2.1 Fluid–structure system

Figure 1 presents a brief schematic of the current fluid–structure interaction problem showing an overview of the experimental arrangement together with definitions of the important geometric and flow parameters. The sphere is elastically mounted in the direction transverse to the oncoming flow. The distance from the free surface is varied in terms of the parameter $h^{\ast }=h/D$ – the immersion ratio – where $h$ is the distance measured from the top of the sphere surface to the undisturbed free surface directly above and $D$ is the sphere diameter.

Table 1 shows the set of non-dimensional parameters relevant for the current study. In problems involving flow-induced vibrations (FIV) of bluff bodies, the response dynamics of the system is often characterised by the normalised vibration amplitude ( $A^{\ast }$ ) and frequency response ( $f^{\ast }$ ) versus reduced velocity. In the current study, the normalised amplitude response is defined as $A^{\ast }=\sqrt{2}A_{rms}/D$ , where $A_{rms}$ is the root mean square (r.m.s.) of the oscillation amplitude of the body. The reduced velocity is defined as $U^{\ast }=U/f_{nw}D$ , where $U$ is the free-stream velocity and $f_{nw}$ is the natural frequency of the system in quiescent water. Another important parameter in the current fluid–structure system is the mass ratio defined as $m^{\ast }=m/m_{d}$ , where $m$ is the total oscillating mass of the system, and $m_{d}$ is the displaced fluid mass ( $m_{d}=\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}D^{3}/6$ with $\unicode[STIX]{x1D70C}$ being the fluid density).

The governing equation for motion characterising cross-flow VIV of a sphere can be written as

(2.1) $$\begin{eqnarray}m\ddot{y} +c{\dot{y}}+ky=F_{y},\end{eqnarray}$$

where $F_{y}$ represents fluid force in the transverse direction, $m$ is the total oscillating mass of the system, $c$ is the structural damping of the system, $k$ is the spring constant and $y$ is the displacement in the transverse direction. Using the above equation, the fluid force acting on the sphere can be calculated from the directly measured displacement, and its time derivatives.

Table 1.

Non-dimensional parameters used in this study. In the above parameters, $A_{rms}$ is defined as the root mean square of the structural vibration amplitude in the $y$ direction. $D$ is sphere diameter; $D_{r}$ is the support-rod diameter, $f$ is the body oscillation frequency and $f_{nw}$ is the natural frequency of the system in quiescent water; $m$ is the total oscillating mass, $c$ is the structural-damping factor and $k$ is the spring constant; $U$ is the free-stream velocity, and $\unicode[STIX]{x1D708}$ is the kinematic viscosity; $m_{A}$ denotes the added mass, defined by $m_{A}=C_{A}m_{d}$ , where $m_{d}$ is the mass of the displaced fluid and $C_{A}$ is the added-mass coefficient (0.5 for a sphere); $f_{vo}$ is the vortex-shedding frequency of a fixed body.

As a first approximation, it is often assumed that $y(t)$ and $F_{y}(t)$ are both approximately sinusoidal and can be represented by

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle y(t)=A\sin (2\unicode[STIX]{x03C0}ft), & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle F_{y}(t)=F_{o}\sin (2\unicode[STIX]{x03C0}ft+\unicode[STIX]{x1D719}), & \displaystyle\end{eqnarray}$$

where $A$ is the displacement amplitude, $F_{o}$ is the amplitude of $F_{y}$ and $\unicode[STIX]{x1D719}$ is the phase between the fluid force and the body displacement.

As done for VIV of spheres and cylinders by Govardhan & Williamson (Reference Govardhan and Williamson2000), the total transverse fluid force ( $F_{y}$ ) can be split into a potential force $F_{potential}$ (comprising the potential added-mass force) and a vortex force $F_{vortex}$ that is due to the vorticity dynamics. From potential theory, the instantaneous $F_{potential}$ acting on the sphere can be expressed as

(2.4) $$\begin{eqnarray}F_{potential}(t)=-C_{A}m_{d}\ddot{y} (t),\end{eqnarray}$$

with $C_{A}$ the potential added-mass coefficient ( $C_{A}=0.5$ for a sphere). Thus, the vortex force $F_{vortex}$ can be computed from

(2.5) $$\begin{eqnarray}F_{vortex}=F_{y}-F_{potential}.\end{eqnarray}$$

The vortex phase is the phase difference between the vortex force and the body displacement and the total phase $\unicode[STIX]{x1D719}_{total}$ is the phase difference between the total force and the body displacement. In general, phase jumps are associated with a switch from one VIV mode to another, and have even been used to distinguish between different modes (Govardhan & Williamson Reference Govardhan and Williamson2005). The instantaneous relative phases between the two forces are calculated using the Hilbert transform (e.g. see Khalak & Williamson Reference Khalak and Williamson1999). For the piercing sphere cases, the potential force is calculated considering the fraction of the sphere that is submerged and assuming that the added-mass coefficient remains the same.

2.2 Experimental details

The experiments were conducted in the recirculating free-surface water channel of the Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Monash University, Australia. The test section of the water channel has dimensions of $600~\text{mm}$  in width, $800~\text{mm}$  in depth and $4000~\text{mm}$  in length. The free-stream velocity in the present experiments could be varied continuously over the range $0.05\leqslant U\leqslant 0.45~\text{ms}^{-1}$ . The free-stream turbulence level was less than 1 % in the current experiments.

The current hydro-elastic problem was modelled using a low-friction air-bearing system that provided a very low structural damping and almost frictionless motion of the sphere in the transverse direction to the oncoming flow. The structural stiffness was controlled by extension springs that were attached to both sides of a slider carriage. Further details of the hydro-elastic facility can be found in Zhao et al. (Reference Zhao, Leontini, Lo Jacono and Sheridan2014a ,Reference Zhao, Leontini, Lo Jacono and Sheridan b ).

Figure 2.

Experimental schematic (not to scale) showing an overview of the experimental arrangement.

A detailed schematic of the experimental set-up is presented in figure 2, showing important components of the set-up. The sphere models used in the present study were solid spherical balls precision-machined from acrylic plastic with a very smooth surface finish. The accuracy of the diameter was within $\pm 200~\unicode[STIX]{x03BC}\text{m}$ . Different spheres with sizes of $D=40~\text{mm}$ , $D=80~\text{mm}$ and $120~\text{mm}$ were used in the current experiments. The spherical models were supported using a thin cylindrical support rod $3~\text{mm}$ in diameter, manufactured from hardened nitrided stainless steel for extra stiffness and to maintain straightness.

The body displacement was measured using a linear encoder (model: RGH24, Renishaw, UK) with a resolution of $1~\unicode[STIX]{x03BC}\text{m}$ . Since the linear encoder was digital, electromagnetic noise did not affect the accuracy of the displacement signal measurement. This provided highly accurate displacement signals and allowed reliable velocity and acceleration signals to be derived. This enabled an accurate determination of the lift force signal to be derived from the displacement signal using (2.1). In the experimental study by Sareen et al. (Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018), the process was tested through a direct comparison against the lift force determined by a force sensor over a wide range of $U^{\ast }$ . It was found that the lift force derived from the linear encoder displacement signal matched well with that measured using the force sensor, except that they were generally smoother, indicating accurate measurements of the displacement and the lift force. Zhao et al. (Reference Zhao, Leontini, Lo Jacono and Sheridan2014b ) also reported such a comparison. Hence, all the force coefficients reported in the current study are derived from the measured displacement signal. It should be noted that the lift force obtained using the force sensor includes the inertial force associated with the acceleration of the mass below the force sensor (e.g. see Zhao et al. (Reference Zhao, Leontini, Lo Jacono and Sheridan2014b ) and Sareen et al. (Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018)). That term, which is derived by calculating the acceleration from the displacement signal, needs to be subtracted from the measured force to give the lift force. Hence the force sensor by itself does not enable the lift force to be measured directly. Also, the force measurements from the force sensor suffered from a greater degree of noise, presumably from electromagnetic sources; hence, the preference for the force derived from the displacement signal. Note here that this approach is appropriate only because highly accurate digital displacement measurements are possible with the current displacement encoder. Of course, one requires accurate measurements of the relevant parameters of the spring–mass system (total oscillating mass, spring constant and the structural damping).

The data acquisition and the controls of the flow velocity were automated via customised LabVIEW programs. For each data set, the displacement signal was acquired at a sampling frequency of $100~\text{Hz}$ for at least 100 vibration cycles.

The natural frequencies and structural damping of the system in both air and water were measured by conducting free decay tests individually in air and in quiescent water. Most of the experiments reported in this paper were performed for a mass ratio of $m^{\ast }=7.8$ . The structural damping ratio with consideration of the added mass was determined to be $\unicode[STIX]{x1D701}=2.04\times 10^{-3}$ . The Scruton number $Sc$ (mass-damping parameter) for the current study was 0.00134.

To gain insight into the flow dynamics, velocity field measurements using particle image velocimetry (PIV) were undertaken in the cross-plane, 1.5 diameters downstream of the sphere. For this purpose, the flow was seeded with $13~\unicode[STIX]{x03BC}\text{m}$ hollow micro-spheres having a specific weight of $1.1~\text{g}~\text{m}^{-3}$ . A laser sheet of ${\sim}3~\text{mm}$ thickness from a continuous laser (model: MLL-N-532-5W, CNI, China), aligned parallel to the $y$ – $z$ plane, was employed to illuminate the laser plane. A mirror was placed at $45^{\circ }$ angle to the free-stream direction towards the downstream side of the sphere. The mirror was placed more than six diameters away from the sphere. Imaging was performed using a high-speed camera (model: Dimax S4, PCO, AG) with a resolution of $2016\times 2016~\text{pixels}^{2}$ . This camera was equipped with a $105~\text{mm}$ Nikon lens, giving a magnification of approximately $11.34~\text{pixel}~\text{mm}^{-1}$ for the field of view. Velocity fields were deduced using in-house PIV software developed originally by Fouras, Lo Jacono & Hourigan (Reference Fouras, Lo Jacono and Hourigan2008), using $32\times 32~\text{pixel}^{2}$ interrogation windows in a grid layout with $50\,\%$ window overlap. All the vorticity fields shown in the current study were phase-averaged over more than 100 cycles. For each PIV measurement case, a set of $3100$ image pairs were sampled at $10~\text{Hz}$ . Each image set was sorted into $24$ phase bins based on the sphere’s displacement and velocity, resulting in more than $120$ image pairs for averaging at each phase.

In the present study, the VIV response was studied over a wide reduced velocity range of $3\leqslant U^{\ast }\leqslant 20$ . The immersion ratio was varied over a range of $0\leqslant h^{\ast }\leqslant 1$ for the first set of experiments, and $-0.062\leqslant h^{\ast }\leqslant -0.750$ for the next set of experiments with the piercing sphere. The Reynolds number for the current study varied between $5000$ and $30\,000$ .

5.1 Vibration response measurements

The response of a semi-submerged sphere can be divided into two regimes, one with $0<h^{\ast }<-0.5$ , and the other with $-0.5\leqslant h^{\ast }\leqslant -0.75$ , due to distinct differences in the response characteristics. Figure 12 shows the variation of $A_{rms}^{\ast }$ with $U^{\ast }$ for various $h^{\ast }$ values for both these regimes. In regime I, as $h^{\ast }$ is decreased from $-$ 0.062 to $-$ 0.375, the displacement amplitude progressively increases and the synchronisation region sequentially widens.

Figure 12.

Amplitude response of a sphere piercing the free surface. (a) The amplitude response in regime I. (b) The amplitude response in regime II. (c) Variation of the peak amplitude, $A_{max}^{\ast }$ , with $h^{\ast }$ . The dashed arrow in (a,b) represents the direction of increasing $h^{\ast }$ (in magnitude).

In contrast to regime I, the displacement amplitude in regime II decreases with a decrease in $h^{\ast }$ , as can be observed in figure 12(b). The amplitude response for $-0.5<h^{\ast }\leqslant -0.75$ is characterised by two peaks, unlike regime I, where only one peak is evident for the $U^{\ast }$ range tested. As observed from the figure, the drop in amplitude is dramatic between $-$ 0.650 and $-$ 0.688, which was found to be repeatable in follow-up experiments. The amplitudes for $h^{\ast }=-0.688$ and $-$ 0.750 are highly reduced with a suppression of vibrations for $U^{\ast }\geqslant 15$ .

Figure 12(c) shows the variation of the peak amplitude, $A_{max}^{\ast }$ , with $h^{\ast }$ . Intriguingly, for $h^{\ast }=-0.250$ , $-$ 0.375 and $-$ 0.5, the amplitudes become even higher than the amplitude response of a fully submerged sphere (shown with dashed horizontal line). In these cases, the vibrations corresponding to the peak amplitude remain highly periodic; however, the periodicity decreases as the response decreases at higher $U^{\ast }$ values, as also shown in the following discussion.

Figure 13.

Variation of the displacement amplitude ( $y/D$ ) with non-dimensional time ( $t/T$ ) for various $U^{\ast }$ values at an immersion ratio of $h^{\ast }=-0.062$ (left) and $h^{\ast }=-0.750$ (right). The corresponding amplitude response is shown at the top.

In figure 13, the time trace of the displacement signal is shown at selected $U^{\ast }$ values for $h^{\ast }=-0.062$ (regime I) and $h^{\ast }=-0.750$ (regime II). As evident in regime I (left column), the vibrations are highly periodic for $U^{\ast }=6.2$ ; however, for $U^{\ast }=12.4$ in the decreasing amplitude branch of the response, the vibrations are of similar magnitude but the periodicity has decreased. For $U^{\ast }=17.1$ , the vibrations become highly non-periodic and consist of small amplitude intermittent vibrations. This behaviour was found to be typical of all other cases in regime I. In regime II on the other hand, the two peaks are separated by a transition region, where there is a sudden decrease in the amplitude; nevertheless, the vibrations are still locked in.

As evident in the frequency contours plots for these $h^{\ast }$ values shown in figure 14, there are slight differences in the frequency spectrum as well in the two regimes. For $h^{\ast }=-0.062$ , there is an increase in the frequency of vibration (albeit small) as the response progresses from the increasing $A^{\ast }$ branch to the decreasing $A^{\ast }$ branch extending towards a highly non-periodic vibration branch at higher $U^{\ast }$ . On the contrary, for $h^{\ast }=-0.750$ , the frequency remains constant over the entire range of $U^{\ast }$ . The question arises, what causes two peaks in regime II? Insight into the transition between the vibration modes in this scenario can be gained through analysis of the force measurement data, as explained in detail in the following section.

5.2 Force measurements

For 1-DOF VIV, the transition between the principal modes of vibrations is continuous and gradual, unlike for the small mass ratio 2-DOF case. Hence, it is difficult to infer the existence of two modes and their existence boundaries from the amplitude response alone. A careful inspection of the force measurement data can provide insight into the mode transitions. Hence, the total transverse force coefficient, ${C_{y}}_{total}$ , and the vortex force coefficient, $C_{vortex}$ , were derived from the displacement signal and its time derivatives. The total phase, $\unicode[STIX]{x1D719}_{total}$ , and the vortex phase, $\unicode[STIX]{x1D719}_{vortex}$ , were also computed to understand the mode transitions in both these regimes.

In figure 15, the variation of $\unicode[STIX]{x1D719}_{total}$ and $\unicode[STIX]{x1D719}_{vortex}$ with $U^{\ast }$ is shown for three different $h^{\ast }$ values in both the regimes. In regime I, both $\unicode[STIX]{x1D719}_{total}$ and $\unicode[STIX]{x1D719}_{vortex}$ increase from lower values to up to ${\sim}170^{\circ }$ , indicating the presence of mode I and II vibrations. In contrast, $\unicode[STIX]{x1D719}_{vortex}$ in regime II jumps up to ${\sim}170^{\circ }$ during lock-in and remains constant over the entire $U^{\ast }$ range. For $h^{\ast }=-0.625$ and $h^{\ast }=-0.500$ , where two peaks in the amplitude response were observed (see figure 12), the total phase crosses $90^{\circ }$ twice, in line with the presence of two peaks in the amplitude response. However, it is difficult to infer from this what is the physical basis of the two peaks. A careful inspection of the $C_{vortex}$ variation helps to make the picture clearer.

Figure 14.

Variation of the frequency response (c,d) correlated with the amplitude response (a,b) for two different semi-submerged cases: (a,c) $h^{\ast }=-0.062$ , with the sphere slightly piercing the surface; (b,d) $h^{\ast }=-0.750$ , where only a small section of the sphere is submerged.

Figure 15.

Variation of the total phase, $\unicode[STIX]{x1D719}_{total}$ , (a,b) and the vortex phase, $\unicode[STIX]{x1D719}_{vortex}$ , (c,d) for three different $h^{\ast }$ values in regime I (a,c) and regime II (b,d).

Figure 16.

Variation of the vortex force coefficient ( $C_{vortex}$ ) with reduced velocity ( $U^{\ast }$ ) for various immersion ratios: (a) regime I; (b) regime II.

Figure 17.

Variation of the total transverse force coefficient, ${C_{y}}_{total}$ , with $U^{\ast }$ for varying immersion ratios $h^{\ast }$ .

Figure 16 shows the variation of $C_{vortex}$ for various immersion ratios for both the regimes. As evident in figure 16(a), the vortex force coefficient jumps up abruptly at the vibration lock-in, at approximately $U^{\ast }\approx 5$ in these cases, and then starts decreasing. However, for $U^{\ast }\approx 8{-}10$ , $C_{vortex}$ starts increasing again, indicating the transition to mode II of the vibration response. This trend was also observed by Govardhan & Williamson (Reference Govardhan and Williamson2005) for the 1-DOF fully submerged sphere VIV case. The two peaks in the vortex force indicates these two vibration modes. On the other hand in regime II, it can be observed that the first peak is smaller than the second peak. Also, the transition from the first peak to the second occurs at lower $U^{\ast }$ values compared to regime I for $h^{\ast }=-0.625$ and $-$ 0.650. Beyond the transition, $C_{vortex}$ gradually increases to a saturation value and then decreases to a lower value with no other peak for all the cases in regime II. It can be conjectured here that there are only two fundamental modes of vibrations for all these cases and there is no mode III, at least within the $U^{\ast }$ range studied. The two peaks in regime II indeed correspond to modes I and II.

Figure 17 shows the variation of the total transverse force coefficient, ${C_{y}}_{total}$ , with increasing $U^{\ast }$ for various $h^{\ast }$ values. The total force also follows a similar trend. However, it is not easy to demarcate the two modes and their transitions just by looking at this plot as there is no other peak evident (significantly bigger) for regime II for higher $U^{\ast }$ values.

It is an interesting finding that although for a 1-DOF sphere VIV, there is no clear demarcation between the two modes for the fully submerged case, the VIV response of a semi-submerged sphere can show a transition region between the two where the amplitude drops, in line with the transition between the two vibration modes seen in low mass ratio tethered sphere studies.

5.3 Effect of Froude number

The Froude number ( $Fr$ ) is an important parameter in problems involving proximity to a free surface. It is defined as the ratio of the inertial force to the gravitational force, and it quantifies the deformation at the free surface. In the current problem, $Fr$ can be defined as $Fr=U/\sqrt{gD}$ for a fixed $h^{\ast }$ case.

Figure 18.

Effect of Froude number on the amplitude response curves for the case when the sphere is just touching the free surface. (a) Mass ratio $m^{\ast }=7.8$ for spheres of diameter 80 mm and 120 mm. (b) Mass ratio $m^{\ast }=28.7$ for spheres of diameter 40 mm and 80 mm. Note that the Froude number varies with reduced velocity as well as sphere diameter.

In the current study, the Froude number is based on the sphere diameter, and $Fr$ varies over the range of $0.05\leqslant Fr\leqslant 0.45$ for both scenarios: where the sphere is close to the free surface; and also where the sphere pierces the free surface. From previous studies with circular cylinders, it can be conjectured that in this range of Froude numbers, there will no large surface deformations at the free surface leading to any significant wave–wake interactions. Nevertheless, some tests were performed to check the sensitivity of the VIV response to a change in Froude number in this range.

The response was measured for three spheres of different sizes: $D=40$ , 80 and 120 mm. Since changing the size of the sphere changes $m^{\ast }$  due to change in the displaced fluid mass and the sphere mass, extra weights were placed on top of the air-bearing rig to maintain the same mass ratio, when possible. Figure 18(a) shows the response of a sphere of size $120~\text{mm}$ compared to that for a $80~\text{mm}$ sphere, for $m^{\ast }=7.8$ and $h^{\ast }=0$ . The Froude number range changed from $0.06\leqslant Fr\leqslant 0.45$ (for the $80~\text{mm}$ sphere) to $0.03\leqslant Fr\leqslant 0.2$ (for the $120~\text{mm}$ sphere); however, no significant change in the shape of the response curve was found. Note here that $Fr$ scales with the $U$ as well, so it increases with increasing $U^{\ast }$ . Similarly, in figure 18(b), the response of a sphere of $D=40~\text{mm}$ is shown compared to that for the $80~\text{mm}$ sphere, for $m^{\ast }=28.7$ and $h^{\ast }=0$ . The Froude number range in this case changed from $0.06\leqslant Fr\leqslant 0.45$ (for the $80~\text{mm}$ sphere) to $0.11\leqslant Fr\leqslant 0.6$ (for the $40~\text{mm}$ sphere); however, again no significant change in the response curve shape was seen. Despite this, these two figures do show a slight increase in peak amplitude as the diameter of the sphere is increased, or the Froude number is reduced. However, overall, it can be inferred that the VIV response is relatively insensitive to the Froude number in this Froude number range.

For a high Froude number range of $Fr\geqslant 2$ , significant wave–wake interactions can be expected for the sphere. Of course, for a 3-D flow past a sphere at this higher Reynolds number, capturing the time-varying surface deformations and their effect on the 3-D structures would be complicated and this remains open for future studies.

In summary, the current results are representative of the general problem of sphere VIV close to or piercing the free surface for a low $Fr$ range.

5.3.1 Effect of mass ratio

Due to nature of the problem set-up, the displaced fluid mass changes with each immersion ratio for a semi-submerged sphere. This changes the mass ratio and the effective added-mass coefficient for the sphere with each $h^{\ast }$ as shown in figure 19(a,b).

Figure 19.

(a) Change in the mass ratio as the immersion ratio is varied for the semi-submerged sphere. (b) Variation in the effective added-mass coefficient at different immersion ratios as a function of the reduced velocity.

When the peak amplitude response of a sphere with either 1-DOF ( $y$ -only motion) or 2-DOF ( $x$ – $y$ motion) is plotted against the mass-damping parameter, it collapses the data onto a single curve. Such a curve is known as a Griffin plot, as shown in figure 20. The open circles are data reported by Govardhan & Williamson (Reference Govardhan and Williamson2005) for 1-DOF and 2-DOF cases. The peak amplitude for a fully submerged sphere in the current study, plotted as the solid diamond symbol, also falls on the same curve, noting that in practice there is a slight Reynolds number dependence (see Govardhan & Williamson Reference Govardhan and Williamson2006 for the effect on circular cylinder VIV). For the semi-submerged sphere, the mass damping changes substantially from 0.018 to 0.1 as $h^{\ast }$ varies from $-$ 0.062 to $-$ 0.750. According to figure 20, this should only result in a variation of $A_{max}^{\ast }$ of less than 20 %. However, the situation is not quite as simple as this because the semi-submerged immersed body geometry can no longer be described as spherical once part of it lies above the waterline, and the effect of the free surface is to alter the wake, and hence the vortex forcing, considerably, as vorticity diffuses into the free surface. As can be gauged from the data presented in the previous sections, the current problem of a semi-submerged sphere shows characteristics distinct from the fully submerged case. Hence, further tests were undertaken to determine the sensitivity of the response to $m^{\ast }$ .

Figure 20.

Griffin plot for the VIV of a sphere: variation of $A_{max}^{\ast }$ with the mass-damping parameter $(m+C_{A})\unicode[STIX]{x1D701}$ . Open circle ( $\circ$ ), the data reported by Govardhan & Williamson (Reference Govardhan and Williamson2005); solid diamond, $A_{max}^{\ast }$ for a fully submerged sphere in the current study; solid triangle, $A_{max}^{\ast }$ reported by Sareen et al. (Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018).

Figure 21.

Variation of $A_{rms}^{\ast }$ with $U^{\ast }$ for different mass ratios: (a) $h^{\ast }=-0.062$ ; (b) $h^{\ast }=-0.25$ .

Figure 21(a) shows the variation of vibration amplitude with $U^{\ast }$ for two different $m^{\ast }$ values at $h^{\ast }=-0.062$ with the sphere just piercing the surface, while figure 21(b) shows the effect of the mass ratio for $h^{\ast }=-0.25$ when the sphere has a greater proportion of its volume above the waterline. As evident from these figures, the amplitude response is more sensitive to a change in mass ratio than for the fully submerged case. For instance, for the case when the top of the sphere is just above the water surface at $h^{\ast }=-0.062$ , doubling the mass ratio from $m^{\ast }=7.8$ to 15.6 (and hence the mass-damping ratio from 0.017 to 0.032) results in a reduction in the peak amplitude of approximately 30 %. However, the predicted variation from the Griffin plot (figure 20) for a fully submerged sphere is only a few per cent. The second case, for $h^{\ast }=0.25$ , also shows a much greater reduction than predicted by the Griffin plot variation. Thus, the presence of the free surface has a much stronger effect on the reduction in the peak amplitude with mass-damping ratio. Despite this, the general reduction in the synchronisation range, and the reduction of amplitude at higher $U^{\ast }$ values, is consistent with the trend observed for a fully submerged sphere. Unfortunately, limits on the weight that the air-bearing rig could support meant that it was not possible to quantify the effect further at higher mass-damping ratios. However, this clearly indicates that further future work is required in order to fully understand and characterise mass-damping ratio effects on VIV of a semi-submerged sphere.

Figure 22.

Variation of $A_{max}^{\ast }$ with $h^{\ast }$ highlighting various regimes where different effects are dominant. The marked (in red) optimum $h^{\ast }$ denotes the suggested approximate optimum value of $h^{\ast }$ for studying the VIV response of a fully submerged sphere if the support rod and the free-surface effects are to be minimised. The results are shown for $D^{\ast }=26.6$ .

6.1 Peak response variation with submergence ratio

Figure 22 shows the variation of peak amplitude, $A_{max}^{\ast }$ , with the immersion ratio, $h^{\ast }$ , summarising the results in the current and previous sections. Based on the current study, the response can be conveniently categorised into several regimes where certain effects are dominant. Amongst other things, this figure highlights that the suggested optimum $h^{\ast }$ for studying the response of a fully submerged sphere, if free surface and support-rod effects are to be minimised, corresponds to $h^{\ast }\approx 1D$ . However, there are also strong effects of $D^{\ast }$  on the peak response, and both $h^{\ast }$ and $D^{\ast }$ strongly affect the amplitude response at higher reduced velocities beyond where the peak response occurs.

7.1 Vorticity measurements in the cross-plane

In terms of VIV wake dynamics, the central distinction between a three-dimensional body like a sphere, and its two-dimensional counterpart, the cylinder, is that VIV of a sphere is induced by streamwise vorticity. This is in contrast to the cylinder whose dynamics are mainly induced by spanwise vortex structures in the wake. The generation of lift from the presence of streamwise vortex structures in the former case is analogous to the induced force on an aircraft wings due to counter-rotating trailing tip vortices. Govardhan & Williamson (Reference Govardhan and Williamson2005) demonstrated this by comparing the direct force measurements from a force sensor to the vortex force calculated using the knowledge of strength and spacing of the streamwise vortices. Therefore, the principal conclusion one may draw is that most of the transverse force on the sphere is associated with the streamwise vorticity, and with the knowledge of the strength of the upper vortex and the spacing between the vortex pair, one may deduce the vortex force acting on the sphere. As performed for a sphere undergoing VIV with 1-DOF by Govardhan & Williamson (Reference Govardhan and Williamson2005), the vortex force (or the lift force) acting on the sphere in this case can be given by the expression

(7.1) $$\begin{eqnarray}F_{vortex}=-\unicode[STIX]{x1D70C}U_{v}\unicode[STIX]{x1D70F}b,\end{eqnarray}$$

where $U_{v}$ is the convection speed of the trailing vortices, $\unicode[STIX]{x1D70F}$ is the strength of the upper vortex and $b$ is the spacing between the vortices.

Particle image velocimetry (PIV) measurements in a plane normal to the flow can reveal important insights into the temporal evolution of streamwise vorticity as the vortex loops pass through the cross-plane. Hence, in the current study, the measurement of the streamwise vorticity has been carried out in a cross-plane at a distance of $1.5D$ from the sphere rear surface similar to the vorticity measurements by Govardhan & Williamson (Reference Govardhan and Williamson2005). Figure 23 shows the vorticity contour plots phase-averaged (over more than 100 cycles) at two different phases of the oscillation cycle for a fully submerged sphere ( $h^{\ast }=1$ ). The plots are shown for a reduced velocity of $U^{\ast }=10$ , which corresponds to the peak amplitude in the VIV response (the heart of mode II). The top boundary in all the contour plots reported in the paper marks the free-surface boundary.

Figure 23.

Streamwise vorticity plots for a fully submerged sphere VIV at $U^{\ast }=10$ taken $1.5D$ downstream of the sphere. The above phase-averaged plots from cross-sectional digital particle image velocimetry (DPIV) measurements show mean vorticity contours at approximately the two opposite extremes of the displacement cycle. The position of the sphere is shown by the dashed circle. Blue contours show anticlockwise vorticity and red contours show clockwise vorticity. Contour levels vary in eight steps in the range $\unicode[STIX]{x1D714}D/U\in [-3.33,3.33]$ . The top boundary indicates the location of the free surface.

As clearly observed from figure 23(a,b), the streamwise vorticity consists of a dominant counter-rotating vortex pair symmetric across the horizontal plane, which is consistent with the formation of vortex loops on both the sides of the wake (Sakamoto & Haniu Reference Sakamoto and Haniu1990; Brücker Reference Brücker1999; Govardhan & Williamson Reference Govardhan and Williamson2005). Streamwise vorticity measurements by Govardhan & Williamson (Reference Govardhan and Williamson2005) also revealed a counter-rotating streamwise vortex pair similar to the current study. As the sphere oscillates from one side to the other, the observed vorticity changes sign. In figure 23(a), the blue vortex (anticlockwise vorticity) is above the red vortex (clockwise vorticity). On the other hand, when the sphere traverses towards the opposite side, the vorticity changes sign with the red vortex now above the blue vortex. This is congruent with the fact that as the sphere oscillates from one side to the other, hairpin loops from opposite sides are shed downstream into the wake. Vorticity measurements in a cross-plane, $1.5D$ downstream of the sphere by Govardhan & Williamson (Reference Govardhan and Williamson2005), also reported similar findings for a fully submerged sphere.

Figure 24.

Streamwise vorticity measurements (phase-averaged) for $U^{\ast }=10$ for a sphere with an immersion ratio of $h^{\ast }=0.125$ at marked phases over a displacement cycle. The corresponding displacement phases are depicted in the plot above. See figure 23 for further information.

The question arises: how do these counter-rotating vortex pairs (depicting propagating streamwise vortex pairs) change when the sphere moves closer to the free surface? In order to answer this question, similar streamwise vorticity measurements at $1.5D$ downstream were undertaken for several different immersion ratios. Figure 24(a–d) shows vorticity contour plots at several phases of the oscillation cycle for a sphere close to a free surface, in this case for an immersion ratio of $h^{\ast }=0.125$ . As evident from the figure, the orientation of the vortex pair has changed dramatically due to proximity to the free surface, indicating that the vortex loops change orientation and the horizontal plane through the sphere centre no longer acts as a plane of symmetry.

In figure 24 when the sphere traverses to one side, the vortex closer to the free surface (red) is reduced in size remarkably; however, as the sphere returns to the other side, the red vortex is seemingly restored and instead the size of the blue vortex is reduced substantially. This means that the anticlockwise (blue) vorticity is the dominant vorticity during the first half of the cycle and the clockwise (red) vorticity is dominant in the next half. These observations of vorticity transformation via turning, stretching and diffusion in the current results are perhaps related to the case of a vortex ring approaching a free surface as reported by Zhang, Shen & Yue (Reference Zhang, Shen and Yue1999). They demonstrated that the mechanisms of vorticity transformation via turning, stretching and diffusion take on distinct roles in the two surface layers, i.e. an inner viscous layer and an outer blockage layer. Similar observations of disconnection and termination of vortex lines at the free surface have been made previously by several studies, e.g. Bernal & Kwon (Reference Bernal and Kwon1989), Gharib & Weigand (Reference Gharib and Weigand1996) and Zhang et al. (Reference Zhang, Shen and Yue1999) for the case of an approaching vortex ring near a free surface, Reichl et al. (Reference Reichl, Hourigan and Thompson2005) for the case of a fixed cylinder near a free surface, Ohring & Lugt (Reference Ohring and Lugt1991) for a vortex pair near a free surface and also in the review study by Sarpkaya (Reference Sarpkaya1996) on the vorticity and free-surface interactions.

In the current case, where the vortex pair is streamwise orientated during the initial evolution, the dynamics may be even simpler. As the upper vortex of the pair advects downstream, the stress-free free-surface condition means that a velocity gradient cannot be maintained, i.e. the vortex diffuses into the free surface (e.g. Wu Reference Wu1995; Brøns, Thompson & Leweke Reference Brøns, Thompson, Leweke and Hourigan2014). Thus, the pair becomes unequal in strength. This causes rotation in the cross-plane through mutual induction, allowing the lower vortex of the pair to move outwards and upwards. During the second half of the cycle, the newly formed vortex pair originating from the opposite side experiences the same dynamics, except with the signs swapped and positions reversed. One can also notice asymmetric deformation of the vortex pair due to unequal induced strains due to the presence of the free surface.

It can be concluded here that as the sphere traverses from one side to the other (in one displacement cycle), the strength of the upper vortex decreases in the cross-plane. This reduction in the strength of the upper vortex suggests a reduction in the vortex force (or lift) acting on the sphere (see (7.1)). Since the transverse force acting on the sphere decreases in this case, the vibration amplitude could be expected to decrease. This is consistent with the amplitude response measurements shown earlier in figure 5. There is considerable reduction in the amplitude response of the sphere for $h^{\ast }=0.125$ at $U^{\ast }=10$ compared to the fully submerged sphere.

Figure 25.

Streamwise vorticity measurements (phase-averaged) for $U^{\ast }=10$ for a sphere with an immersion ratio of $h^{\ast }=-0.25$ at several phases through a displacement cycle. See figure 23 for further details.

An interesting case is where the sphere pierces through the free surface. Figure 25(a–d) shows cross-stream vorticity contour plots for $h^{\ast }=-0.25$ at $U^{\ast }=10$ . Notably, at a distance $1.5D$ downstream, the upper vorticity has completely disappeared. In contour panels (a) and (b), only anticlockwise vorticity is present in the cross-plane. For panels (c,d), only clockwise vorticity is present. Of course, there may still be some vorticity of opposite sign in the region (in the same plane) not captured by the PIV imaging plane. Also, these observations indicate that the formation length of the vortices is significantly longer compared to a fully submerged case, and large elliptic deformation is evident. From these vorticity contour plots, it seems likely that there are no hairpin vortex loops that develop in the wake past the sphere for a piercing sphere case. Hence, the VIV dynamics in this case are very different to that of the fully submerged case. For a partially submerged sphere (in this case, $h^{\ast }=-0.25$ , i.e. only 75 % of the sphere is submerged), as the sphere traverses from one side to the other, only one streamwise vortex is seen in the cross-plane with alternate signs, in contrast to the fully submerged sphere, where a counter-rotating vortex pair is seen with alternate signs. However, for this case, the vibrations are larger compared to the fully submerged case, as shown in the amplitude response measurements in figure 12. To obtain the full picture of the vortex dynamics for this case, with implications for the sphere forcing, would seem to require capturing the vorticity formation closer to the sphere. This is difficult to capture with the current PIV imaging set-up.

In order to obtain a better perspective of the wake structures, spatio-temporal reconstructions of the sphere wake at a Reynolds number of ${\sim}Re=1.5\times 10^{4}$ are shown in figure 26, employing 24 cross-stream vorticity fields, each phase-averaged over more than 100 sphere oscillation cycles. A convection velocity of $U$ (free-stream velocity of $0.196~\text{ms}^{-1}$ ) was assumed to build the three-dimensional images. As is evident in figure 26(a), the wake comprises a two-sided chain of streamwise vortex loops. Govardhan & Williamson (Reference Govardhan and Williamson2005) also reported similar wake structures for the sphere wake, but at a lower Reynolds number of $Re=3000$ . On the other hand, for the case when the sphere is much closer to the free surface, at $h^{\ast }=0.125$ in figure 26(bi), the loops twist to attach to (and diffuse into) the free surface alternatively. This observation is clearer in figure (bii), which shows the same structures but from a different perspective view (zoomed in view from the bottom).

Figure 26.

Spatio-temporal reconstructions of the streamwise vorticity crossing the transverse plane at a distance $1.5D$ from the sphere rear surface for (a) $h^{\ast }=1$ , and (b) $h^{\ast }=0.125$ , based on the phase-averaged streamwise vorticity. Figures (bi) and (bii) show two different perspective views of the same case, $h^{\ast }=0.125$ . The wake is shown for $U^{\ast }=10$ , corresponding to a Reynolds number of ${\sim}1.5\times 10^{4}$ . Blue indicates anticlockwise vorticity, and red clockwise vorticity (both in the $x$ – $y$ plane). The $x$ – $y$ plane at the $z/D=1.5$ boundary of the visualisation box indicates the location of the free surface. A convection velocity of $U$ (free-stream velocity) of $0.196~\text{ms}^{-1}$ was used for the reconstructions.

7.2 Some comments on surface patterns and deformations

In order to gain some qualitative insight into the free-surface deformation patterns, an ultraviolet lamp was placed just above the water channel facing downwards towards the wake. A Nikon D7000 camera was placed upstream to capture the free-surface patterns. In quiescent water, the UV lamp reflection appeared as a straight horizontal band, and as the free surface deforms, this deformation can be captured by visualising the UV reflection patterns.

Figure 27.

Images showing the surface distortion for $h^{\ast }=0$ at $U^{\ast }=5$ (a) and $U^{\ast }=12$ (b), and for $h^{\ast }=0.25$ (c) and $h^{\ast }=-0.25$ (d) at $U^{\ast }=15$ . The UV lamp was placed above the water channel facing downwards towards the wake. The free-surface patterns were viewed using a Nikon camera placed upstream.

Figure 27(a) shows the UV lamp pattern for $h^{\ast }=0$ and $U^{\ast }=5$ . In this case, there is minimal free-surface deformation, hence the UV reflection appears as a horizontal band with little distortion that would be caused by surface waves. However, at a relatively higher $U^{\ast }$  value of 12 (figure 27 b), small surface ripples are clearly visible with a standing wave at the front of the surface. On the other hand, for $h^{\ast }=0.25$ , as shown in figure 27(c), there is very small deformation (unlikely to be significant); however, patterns resembling Kelvin-type waves are visible on the surface, which are probably due to the piercing support rod. For a piercing sphere case, as shown in figure 27(d), the patterns are very different, with appreciable surface deformation seen as a standing wave structure.

In most of these cases, the surface wave structures are of low amplitude of less than two per cent of the sphere diameter. However, for $h^{\ast }=0.125$ , they exceeded ${\sim}4\,\%$ for $U^{\ast }\gtrsim 15$ . The limiting case of $h^{\ast }=0$ had deformations in the vicinity of the submerged sphere of ${\sim}10\,\%$ at $U^{\ast }\gtrsim 15$ . The surface deformations were estimated by measuring the approximate dip in the free surface in close proximity to the sphere using scaled images similar to those shown in figure 27. The observation of generally small surface deformation over most of the parameter space investigated is consistent with the relatively small effect of Froude number observed in § 5.3. Much higher Froude numbers are expected to lead to significantly larger surface deformations, in turn leading to nonlinear wave–wake interactions and complicated dynamics; however, even for the small Froude numbers examined in this paper, the proximity to the free surface has a strong effect on the VIV and the wake dynamics.