3.3.1. Vibration response, fluid forces and phases

Figure 13 displays the $A^*_{10}$ amplitude response along with the logarithmic-scale power-spectral density (PSD) contour plots of the frequency response of the oblate spheroid oscillation $(f^{*}_{y}=f_{y}/f_{\textit{n}w})$ , the total transverse fluid force $(f^{*}_{C_y}=f_{C_y}/f_{\textit{n}w})$ and the vortex force component $(f^{*}_{C_v}=f_{C_v}/f_{\textit{n}w})$ , as a function of $U^{*}$ for the lowest mass-damping parameter tested, $(m^* + C_{A})\zeta =0.1263$ .

Figure 13. ( $a$ ) Amplitude response and $(b$ – $d)$ logarithmic-scale normalised frequency power spectral density (PSD) contours as a function of the reduced velocity for the case of $\epsilon = 2.00$ with $(m^* + C_{A})\zeta =0.1263$ . In panels $(b$ )–( $d)$ , vertical dashed lines represent the boundaries of the response regimes. Note the following abbreviations: Mode I (M-I), Mode II (M-II), transition (T) and galloping-like (G-I).

Figure 13( $a$ ) reveals that two main vibration regimes can be identified: VIV and galloping-like (vibration). The VIV regime occurs over the range $3.6 \leqslant U^* \leqslant 9.0$ and the galloping-like regime over the range $9.4 \leqslant U^* \leqslant 12.0$ .

The onset of the VIV regime is marked by a minor jump in the vibration amplitude, from $A^*_{10} \leqslant 0.05$ at $U^*=3.6$ to $A^*_{10}=0.15$ at $U^*=3.7$ (figure 13 $a$ ). The start of the lock-in region is very similar to the case of a non-rotating sphere, where the onset is within the range $4.0 \leqslant U^* \leqslant 5.0$ (see Govardhan & Williamson Reference Govardhan and Williamson2005; Sareen et al. Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a ). Furthermore, over the range $3.7 \leqslant U^* \leqslant 5.5$ , the body vibration frequency ( $f^*_{y}$ ), the transverse force frequency ( $f^*_{C_y}$ ) and the vortex force frequency ( $f^*_{C_v}$ ) show a single dominant frequency component close to the natural frequency of the system ( $f_{\textit{n}w}$ ) (figure 13 b,c,d). This implies the synchronisation between the displacement frequency and the vortex shedding frequency. Thus, the first mode of vortex formation is dominated by VIV, termed Mode I (M-I).

Figure 14. Variation of ( $a$ ) transverse lift force coefficient $(C^{\textit{rms}}_{y})$ and vortex force coefficient $(C^{\textit{rms}}_{v})$ , and $(b)$ total phase $(\phi _{t})$ and vortex phase $(\phi _{v})$ , all as functions of the reduced velocity $U^*$ for $\epsilon = 2.00$ and $(m^* + C_{A})\zeta =0.1263$ .

Figure 14 displays the variation of the root-mean-square values of the total transverse force, $C^{\textit{rms}}_{y}$ , the vortex force, $C^{\textit{rms}}_{v}$ , the total phase ( $\phi _{t}$ ) and vortex phase ( $\phi _{v}$ ) as a function of $U^*$ . The start of M-I is induced by a sudden jump of the fluid force coefficients at $U^*=3.6$ and then progressively rises until a first magnitude peak is reached. The total phase ( $\phi _{t}$ ) is almost constant at $30^{\circ }$ over the range $3.6 \leqslant U^* \leqslant 5.5$ , while the vortex phase ( $\phi _{v}$ ) varies slightly between $30^{\circ }$ and $50^{\circ }$ over the same range. This is similar compared to a non-rotating sphere undergoing VIV, where M-I of VIV is associated with a vortex phase $\phi _{v} \sim 50^{\circ }$ (see Govardhan & Williamson Reference Govardhan and Williamson2005; Rajamuni, Thompson & Hourigan Reference Rajamuni, Thompson and Hourigan2018; Sareen et al. Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a ).

With a further increase in the reduced velocity ( $U^* \geqslant 5.5$ ), the vibration amplitude transitions towards a plateau, with $A^*_{10}$ very close to $1.0$ , marking the appearance of another vortex mode of formation, termed Mode II (M-II) (see figure 13). This mode displays a distinct frequency response compared with M-I; in particular, there is a significant contribution of a superharmonic at three times the body oscillation frequency, $3f^*_{y}$ , in the total transverse and vortex force frequency responses, starting at $U^*=5.5$ . The end of M-II is marked by a noticeable step (i.e. kink) in the frequency responses of $f^*_{C_y}$ and $f^*_{C_v}$ at $3f^*_{y}$ for the value of $U^*=9.0$ . It is noteworthy that for M-II, the highest significant frequency component is still the one equal to the displacement frequency, $f^*_{y}$ , suggesting that the FIV responsible for the vibration is associated with VIV.

Figure 14 reveals that the value of $C^{\textit{rms}}_{y}$ decreases progressively from its peak $0.35$ at $U^*=5.5$ towards $0.15$ at $U^*=7.0$ , indicative of a transition between VIV modes. In addition, there is a second peak displayed by the vortex force coefficient ( $C^{\textit{rms}}_{v}$ ) at $U^*=8.6$ that marks the peak amplitude of vibration while in M-II. For this mode, $\phi _{t}$ grows almost linearly from $\sim 20^{\circ }$ to $155^{\circ }$ in the range $5.5 \leqslant U^* \leqslant 9.0$ . Similarly, $\phi _{v}$ displays a marked jump at $U^*=5.5$ and then grows until it reaches a plateau value at $160^{\circ }$ . This is consistent with the phase relationship seen for VIV of a sphere, consistent with this oscillation remaining VIV-dominated (see Govardhan & Williamson Reference Govardhan and Williamson2005).

For $U^* \geqslant 9.2$ , the dynamic response displays a trend similar to a transverse galloping response of a square, D-section or elliptical cylinder (see Zhao et al. Reference Zhao, Leontini, Lo Jacono and Sheridan2014b , Reference Zhao, Hourigan and Thompson2018a ; Lo et al. Reference Lo, Hourigan, Thompson and Zhao2023), where the amplitude of vibration climbs almost linearly with increasing reduced velocity beyond a critical threshold value, $U^*_{\textit{crit}}=9.2$ . Over this vibration regime, the amplitude of vibration amplifies from $1.3$ at $U^*=9.2$ to $2.43$ at $U^*=12$ , representing an enhancement of $87\,\%$ of the amplitude of vibration. Over the range $9.4 \leqslant U^* \leqslant 12.0$ , both fluid force components, $f^*_{C_{y}}$ and $f^*_{C_{v}}$ , reveal strong harmonic contributions at one and three times the displacement frequency (i.e. $f^*_{C_y}=f^*_{C_v}=1$ and $f^*_{C_y}=f^*_{C_v}=3$ ), suggesting that a strong third harmonic contribution can be associated with the high amplitude of vibration. Previous studies of the transverse FIV of bluff bodies (see Zhao et al. Reference Zhao, Leontini, Lo Jacono and Sheridan2014b , Reference Zhao, Hourigan and Thompson2018a ,Reference Zhao, Nemes, Lo Jacono and Sheridan c ; Wang et al. Reference Wang, Du, Zhao and Sun2017) or in-line FIV (see Bourguet & Lo Jacono Reference Bourguet and Lo Jacono2015; Zhao et al. Reference Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018b ) suggested that large-amplitude vibrations are attributable to harmonic synchronisation. For the spheroid case, the wake-body synchronisation involves the vortex and total transverse force harmonics, which leads to a galloping-like oscillation response termed Mode G–I.

Figure 14 shows that a sharp drop in $\phi _{t}$ , from $150^{\circ }$ at $U^*=9.0$ to $90^{\circ }$ at $U^*=9.4$ , characterises the transition between VIV and galloping-like. Beyond that, $\phi _{t}$ and $\phi _{v}$ remain almost linear at $90^{\circ }$ and $145^{\circ }$ , respectively. Similarly, for Mode G-I, both fluid force coefficients, ${C_{y}}^{\textit{rms}}$ and ${C_{v}}^{\textit{rms}}$ , remain nearly constant at $0.13$ and $0.18$ , respectively, over the entire range $9.2 \leqslant U^* \leqslant 12.0$ .

3.3.2. Wake modes

In this subsection, PIV measurements are analysed to characterise the wake modes in terms of their vorticity patterns.

Figure 15 illustrates phase-averaged PIV snapshots for Mode I for a sequence of increasing reduced velocities: $U^* \in [4.5, 5.0, 5.5]$ . Four phase-averaged PIV snapshots (selected from 48 phases) are presented in the columns. The images are structured as follows: the first column from left to right shows the wake dynamics of the oblate spheroid at the equilibrium position, the second column corresponds to the oblate spheroid at the highest displacement point, the third column is associated with another equilibrium position and the fourth column displays the wake structure at the lowest position. Due to the 3-D and turbulent nature of the flow at high Reynolds numbers, the wake vortex shedding is complex; however, several phase-average coherent structures can be discerned in the laser plane, and the average flow dynamics can be inferred from the visualisations. In this study, the nomenclature for the wake patterns is based on the terminology introduced by Williamson & Roshko (Reference Williamson and Roshko1988). Note that the interpretations supplied here result from viewing the wake animations; these interpretations are more difficult to arrive at from the presented sets of still images in the figures. The movie images are included in the supplementary material to help the reader understand the wake structure formation for the oscillating oblate spheroid.

Figure 15. Wake patterns visualised from PIV spot measurements in the Mode-I regime for $\epsilon = 2.00$ and $(m^* + C_{A})\zeta =0.1263$ at ( $a$ ) $U^*=4.5$ , ( $b$ ) $U^*=5.0$ , and ( $c$ ) $U^*=5.5$ . The normalised vorticity range is $\omega ^{*}=\omega b/U$ $\in [-3, 3]$ , with $\omega$ being the vorticity. The horizontal dashed line at $y/b=0$ denotes the centreline of the zero flow condition, facilitating a clearer observation of the wake deflection. At the same time, the vertical red bar located at $x/b$ signifies the vibration ranges. The fluid flow in the images is from left to right. The blue is for negative vorticity and the red is for positive vorticity. The red point on the sine waves shows the corresponding position of the spheroid during the oscillation cycle. Note that the oscillations during the synchronised regimes shown here are assumed to be sinusoidal; thus, the sinusoid symbolically represents the displacement of the oblate spheroid. Supplementary movies are included available at https://doi.org/10.1017/jfm.2025.866 to help the reader understand the wake structure formation for the oscillating oblate spheroid.

As illustrated in figure 15, the low-damped case of $(m^* + C_{A})\zeta =0.1263$ in the M-I regime exhibits a 2(P + S) wake mode, characterised by two pairs of counter-rotating vortices (P) along with two single vortices (S) of opposite-sign shed per body oscillation cycle. For $U^*=4.50$ , when the body moves upwards towards its top position, the wake is deflected in the opposite direction (i.e. downwards), composed of a pair of counter-rotating vortices $P_{1}$ , and a single vortex of negative vorticity $S_{1}$ ; similarly, when the oblate spheroid moves towards its lowest position, the wake is deflected upwards, displaying another pair of counter-rotating vortices $P_{2}$ along with a single vortex $S_{2}$ . The wake shows the half-period flip symmetry, similar to that of an oscillating sphere or cylinder experiencing VIV (see Blackburn, Marques & Lopez Reference Blackburn, Marques and Lopez2005; Rajamuni et al. Reference Rajamuni, Thompson and Hourigan2018; Sareen et al. Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a ; Eshbal et al. Reference Eshbal, Rinsky, David, Greenblatt and van Hout2019b ; Mishra, Bhardwaj & Thompson Reference Mishra, Bhardwaj and Thompson2024). As the reduced velocity is increased to $U^*=5.5$ , the detachment of the shear layer occurs closer to the afterbody; however, the deflection angle of the wake appears similar.

Figure 16. Sample time traces of the body displacement $(y^*)$ , and the coefficients of the total transverse force ( $C_{y}$ ) and the vortex force ( $C_{v}$ ) within the M-I regime, along with the corresponding frequency PSD plots for $\epsilon = 2.00$ and $(m^* + C_{A})\zeta =0.1263$ at ( $a$ ) $U^*=4.5$ , ( $b$ ) $U^*=5.0$ , and ( $c$ ) $U^*=5.5$ . The time traces are plotted as a function of the dimensionless time $\tau = t f_{\textit{n}w}$ and the power spectrum as a function of $f^*$ . The red dots in the power spectrum highlight multiple harmonic contributions.

Figure 16 exhibits sample time traces of the fluid forces and the spheroid displacement as a function of the dimensionless time $\tau =t/T_{\textit{n}w} = t f_{\textit{n}w}$ , alongside the power spectral density (PSD) of these signals as a function of the dimensionless frequency, $f^*=f/f_{\textit{n}w}$ , for the corresponding $U^*$ displayed in figure 15. Figure 16 shows that both the total transverse force ( $C_{y}$ ) and the vortex force ( $C_{v}$ ) are generally in phase with the body displacement ( $y^*$ ) over the range of velocities associated with M-I. This agrees with the results presented in figure 14 where both fluid force phases are below $90^{\circ }$ . The PSD plots reveal that the most powerful frequency contribution is attributed to $f^*=1$ .

Figure 17. Wake patterns visualised from PIV spot measurements in the M-II regime for $\epsilon = 2.00$ and $(m^* + C_{A})\zeta =0.1263$ at ( $a$ ) $U^*=6.0$ , ( $b$ ) $U^*=7.0$ and ( $c$ ) $U^*=8.0$ . The red point on the sine waves shows the corresponding position of the spheroid within the oscillation cycle. The movie images are included in the supplementary material to help the reader understand the wake structure formation for the oscillating oblate spheroid.

As shown in figure 17, for M-II, the oblate spheroid wake depicts a 2(P + S) structure, similar to the one exhibited in M-I. For example, figure 17( $c$ ) displays the wake structure of the vortex shedding at $U^*=8.0$ . The wake deflection angle increases along with a more elongated shear layer and stronger interaction between opposite-signed shear layers relative to M-I as shown in figure 15. As the spheroid ascends, it sheds a single vortex, $S_{1}$ , followed by a pair of counter-rotating vortices, $P_{1}$ . Conversely, when the spheroid descends, another single vortex, $S_{2}$ , is shed before a pair of counter-rotating vortices, $P_{2}$ . The primary difference from M-I lies in the deflection angle of the wake and the timing of the formation of the vortical structures.

Figure 18. Sample time traces of the body displacement $(y^*)$ , and the coefficients of the total transverse force ( $C_{y}$ ) and the vortex force ( $C_{v}$ ) within the M-II regime, along with the corresponding frequency PSD plots for $\epsilon = 2.00$ and $(m^* + C_{A})\zeta =0.1263$ at ( $a$ ) $U^*=6.0$ , ( $b$ ) $U^*=7.0$ and ( $c$ ) $U^*=8.0$ . The red dots in the power spectrum highlight multiple harmonic contributions.

Figure 18 demonstrates that both fluid force coefficients (i.e. the total transverse and vortex forces) are evidently out of phase with respect to the body displacement. This is in contrast to the M-I case shown in figure 16. The PSD plots indicate the presence of two peaks associated with the frequency harmonic components of $f^*=1$ and $f^*=3$ , characterised by an approximately equal power contribution for this vibration regime.

Figure 19 reveals the sample PIV vorticity fields of the galloping-like regime G-I for $U^*\in [10.0, 11.0, 12.0]$ . The wake pattern associated with this vibration regime is 2(3P+ S), characterised by four pairs of opposite-sign vortices (P) and two single vortices of opposite vorticity $(S)$ shed per body oscillation cycle. Compared with the wake structure displayed in M-I and M-II, there is a significant enlargement (i.e. longer) of the shear layers combined with larger body vibrations. Furthermore, there is a stronger interaction between opposite-signed shear layers, resulting in a higher number of vortical structures shed per oscillation cycle compared with VIV modes. The number of distinct vortex structures per cycle may increase for higher values of the reduced velocity, although this was not tested.

Figure 19. Wake patterns visualised from PIV spot measurements within the G-I regime for $\epsilon = 2.00$ and $\zeta =0.0050$ at ( $a$ ) $U^*=10.0$ , ( $b$ ) $U^*=11.0$ and ( $c$ ) $U^*=12.0$ . The red point on the sine waves shows the corresponding position of the spheroid during the oscillation cycle. The movie images are included in the supplementary material to help the reader understand the wake structure formation for the oscillating oblate spheroid.

Figure 20 exhibits the sample time traces of the structural (body displacement) and fluid (transverse and vortex force) dynamics corresponding to the PIV visualisation shown in figure 19. For Mode G-I, both $C_{y}$ and $C_{v}$ display two dominant frequencies that have approximately equal power contributions, matching the body vibration frequency at $f^*=1$ and a harmonic at $f^*=3$ . In this case, the total phase remains close to $90^{\circ }$ , suggesting that the transverse total force is nearly in phase with the body’s velocity; therefore, supporting the body vibration and inducing a galloping-dominated response.

It is noteworthy that the second harmonic in the fluid forcing ( $f^*_{C_y}$ and $f^*_{C_v}$ ) seems to correlate with the number of vortex pairs (i.e. the number of ‘P’) in the wake pattern. For instance, a relatively weak second harmonic is observed in the 2(P + S) patterns in M-I and M-II. However, as demonstrated by Cordero Obando et al. (Reference Cordero Obando, Thompson, Hourigan and Zhao2024), in the 2(2P+ S) pattern observed in the V-II regime for $\epsilon = 3.20$ , the second harmonic of $f^*_{C_y}$ exceeds the third harmonics in strength. Conversely, in the 2(3P+ S) mode during the G-I regime of $\epsilon = 2.00$ in the present study, the third harmonic appears to be second-strongest frequency component, while the second harmonic becomes negligible (see figure 20).

Figure 20. Sample time traces of the body displacement $(y^*)$ , and the coefficients of the total transverse force ( $C_{y}$ ) and the vortex force ( $C_{v}$ ) within the G-I regime, along with the corresponding frequency PSD plots for $\epsilon = 2.00$ and $(m^* + C_{A})\zeta =0.1263$ at ( $a$ ) $U^*=10.0$ , ( $b$ ) $U^*=11.0$ and ( $c$ ) $U^*=12.0$ .

3.4.1. Quasi-steady analysis

This section examines the potential for galloping through a quasi-steady analysis, focusing on the lift force ( $F_{L}$ ), drag force ( $F_{D}$ ) and transverse force ( $F_{y}$ ) acting on a static body as the angle of incidence, $\alpha$ , changes. The approach assumes that the different phases of vibration are quasi-steady, meaning there is little to no phase difference between the fluid forces and the body’s velocity. Under this assumption, the instantaneous excitation force acting on the moving structure is closely approximated by the static force at the corresponding instantaneous angle of attack (Naudascher & Rockwell Reference Naudascher and Rockwell2005).

For an elastically mounted bluff body undergoing FIV with a transverse velocity, $\dot {y}$ , the relative velocity between the moving body and the free stream flow has a magnitude, $U_{\textit{rel}} = \sqrt {U + \dot {y}^{2}}$ , and an angle, tan $(\alpha ^{\prime }) = \dot {y}/U$ (figure 21).

Figure 21. Sketch of the relative velocity and fluid forces experienced by an oblate spheroid moving transversely in a free stream flow. Note that $\dot {y}$ is the body’s velocity, $U_{\textit{rel}}$ is the relative velocity and $\alpha ^{\prime }$ is the relative (effective) angle of attack; $C_{L}$ and $C_{D}$ are the lift and drag coefficients with respect to $U_{\textit{rel}}$ , respectively, while $C_{y}$ and $C_{x}$ are the transverse and streamwise force coefficients, respectively.

Figure 22. (a) $C_{L,q}$ and (b) $C_{D,q}$ , which are the lift and drag coefficients, respectively, obtained from static tests as a function of the inclination angle $\alpha$ for the $\epsilon = 2.00$ spheroid at various Reynolds numbers.

Figure 22 displays the variations of the mean quasi-steady coefficients of the lift ( $C_{L,q}$ ), and drag ( $C_{D,q}$ ) as a function of inclination angle of attack in the range of $-10^{\circ } \leqslant \alpha \leqslant 90^{\circ }$ at different fixed Reynolds numbers in the range $7500 \leqslant Re \leqslant 19\, 840$ . It can be seen that the drag coefficient ( $C_{D,q}$ ) attains a maximum value of $0.85$ at an incidence angle of $\alpha =0^{\circ }$ , progressively decreasing to a global minimum of $0.25$ at $\alpha =90^{\circ }$ . Notably, the drag coefficient variation seems to follow a common trend independent of the Reynolds number value within the tested range. However, the lift coefficient ( $C_{L,q}$ ) starts at $0$ for $\alpha =0^{\circ }$ and reaches an absolute value of $0.42$ at $\alpha =58^{\circ }$ . Within this range of inclination angle, the trend followed by $C_{L,q}$ seems to be uncorrelated with the value of $Re$ . Furthermore, beyond $\alpha \geqslant 58^{\circ }$ , there is an inflexion point for $Re \geqslant 15\,000$ , marked by a sudden jump in the magnitude of $C_{L,q}$ from $0.42$ to $0.67$ between $58^{\circ }$ and $59^{\circ }$ , and then a decrease in the magnitude of $C_{L,q}$ until reaching 0 at $\alpha =90^{\circ }$ . This change in the shape of $C_{L,q}$ denotes a change in the regime flow, most likely associated with a change in the wake structure.

Figure 23. Variation of the maximum relative angle of attack, $\alpha ^{\prime }_{\textit{max}}$ , as a function of the reduced velocity, $U^*$ , for $\epsilon = 2.00$ with various values of $(m^* + C_{A})\zeta$ .

When compared with figure 23 for the oscillating case, it can be seen that the range of $U^*$ associated with a galloping-like response displays an almost constant value of maximum relative angle of attack ( $\alpha ^{\prime }_{\textit{max}}$ ) between $45^{\circ }$ and $50^{\circ }$ , a value that is close to that associated with the inflexion point for the static case. Note that the instantaneous relative angle of attack is calculated as $\alpha ^{\prime }(t) = \tan ^{-1} ({\dot {y}(t)}/{U} )$ .

To assess if the oblate spheroid with an aspect ratio of $\epsilon =2.00$ is susceptible to galloping oscillations under soft excitation (i.e. small-amplitude excitation from rest), a stability criterion can be derived from the following equation of motion:

(3.2) \begin{equation} m\ddot {y} + 2m\zeta \omega _{n}\dot {y}+m{\omega _{n}}^{2}y = \frac {\pi }{8} \rho U^{2}b^{2}C_{y}, \end{equation}

where $\omega _{n} = 2\pi f_{nw}$ is the natural angular frequency of the system. Based on the quasi-steady theory by Parkinson & Smith (Reference Parkinson and Smith1964), the transverse fluid force coefficient ( $C_{y}$ ) for small amplitude excitation can be approximated as a first-degree polynomial (Naudascher & Rockwell Reference Naudascher and Rockwell2005):

(3.3) \begin{equation} C_{y} = a_{1}\frac {\dot {y}}{U}. \end{equation}

Substituting (3.3) in (3.2) yields

(3.4) \begin{equation} \ddot {y} + \left(\frac {2\zeta \omega _{n}}{m} - \frac {\pi \rho U b^{2}}{8m} a_{1}\right)\dot {y} + \omega _{n}^{2} y = 0. \end{equation}

According to (3.4), the dynamical system becomes unstable if the combined damping term (i.e. the sum of structural and aerodynamic damping) is negative:

(3.5) \begin{equation} \frac {2\zeta \omega _{n}}{m} - \frac {\pi \rho U b^{2}}{8m} a_{1} \lt 0. \end{equation}

Here, the sign of the damping term is determined solely by $a_{1}$ , as all other quantities are positive. Transverse galloping from rest occurs if $a_{1} = \beta _{qs} = {\partial {C_{y}}/\partial {\alpha }} = -({\partial {C_{L,q}}/\partial {\alpha }}+C_{D,q}) \geqslant 0$ , a condition known as the Den Hartog criterion (Naudascher & Rockwell Reference Naudascher and Rockwell2005). It is noteworthy that $a_{1}$ represents the slope of the transverse force coefficient ( $C_{y}$ ) at zero angle of attack, $\alpha = 0^{\circ }$ . Based on the Den Hartog criterion, the oblate spheroid does not exhibit galloping from rest, as $a_{1} \lt 0$ . Specifically, at $\alpha =0^{\circ }$ , $a_{1}$ has a value of $-0.1$ (see figure 24). The lift and the drag forces contribute to damping, confirming that the oblate spheroid is not susceptible to soft galloping.

Figure 24. Variation of the quasi-steady stability criteria $\beta _{qs}$ computed from the quasi-steady force coefficients as a function of the incidence angle of attack, $\alpha$ .

It is also insightful to consider the potential for the oblate spheroid to exhibit hard galloping, defined as the onset of vibration triggered by a finite-amplitude disturbance (Novak Reference Novak1972). For this analysis, an expression for the quasi-steady transverse force coefficient ( $C_{y,q}$ ) is required as a function of the measured quasi-steady lift ( $C_{L,q}$ ) and quasi-steady drag ( $C_{D,q}$ ):

(3.6) \begin{equation} C_{y,q}(\alpha ) = -\frac {1}{\cos (\alpha )}\left (C_{L,q} + C_{D,q} \tan (\alpha )\right )\!. \end{equation}

A key factor to consider is the instantaneous quasi-steady power coefficient, $C_{p,q}$ , which represents the energy transfer from the fluid to the oblate spheroid due to the transverse force. If $C_{p,q} \gt 0$ , the transverse force amplifies the vibration of the structure, while negative values indicate that any vibration due to the transverse force will be damped. The power coefficient was computed as follows:

(3.7) \begin{equation} C_{p,q}(\alpha ) = C_{y,q}\dot {y} = U \tan (\alpha ) C_{y,q}. \end{equation}

Figure 25 shows the variation of the quasi-steady lift ( ${C}_{L,q}$ ), drag ( ${C}_{D,q}$ ), transverse ( ${C}_{y,q}$ ) and power ( ${C}_{p,q}$ ) coefficients against the inclination angle of attack, $\alpha$ , at a Reynolds number of $Re=19\,800$ . According to (3.6), the lift contributes positively to the transverse force, while the drag opposes the motion. For all the angles of attack tested ( $0^{\circ } \leqslant \alpha \leqslant 55^{\circ }$ ), the lift ( $C_{L,q}$ ) remains negative in the range ( $-0.4 \leqslant C_{L,q} \leqslant 0.4$ ), while the drag ( $C_{D,q}$ ) remains positive in the range ( $0.5 \leqslant C_{D,q} \leqslant 0.9$ ). For all the values of $\alpha$ , the drag dominates the lift in (3.6) (see figure 25 $c$ ), generating a negative power contribution (see figure 25 $d$ ), therefore attenuating the motion. Thus, based on this quasi-steady analysis, no energy from the flowing fluid is added to the structure. Consequently, based on quasi-steady analysis, the oblate spheroid is not susceptible to developing hard galloping either.

Figure 25. Variation of the $(a)$ lift $C_{L,q}$ , $(b)$ drag $C_{D,q}$ , $(c)$ transverse $C_{y,q}$ and $(d)$ power $C_{p,q}$ coefficients as a function of the inclination angle $\alpha$ for the quasi-steady model, obtained from force measurements on the fixed spheroid of $\epsilon =2.00$ for a Reynolds number of $Re=19\,840$ .

3.4.2. Instantaneous force coefficients

This subsection presents the force and power coefficients derived from the FIV experiments and contrasts them with the quasi-steady predictions outlined in § 3.4.1. Figure 26 presents the variation of the instantaneous fluid force and power coefficients as a function of the relative angle of attack, $\alpha '$ , for different values of reduced velocity $U^* = 6,7,8,9,10,11$ and $12$ . These values of $U^*$ are associated with different dynamic response regimes. The lift ( $C_{L}$ ) and drag ( $C_{D}$ ) coefficients were computed as a function of the relative angle of attack ( $\alpha '$ ), and the transverse ( $C_{y}$ ) and streamwise ( $C_{x}$ ) coefficients:

(3.8) \begin{align} C_{L}(\alpha ') & =\dfrac {C_{y}\cos (\alpha ') - C_{x}\sin (\alpha ')}{1 + \tan ^{2}(\alpha ')} \end{align}

(3.9) \begin{align} C_{D}(\alpha ') & = \dfrac {C_{y}\sin (\alpha ') + C_{x}\cos (\alpha ')}{1 + \tan ^{2}(\alpha ')}. \\[12pt] \nonumber \end{align}

Figure 26. Variation of the instantaneous $(a)$ lift ( $C_L$ ), $(b)$ drag ( $C_D$ ), $(c)$ transverse force ( $C_y$ ) and $(d)$ power coefficients ( $C_p$ ) obtained from an elastically mounted oblate spheroid oscillating as a function of the relative angle of attack ( $\alpha {'}$ ) at different values of $U^*$ . Note that the quasi-steady force coefficients for $Re=19\,840$ are plotted to compare both approaches.

First, it is evident that the instantaneous lift coefficient ( $C_{L}$ ) exhibits a general trend across all values of $U^*$ , ranging approximately in the range $-0.6 \leqslant C_{L} \leqslant 0.6$ . Compared with the quasi-steady lift ( $C_{L,q}$ ), both show good agreement, indicating that the lift variations are primarily due to quasi-steady effects and are relatively insensitive to changes in Reynolds number. Conversely, the instantaneous drag coefficient ( $C_{D}$ ) is influenced by $U^*$ (i.e. Reynolds number), leading to significant variations depending on the dynamic response regime. Compared with the quasi-steady drag ( $C_{D,q}$ ), there is a noticeable difference between their values depending on the dynamic response regime in which the instantaneous force measurement is taken. This suggests that in addition to a quasi-steady effect, there is a vortex-shedding effect that modifies the behaviour of the drag. Regarding energy transfer, certain values of the relative angle of attack exhibit a positive energy contribution (see figure 26 d). This enhances the oblate spheroid motion and sustains large amplitude vibrations.

Figure 27. Variation of the instantaneous stability criteria $\beta _{\textit{inst}}$ computed from the instantaneous force coefficients as a function of the relative angle of attack, $\alpha {'}$ . Note that the variation of the quasi-steady stability criteria ( $\beta _{qs}$ ) is included along with the instantaneous values of the stability criteria ( $\beta _{\textit{inst}}$ ).

Additionally, the analysis of the instantaneous stability criterion merits particular attention. Figure 27 illustrates the stability criterion of Den Hartog, derived from the instantaneous fluid force coefficients as a function of the relative angle of attack for different values of reduced velocity ( $U^*$ ). Notably, the instantaneous stability parameter, $\beta _{\textit{inst}} = {\partial {C_y}/\partial {\alpha ^{\prime }} = - {\partial C_L/\partial \alpha ^{\prime }} - C_D}$ , exhibits both symmetry and an evident trend; its magnitude increases with the reduced velocity. This trend highlights that as $U^*$ rises, the body becomes progressively more susceptible to galloping-like oscillations due to soft excitation. Consequently, the system transitions to an unstable regime from the perspective of a soft oscillator, indicating the potential for galloping to occur from rest.

The quasi-steady model does not capture the effects of vortex shedding, which occurs as a single vortex is shed each half-cycle of motion, as shown in figure 19, and is likely to be contributing to the motion of the body. Instead, the quasi-steady model focuses on the geometry of the body and movement-induced excitation, neglecting vortex shedding effects that significantly influence the fluid–structure interaction. This explains the discrepancies between the quasi-steady model and instantaneous force measurements. In summary, the energy transfer analysis based on the instantaneous force measurements suggests that the oblate spheroid is susceptible to a galloping-like response. Therefore, from the quasi-steady perspective, this spheroid case ( $\epsilon =2.00$ ) is not susceptible to classical transverse galloping from rest. However, based on the extrapolation of the stability criterion (Den Hartog’s) applied on instantaneous fluid forcing, this spheroid can exhibit galloping oscillations, as indicated by the $\alpha ^{\prime }_{\textit{max}}$ values and the amplitude growth at high $U^{*}$ . This consideration motivates our use of the term ‘galloping-like’ rather than classical transverse galloping.

3.5.1. Vibration response

The case of $(m^* + C_{A})\zeta =0.4291$ represents a high mass-damping parameter configuration exhibiting a dynamic response composed of VIV only. Figure 28( $a$ ) demonstrates that the amplitude of vibration is typical of a VIV response that has been observed in the case of elastically mounted spheres immersed in water or air (see Govardhan & Williamson Reference Govardhan and Williamson2005; Rajamuni et al. Reference Rajamuni, Thompson and Hourigan2018; Sareen et al. Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a ). Over the lock-in region, the maximum amplitude of vibration is $A^*_{10}=0.6$ at $U^*=6.4$ . The frequency response of the oblate spheroid displacement (figure 28 $b$ ) reveals that over the range $4.8 \leqslant U^* \leqslant 8.0$ , the displacement frequency ( $f^*_{y}$ ) is equal to the natural frequency of the system ( $f_{\textit{n}w}$ ). Based on the frequency responses of the fluid forces, two distinct vibration regimes can be associated with the VIV region: M-I and M-II. Mode I extends over the range $4.8 \leqslant U^* \leqslant 5.8$ , where the frequency response of the fluid forces displays the stronger contribution at $f^*_{C_y}= f^*_{C_v}=1$ and a weaker third harmonic contribution at $f^*_{C_y}=f^*_{C_v}=3$ .

Figure 28. ( $a$ ) $A^{*}_{\textrm {{10}}}$ amplitude response and $(b$ – $d)$ logarithmic-scale PSD contours of normalised frequency responses as a function of reduced velocity for the $\epsilon = 2.00$ case with $(m^* + C_{A})\zeta =0.4291$ . In panels $(b$ )–( $d)$ , the vertical dashed lines represent the boundaries of the response regimes.

Figure 29. Variation of the ( $a$ ) transverse lift force coefficient $(C^{\textit{rms}}_{y})$ and vortex force coefficient $(C^{\textit{rms}}_{v})$ , and $(b)$ total phase $(\phi _{t})$ and vortex phase $(\phi _{v})$ , all as functions of the reduced velocity $U^*$ for the $\epsilon = 2.00$ case with $(m^* + C_{A})\zeta =0.4291$ .

In figure 29( $b$ ), it can be seen that the values of $C^{\textit{rms}}_{y}$ and $C^{\textit{rms}}_{v}$ suddenly jump from 0.05 to 0.15 in the range $4.8 \leqslant U^* \leqslant 5.0$ , marking the onset of M-I. Within this $U^*$ range, $\phi _{v}$ grows from $\sim$ $50^{\circ }$ to $\sim 75^{\circ }$ . Additionally, $\phi _{t}$ leads by $30^{\circ }$ , maintaining the same trend as $\phi _{v}$ .

With further increase in the reduced velocity, at $U^* \geqslant 6.0$ , there is the appearance of a weak second harmonic in the spectra of the fluid forces, indicating a transition between VIV modes (see figure 28 $c$ , $d$ ). Based on the variation of the fluid forces and phases in figure 29, the onset of this second vibration is marked by a jump in the value of the total and vortex phases, where $\phi _{t}$ increases from $\sim 75^{\circ }$ to $\sim 145^{\circ }$ and $\phi _{v}$ from $\sim 110^{\circ }$ to $\sim 150^{\circ }$ . The maximum amplitude of vibration is $A^*_{10}=0.6$ at $U^*=6.4$ (see figure 29 a); this point is associated with a global maximum in the vortex force coefficient, $C^{\textit{rms}}_{v}=0.33$ . The end of the VIV region is marked by a sudden drop in the values of the total and vortex phases from ${\sim} 150^{\circ }$ to ${\sim} 90^{\circ }$ at $U^*=8.0$ , marking the onset of the desynchronisation region.

3.5.2. Wake modes

Figure 30 shows phase-averaged PIV snapshots for M-I for two different reduced velocities of $U^*=[5, 6]$ for the high-damped case of $(m^* + C_{A})\zeta =0.4291$ , where the FIV response exhibits VIV only. Increasing the damping ratio does not affect the wake pattern seen for M-I, which is still a 2(P + S) pattern, characterised by a pair (P) of opposite-sign vortices along with one single vortex (S) shed during half of an oscillation cycle. However, an increase in the damping ratio induces a notable decrease in the vibration range.

Figure 30. Wake patterns visualised from PIV spot measurements within the M-I regime for the $\epsilon = 2.00$ case with $(m^* + C_{A})\zeta =0.4291$ at ( $a$ ) $U^*=5.0$ and ( $b$ ) $U^* = 6.0$ . The red point on the sine waves shows the corresponding position of the spheroid during the oscillation cycle. The movie images are included in the supplementary material to help the reader understand the wake structure formation for the oscillating oblate spheroid.

Figure 31. Sample time traces of the body displacement $(y^*)$ , and the coefficients of the total transverse force ( $C_{y}$ ) and the vortex force ( $C_{v}$ ) within the M-I regime, along with the corresponding frequency PSD plots for the $\epsilon = 2.00$ case with $(m^* + C_{A})\zeta =0.4291$ at ( $a$ ) $U^* = 5.0$ and ( $b$ ) $U^*=6.0$ .

Figure 31 shows that the displacement and fluid force coefficients are highly periodic in this mode, with the fluid force coefficients, $C_{y}$ and $C_{v}$ , being in phase with the body displacement. The PSD plots reveal a behaviour similar to compared with those for $(m^* + C_{A})\zeta =0.1263$ (figure 16), where the main frequency contribution is at $f^*=1$ , while noting an additional strong component at $f^*=3$ .

Figure 32. Wake patterns visualised from PIV spot measurements within M-II regime for the $\epsilon = 2.00$ case with $(m^* + C_{A})\zeta =0.4291$ at $U^*=7.0$ . The red point on the sine waves shows the corresponding position of the spheroid during the oscillation cycle. The movie images are included in the supplementary material to help the reader understand the wake513 structure formation for the oscillating oblate spheroid.

Figure 32 illustrates the wake structure displayed by the spheroid for M-II at $U^*=7.0$ . In this case, even though the damping ratio has been incremented, the wake still exhibits a 2(P + S) wake pattern, as seen for $(m^* + C_{A})\zeta =0.1263$ (see § 3.3). Analogous to what has been discussed for the low damping case, the main difference with M-I is a greater deflection angle of the wake, along with an elongation of the shear layer. During the first half of the oscillation cycle, a pair of counter-rotating vortices (P $_{1}$ ) is shed along with a single vortex (S $_{1}$ ); it is worth noting that the vortices have similar intensities. In the second half of the oscillation cycle, there is a second pair of counter-rotating vortices (P $_{2}$ ) along with a single vortex (S $_{2}$ ) shed. The wake is very similar to the topology observed in figure 15 at $U^*=8.0$ , suggesting that M-II persists, even though the structural damping ratio has been increased significantly.

Figure 33. Sample time traces of the body displacement $(y^*)$ , and the coefficients of the total transverse force ( $C_{y}$ ) and the vortex force ( $C_{v}$ ) within the M-II regime, along with the corresponding frequency PSD plots for the $\epsilon = 2.00$ case with $(m^* + C_{A})\zeta =0.4291$ at $U^*=7.0$ .

Figure 33 shows that the transverse and vortex forces are notably out of phase with the body displacement, similar to the behaviour observed for the low-damped case $(m^* + C_{A})\zeta =0.1263$ . The frequency PSD plots display the presence of a main peak matching the body vibration frequency at $f^*=1$ and an additional third harmonic contribution of weak power.

Figure 34 summarises the effect of damping ratio on the amplitude response through a contour map in $U^*-(m^* + C_{A})\zeta$ parameter space. This clearly demonstrates that high vibration amplitudes are observed for low damping ratio and that as the damping ratio is increased, there is both a reduction in the peak amplitude as well as a reduction in the synchronisation $U^*$ range. Together with the measured variations of the fluid forces and phases, it is possible to identify the regions of parameter space occupied by Modes I, II and G-I.

Figure 34. FIV amplitude response contour map for an oblate spheroid of $\epsilon = 2.00$ in $(U^*, (m^* + C_{A})\zeta , G)$ parameters space. Note that $G$ corresponds to the gap distance in ${\textrm{mm}}$ , which is directly related to the mass-damping parameter, $(m^* + C_{A})\zeta$ , shown on the right axis. The vertical dash-dotted lines display the boundary between VIV and galloping. The region within the red dot boundary corresponds to M-I, the yellow dots boundary ( ) demarcates M-II and the orange dot boundary corresponds to mode G-I.

3.7.1. An optimal damping ratio for maximum average power coefficient

For an elastically mounted sphere undergoing structural vibration, it is possible to harvest energy from the flow by adding a structural damper. In practice, this damping can be supplied by an electrical generator. The instantaneous power output of the spheroid can be calculated as $P = F_{y}\boldsymbol{\cdot }\dot {y}$ , where $\dot {y}$ is the velocity of the spheroid. Assuming the structural vibration is periodic, this can be approximated by $P = c\dot {y}^2$ (Zhao et al. Reference Zhao, Thompson and Hourigan2022a ). A lower damping ratio may result in higher vibration amplitude and body velocity $\dot {y}$ ; however, this does not necessarily correspond to a higher output power. Furthermore, a power output coefficient can be defined as

(3.10) \begin{equation} C_{p} = \dfrac {P}{\rho {U}^3 b^2 \pi /8}, \end{equation}

where $\rho {U}^3 b^2 \pi /8$ is the fluid power passing through the projected spheroid surface area. Equation (3.10) demonstrates the energy that can be potentially extracted from the surrounding fluid flow via structural vibration. For a bluff body undergoing FIV, the time-average power coefficient can be expressed as

(3.11) \begin{equation} \overline {C_{p}} = \frac {1}{T} \int _{0}^{T} C_{p}(t) \,{\textrm{d}}t. \end{equation}

Here, $T$ is the integration time spanning many oscillation cycles.

Figure 38 illustrates the variation of the time-average power coefficient as a function of reduced velocity for various values of the mass-damping parameter and aspect ratio. For all cases ( $\epsilon =1.00, 2.00, 3.20$ ), there is a specific damping ratio where the power coefficient reaches its maximum. As shown in figure 39, the optimal damping ratio for energy harvesting varies slightly with geometry. For the sphere case, the maximum power output occurs at $(m^* + C_{A})\zeta = 0.1401$ . For $\epsilon =2.00$ , the optimal value is $\zeta =0.2679$ , while the thinnest oblate spheroid ( $\epsilon =3.20$ ) achieves its peak performance at $\zeta =0.2726$ .

Figure 38. Variations the time-average power coefficient ( $\overline {C_{p}}$ ) as a function of reduced velocity for $(a)$ $\epsilon =1.00$ , $(b)$ $\epsilon =2.00$ and $(c)$ $\epsilon =3.20$ , with various values of $(m^* + C_{A})\zeta$ .

Interestingly, while the optimal mass-damping parameter across different geometries remain similar, the maximum values of $\overline {C_p}$ vary significantly. The sphere achieves a peak of $0.021$ at $U^*=8.6$ , the oblate spheroid ( $\epsilon =2.00$ ) reaches $0.10$ at $U^*=6.0$ , and the thinnest spheroid ( $\epsilon =3.20$ ) attains the highest value of $0.175$ at $U^*=6.8$ . Additionally, as the afterbody size decreases, less damping is required to minimise or attenuate the potential of energy harvesting from FIV.

The oblate spheroid with $\epsilon =2.00$ demonstrates robust performance, maintaining $\overline {C_p}_{\textit{max}}$ values in the range $0.075 \leqslant \overline {C_p}_{\textit{max}} \leqslant 0.10$ over a broad range of mass-damping parameters ( $0.1263 \leqslant (m^* + C_{A})\zeta \leqslant 0.2012$ ). In contrast, the thinnest spheroid ( $\epsilon =3.20$ ) achieves a higher power coefficient but within a narrower range of the mass-damping parameter, highlighting the trade-off between peak performance and operational range. As demonstrated by Cordero Obando et al. (Reference Cordero Obando, Thompson, Hourigan and Zhao2024), the energy extraction performance of spheroids is influenced by two primary parameters: the damping ratio, which optimises energy extraction from the flow, and the aspect ratio.

Figure 39. Variation of the maximum average power coefficient, $\overline {C_{p}}_{\textit{max}}$ , as a function of the damping ratio, $\zeta$ , for various aspect ratios, $\epsilon$ .

It can be concluded that the maximum average power coefficient is found over the range of reduced velocity $U^*$ corresponding to the VIV region regardless of the value of $(m^* + C_{A})\zeta$ ; however, the power output decreases quickly moving away from the corresponding $U^*$ value. Over the galloping-dominated $U^*$ range, the average power coefficient $\overline {C_{P}}$ remains almost constant. Consequently, the potentially extractable power from the galloping region may be more suitable for engineering applications. Galloping behaviour has been advantageous for practical applications (see Barrero-Gil, Alonso & Sanz-Andres Reference Barrero-Gil, Alonso and Sanz-Andres2010; Vicente-Ludlam, Barrero-Gil & Velazquez Reference Vicente-Ludlam, Barrero-Gil and Velazquez2014; Hémon et al. Reference Hémon, Amandolese and Andrianne2017) due to the stability of power output. In comparison, the maximum power coefficient for a circular cylinder undergoing VIV is ${\sim} 0.18$ (Bernitsas et al. Reference Bernitsas, Raghavan, Ben-Simon and Garcia2008; Soti et al. Reference Soti, Zhao, Thompson, Sheridan and Bhardwaj2018). While the maximum power coefficient of the spheroid is below that of the circular cylinder, which has been used as a renewable energy harvesting device (Bernitsas et al. Reference Bernitsas, Raghavan, Ben-Simon and Garcia2008), the current investigation indicates an improvement in $\overline {C_p}$ over that of a sphere. Indeed, Cordero Obando et al. (Reference Cordero Obando, Thompson, Hourigan and Zhao2024) shows that increasing the aspect ratio of the spheroid to 3.20, increases the power coefficient to 0.165, which is almost comparable to the circular cylinder case. Of note, the Betz limit – derived from an actuator disk theory using the rotor swept area as the reference area (see Betz Reference Betz1966) – establishes an upper limit of $16/27$ for the power coefficient of such devices. Of course, adding elastically mounted spheroids in the wake of those upstream may increase the energy extraction potential – as occurs in wind turbine-based wind farms. It may also be possible to place coupled spheroids aligned cross-stream to extract further energy – again noting that adding blades increases the efficiency of wind turbines. While these aspects are not within the scope of the current study, it is of interest that such a compound device already exists for elastically mounted circular cylinders (Bernitsas et al. Reference Bernitsas, Raghavan, Ben-Simon and Garcia2008; Kim & Bernitsas Reference Kim and Bernitsas2016).