀ + T/2 and (d) t = t ₀ + 3T/4.

Figure 12.

Phase-averaged contours of vorticity, ${\langle \omega \rangle _p}D/U$, in the upper branch $({U^\ast } = 5.74,\; Re = 5660)$ for the cylinder end condition of the unattached endplate placed at the small gap ratio of g* = 2 %. The patterns correspond to intermittently switching 2P_O/2P vortex shedding modes. Here, ${t_0}$ is the time reference point when the cylinder passes the centreline (i.e. y/D = 0 location) in the positive y direction, and each consecutive plot is separated by a quarter period in the oscillation cycle; (a) t = t ₀, (b) t = t ₀ + T/4, (c) t = t ₀ + T/2 and (d) t = t ₀ + 3T/4.

In figure 11(a), as the cylinder moves up, two vortices, marked as vortex A and B, are distinguishable near the cylinder. Vortex A is a remnant of the previous cycle, fully formed and separated at time ${t_0}$, while vortex B is newly forming by the clockwise vorticity convecting around the cylinder. As the cylinder accelerates downward from figure 11(b,c), the clockwise vorticity convecting from the cylinder continues to get fed into vortex B, resulting in a stronger vorticity region in figure 11(c). Meanwhile, the counterclockwise vortex from the cylinder forms a new vortex, marked as vortex C in figure 11(c). As the cylinder moves to figure 11(d), vortex B separates, while the newly forming vortex C remains attached to the cylinder. Similarly, as the cylinder accelerates in the +y direction from figure 11(d) back to figure 11(a), vortex C grows in size with the counterclockwise vorticity shedding from the cylinder, resulting in a stronger vorticity region (marked as vortex A in figure 11a), which ultimately separates from the cylinder when the cylinder passes the centreline. Over this entire oscillation cycle, two single vortices per cycle appear in the wake, producing the narrow vortex street known as the 2S mode.

In the upper branch given in figure 12, with an increase in the wavelength of the cylinder's trajectory, $\lambda $, the vortices convect to the wake region much sooner in the oscillation cycle than they do in the initial branch. Therefore, at the y/D = 0 location in figure 12(a), the vortex A from the previous cycle is already convected further downstream, while the newly forming vortex B is right behind the cylinder. As the cylinder moves up (from figure 12a,b), the shear layer with counterclockwise vorticity keeps rolling directly into vortex B, creating a stronger region of vorticity. This vortex is separated as the cylinder reaches its maximum displacement in figure 12(b). Notice that, in comparison, the separation of the first new vortex of the cycle (vortex B) happens much later in the initial branch (i.e. it occurs in figure 11(c), which is a quarter cycle later than that in the upper branch). Likewise, the vortex forming from the opposite side of the cylinder, vortex C, convects around the cylinder more efficiently in the upper branch compared with the initial branch due to two effects. First, $\lambda $ is larger, giving the shear layer vorticity effectively more time to convect, and secondly, the large vorticity of vortex B essentially pulls it towards the wake. Again, as the cylinder goes down (from figure 12c,d), the shear layer with counterclockwise vorticity from the cylinder rolls into vortex C, which separates at the maximum negative displacement point in figure 12(d). In comparison, this separation does not happen until the cylinder returns to its y/D = 0 position in the initial branch (figure 11a).

After a first glance at figure 12, one might think the overall vortex shedding mode of the upper branch seems similar to the 2S mode of the initial branch observed previously in figure 11, only that the vortex street is much wider as the cylinder displacement is significantly larger and the vortices are shed at the displacement extrema rather than the centreline (around y/D = 0). However, the phased-averaged results presented in figure 12 cannot depict the complete picture for the upper branch as the vortex shedding mode in this branch is known to alternate between the 2P and 2P_o modes (Morse & Williamson Reference Morse and Williamson2009). Previously, in figure 7, the phase difference between the total transverse force and the body oscillations in the upper branch was observed to alternate between being around 180° and much lower values, which is a result of this alternation. Therefore, figure 13 is provided, which shows the contours of instantaneous vorticity, $\omega D/U$, for three separate instances (${t_1},{t_2}$ and ${t_3}$). In these instances, the cylinder is at its maximum negative displacement point, just starting to move upwards in the +y direction. At time ${t_1}$, as seen in figure 13, a pair of vortices (marked as A and B) are shed, and another pair is formed on the opposite side of the cylinder. Hence, ${t_1}$ falls into the period during which 2P vortex shedding occurs. Notice also that, at this time, the phase difference between the total transverse force and the body motion is around 180°, as marked on the time history of the phase angle difference given in the same figure. As the most distinctive feature of the 2P_o mode from the 2P mode, it is known that the 2P_o mode exhibits a much weaker second vortex compared with the first vortex in each vortex pair (Morse & Williamson Reference Morse and Williamson2009). Times ${t_2}$ and ${t_3}$, in figure 13, show the case of 2P_o vortex shedding, where the second vortex is much weaker than the first. It should be noted that, during the tests, the second weaker vortex in the 2P_o mode was observed to have varying strength from cycle to cycle. Sometimes, it could be detected (such as the instant at ${t^\ast } = {t_2}$ in figure 13), while other times, it could not even be perceived within the PIV resolution (such as the instant at ${t^\ast } = {t_3}$ in figure 13). Also, a further noticeable feature of the 2P_o mode is that the phase difference of the lift force from the oscillatory motion is significantly lower in these instances, as marked on the phase angle history in figure 13. A general inspection of figure 13 shows that the phase difference between the transverse force and the cylinder's motion changes between high and low values, which is related to the intermittent switching of vortex structures between the 2P and 2P_o vortex shedding modes in the upper branch. As expressed earlier by (3.7), the amplitude of oscillations is proportional to the sine of the phase angle difference between the total lift force and the cylinder's motion. The varying nature of this phase angle between high and low values in the upper branch results in an average phase difference close to $\varphi = 90\mathrm{^\circ }$, leading to the highest oscillation amplitude for this branch, in accord with (3.7). On the other hand, the average phase difference is around $\varphi = 0\mathrm{^\circ }$ for the initial branch (as seen in figure 7) and $\varphi = 180\mathrm{^\circ }$ for the lower branch (as seen in figure 8), the sine of which results in lower oscillation amplitudes.

Figure 13.

Instantaneous contours of vorticity, $\omega D/U$, in the upper branch $({U^\ast } = 5.74,\; Re = 6660)$ for the cylinder end condition of the unattached endplate placed at the small gap ratio of g* = 2 % (top row) and the time history of the phase difference between the lift force and the body motion (bottom row). Here, ${t_1}$, ${t_2}$ and ${t_3}$ show selected instances where the cylinder is at its maximum negative displacement (i.e. the furthest position in the cycle in the -y direction). The patterns depict that the vortex shedding mode is 2P at ${t_1}$ and switches to 2P_O at ${t_2}$ and ${t_3}$.

In the lower branch, the 2P vortex shedding is clearly distinguishable from figure 14, where the contours of phased-averaged vorticity, ${\langle \omega \rangle _p}D/U$, are provided. In this case, the higher flow velocities mean that the rolled-up shear layers stretch and convect around the cylinder even earlier in the cycle. As a result, vortex B has already reached the opposite side of the wake in figure 14(a). In fact, as the cylinder moves upwards, part of vortex B is attached to the cylinder, and part of it is attracted by vortex A. As a result of the high strain rate within the vorticity region (Govardhan & Williamson Reference Govardhan and Williamson2000), vortex B splits. One part pairs with vortex A and moves downstream, while the other part remains connected to the cylinder, eventually causing the next vortex (vortex C) to split. This arrangement of vortices leads to the 2P mode, in which two pairs of similar-strength vortices are shed downstream in each cycle.

Figure 14.

Phase-averaged contours of vorticity, ${\langle \omega \rangle _p}D/U$, in the lower branch $({U^\ast } = 7.44,\; Re = 7335)$ for the cylinder end condition of the unattached endplate placed at the small gap ratio of g* = 2 %. The patterns indicate the 2P vortex shedding mode. Here, ${t_0}$ is the time reference point when the cylinder passes the centreline (i.e. y/D = 0 location) in the positive y direction, and each consecutive plot is separated by a quarter period in the oscillation cycle; (a) t = t ₀, (b) t = t ₀ + T/4, (c) t = t ₀ + T/2 and (d) t = t ₀ + 3T/4.

To summarize, as the cylinder accelerates in each half-cycle, the shear layers coming from the sides of the cylinder roll up and create vorticity regions, which convect around the cylinder before separating in the next half-cycle. It is the timing of these vortical structures and their interaction with the shear layers that determine the vortex shedding mode. In the initial branch, where the wavelength of the cylinder's trajectory, $\lambda $, has a low value, each vorticity region reaches the wake later in the cycle compared with the other branches. As a result, being unaffected by the preceding vortex that has already formed and shed from the opposite side of the cylinder half a cycle ago, a newly forming vortex in the near wake continues to grow by the shear layer coming from the cylinder with same-sign vorticity and eventually separates. Thereby, in every oscillation cycle, two vortices are shed half a cycle apart, which leads to the 2S vortex shedding mode. In the upper branch, with increased wavelength, λ, the vortices reach the wake relatively sooner in the cycle. Depending on the timing of vortices, one of two bistable vortex structures forms. (i) In one case, the vortices reach downstream of the cylinder with perfect timing, meaning they arrive early enough to merge with the vorticity regions created by the cylinder acceleration and get strong enough to separate at the maxima of the cylinder's motion while still being late enough in the cycle so that they split into a weaker vortex by the previous vortex. This delicate balance in vortex timing results in the 2P_o mode. (ii) The other case is when vortices arrive slightly earlier in the cycle and get split by the previous vortex, yielding the 2P vortex shedding mode. Finally, in the lower branch, where λ is much larger, every newly forming vortex reaches the wake region even sooner. Being affected by the vortex formed and shed earlier, it always splits in half, forming a stable 2P vortex street.

It was observed in figures 3 and 6 that the end condition does not have a significant effect on the initial and upper branches. Hence, as expected, the flow structures for various gap ratios also reflected this behaviour. As they were similar to what has been depicted for the case of g* = 2 %, given in figures 11–13, near-wake flow patterns in the initial and upper branches for other gap ratios are not repeated here. In the lower branch, however, the near-wake flow characteristics significantly differ when the gap ratio is altered. Figures 15 and 16 illustrate the contours of phased-averaged vorticity, ${\langle \omega \rangle _p}D/U$, at g* = 40 % for two different points in the lower branch. The beginning of the lower branch, ${U^\ast } = 7.44$, is shown in figure 15, while a higher reduced velocity $({U^\ast } = 10.4)$ further down in the branch is given in figure 16. Interestingly, for ${U^\ast } = 7.44$, an intermittent 2P_o/2P vortex shedding mode is observed for g* = 40 % (as apparent from the similarity of the phase-averaged vorticity patterns between figures 15 and 12), while a stable 2P mode was prominent at the same ${U^\ast }$ for the smaller gap ratio of g* = 2 %, as shown in figure 14. As the flow velocity is increased, the second vortex gets progressively stronger. Vorticity patterns depicting this trend were obtained but are not shown here for brevity. Towards the end of the lower branch, at ${U^\ast } = 10.4$ (shown in figure 16), a clear and stable 2P vortex shedding is achieved. These findings are also supported by the results from the force analysis presented earlier in figure 9, and, once again, indicate that, when the gap ratio is increased, the transition from the upper branch (with its intermittent 2P_o/2P vortex shedding mode) to the lower branch (with its steady 2P mode) shifts to higher reduced velocities, and this shift occurs gradually over a range of ${U^\ast }$ rather than being abrupt.

Figure 15.

Phase-averaged contours of vorticity, ${\langle \omega \rangle _p}D/U$, in the lower branch $({U^\ast } = 7.44,\; Re = 7335)$ correspond to intermittently switching 2P_O/2P vortex shedding modes for ${g^\ast } = 40\,\%$. Here, ${t_0}$ is the time reference point when the cylinder passes the centreline (i.e. y/D = 0 location) in the positive y direction, and each consecutive plot is separated by a quarter period in the oscillation cycle; (a) t = t ₀, (b) t = t ₀ + T/4, (c) t = t ₀ + T/2 and (d) t = t ₀ + 3T/4.

Figure 16.

Phase-averaged contours of vorticity, ${\langle \omega \rangle _p}D/U$, in the lower branch $({U^\ast } = 10.4,\; Re = 10\;258)$ indicating the 2P vortex shedding mode for ${g^\ast } = 40\,\%$. Here, ${t_0}$ is the time reference point when the cylinder passes the centreline (i.e. y/D = 0 location) in the positive y direction, and each consecutive plot is separated by a quarter period in the oscillation cycle; (a) t = t ₀, (b) t = t ₀ + T/4, (c) t = t ₀ + T/2 and (d) t = t ₀ + 3T/4.

4.2. Three-dimensional effects

In the previous section, the vortex shedding modes encountered at the mid-span of the cylinder have been established for each synchronization branch, and how these modes are affected by the end conditions has been discussed. These results showed that the gap ratio significantly influences the lower branch in particular. To investigate the cause, this section will explore the spanwise variation in flow features for the lower branch. Figure 17 shows the contours of phase-averaged vorticity, ${\langle \omega \rangle _p}D/U$, for the vertical plane behind the cylinder at y/D = −0.6. This data acquisition plane was illustrated earlier in figure 2. In figure 17, the situation for two gap ratios (g* = 2 % and 32 %) is depicted in the lower branch for four consecutive quarter periods in the oscillation cycle, starting from the time ${t_0}$, which marks the time when the cylinder passes the y/D = 0 position with a positive velocity. For the small gap ratio of g* = 2 %, shown in the top row, there is a small counterclockwise vorticity near the base of the cylinder, but it is weak and dissipates quickly without much effect on the downstream flow. On the other hand, for the case of g* = 32 %, shown on the bottom row, the flow stream coming through the gap rolls up behind the cylinder and creates a much stronger vortex. This counterclockwise vortex travels downstream over the oscillation cycle, affecting a much larger area behind the cylinder. The difference in the spanwise extent of the wake region affected by the gap flow can be observed clearly in figure 18, which illustrates the contours of time-averaged streamwise velocity, $\langle u\rangle /U$, for various gap ratios. At the smallest gap ratio of g* = 2 %, the streamwise velocity stays constant across the span, but at higher gap ratios, there is a large-magnitude streamwise flow stream coming through the gap, indicated by the red contours, that stays close to the endplate. Furthermore, due to the formation of the strong counterclockwise vortex discussed in figure 17, there is also a sizable region of lower-magnitude streamwise velocity directly behind the cylinder, indicated by the blue contours. As the gap ratio increases, the spanwise reach of this streamwise velocity (i.e. the blue contours) increases, affecting larger portions of the span downstream of the cylinder. This low-magnitude streamwise velocity region reaches its maximum spatial extent in the spanwise direction when the gap ratio is set to g* = 32 %, where the stream caused by the gap affects around 45 % of the span.

Figure 17.

Phase-averaged contours of vorticity, ${\langle \omega \rangle _p}D/U$, in the vertical plane behind the cylinder. Here, ${t_0}$ is the time reference point when the cylinder passes the centreline with positive velocity, and each consecutive plot is separated by a quarter period of oscillation. The first and second rows show results for a gap ratio of ${g^\ast } = 2\,\%$ and 32 %, respectively, in the lower branch (at ${U^\ast } = 8.70$ and $Re = 8588$). (a,e) $t= t_{0}$, (b,f) $t= t_{0} + T_{n}/4$, (c,g) $t= t_{0} + T_{n}/2$, (d,h) $t= t_{0} + 3T_{n}/4$.

Figure 18.

Time-averaged contours of the streamwise velocity component,$\langle \; u\rangle/U$, in the vertical plane behind the cylinder. The plot shows various gap ratios in the lower branch at ${U^\ast } = 8.70$ and $Re = 8588$. As the gap ratio increases, larger portions of the span are affected by the lower-velocity region (seen as blue contours) behind the cylinder. (a) $g^{\ast} =2 \%$, (b) $g^{\ast} =8 \%$, (c) $g^{\ast} =16 \%$, (d) $g^{\ast} =32 \%$

Figure 19 depicts the contour patterns of phase-averaged vorticity, ${\langle \omega \rangle _p}D/U$, in two horizontal planes z/D = 3.5 and 5.25, along with the contours of phase-averaged streamwise velocity, ${\langle u\rangle _p}D/U$, in the vertical plane y/D = −0.6 for two gap ratios, g* = 2 % and 40 %, of the unattached plate. From figure 19(a), where the gap ratio is only 2 %, it is apparent that there is no spanwise variation in the streamwise flow velocity, and both horizontal planes clearly show a pair of vortices with similar strength, marked as A and B₁, travelling downstream on one side of the cylinder's wake while two additional vortices, marked as B₂ and C, are forming during the given phase, indicative of the 2P vortex shedding mode in both horizontal planes. In figure 19(b), however, where the gap ratio is 40 %, it can be observed from the streamwise velocity contours of the vertical plane that the increased gap ratio between the cylinder and the endplate causes an area of lower streamwise velocity to form in the lower half of the cylinder's wake. For this gap ratio of g* = 40 %, although the higher horizontal plane (z/D = 5.25) is outside of this low-magnitude streamwise velocity region and thus is unaffected by the gap flow, the lower horizontal plane (z/D = 3.5) is well within this region. Consequently, while the higher horizontal plane (z/D = 5.25) has transitioned to the lower synchronization branch and is thereby showing an equally strong pair of vortices (marked as A and B₁) on one side of the wake with the other vortex pair (B₂ and C) of the 2P vortex mode forming, the lower plane (z/D = 3.5) effectively sees lower streamwise velocities and, thereby, it has not transitioned to the lower branch just yet. As a result, the phase-averaged vorticity pattern in the lower horizontal plane is undergoing the 2P_O vortex shedding mode with a much weaker second vortex. The overall force applied to the cylinder is the sum of the forces acting over the entire span. Hence, the spanwise wake region with the lower streamwise velocity, where the 2P_O vortex shedding mode persists, reduces the overall phase difference between the force and the body motion, leading to a higher oscillation amplitude (in accord with (3.7)). Effectively, the area of lower streamwise velocity delays the transition in vortex mode for the large gap ratio of g* = 40 % because a considerable portion of the span is seeing lower velocities. Nevertheless, for higher reduced velocities, larger portions of the span will eventually transition to the lower branch due to increased flow velocities. Accordingly, the large region of lower streamwise velocity forming as a result of large cylinder-endplate gaps explains both the gradual upper-to-lower branch transition associated with large gap ratios (seen in figures 1(b) and 3) and the fact that, at high enough flow velocities, the end condition effects start to diminish (observed in figures 3 and 4).

Figure 19.

Phase-averaged contours of streamwise velocity component, ${\langle u\rangle _p}/U$, in the vertical y/D = −0.6 plane behind the cylinder and phased-averaged contours of vorticity, ${\langle \omega \rangle _p}/U$, at two horizontal planes z/D = 3.5 and 5.25: (a) for the gap ratio of ${g^\ast } = 2\,\%$ and (b) ${g^\ast } = 40\,\%$. The arrow placed in the -y direction on the cylinders shows the direction of movement of the cylinder at the specific phase considered. The measurements are for ${U^\ast } = 8.70$ and $Re = 8588$ in the lower branch. (a) Unattached endplate g* = 2 % and (b) unattached endplate g* = 40 %.