3.1. Effect of m* on vibration response

We present here the effect of m* on vibration and frequency responses at ζ = 0 (figure 4). The cylinder responses can be divided into three branches, namely the initial branch (IB), lower branch (LB) and galloping branch (GB). For m* = 20, the vortex excitation regime (including IB and LB) appears when U_r ≤ 12, whereas GB occurs when U_r > 12. Following Bhatt & Alam (Reference Bhatt and Alam2018), the identifications of IB and LB are made based on the relationship between St and U_r shown in figure 5(a). The Strouhal number St – the dimensionless vortex shedding frequency of the vibrating cylinder – was estimated from the power spectral density functions of the lift signals. Here, St ₀ is used to denote the dimensionless shedding frequency of a fixed cylinder (figure 5a). The IB corresponds to St < St ₀ while the LB to St > St ₀. The boundary between IB and LB is characterized by a jump in St, e.g. between U_r = 5.7 and 5.77 for m* = 20, between U_r = 5.45 and 5.5 for m* = 10 and between U_r = 5.2 and 5.3 for m* = 5 (figure 5a). The value of A* increases with increasing U_r in IB, which is accompanied by a declining St and a constant f_s/f_o = 1.0 (figures 4a,b and 5a). The f_s/f_o = 1.0 indicates the lock-in (figure 4b). On the other hand, f_r grows with increasing U_r (figure 5b).

Figure 4. Dependence on reduced velocity U_r and mass ratio m* of (a) response amplitude A* and (b) f_s/f_o. (c) Zoom-in view of A*–U_r plot in (a) for U_r = 1–12. Here, ζ = 0 and Re = 170.

Figure 5. Variations of (a) Strouhal number St, (b) oscillation frequency ratio f_r, (c) time-mean drag coefficient ${\bar{C}_D}$ and (d) fluctuating lift ${C^{\prime}_L}$ with reduced velocity U_r for different mass ratio m* values. Here, damping ratio ζ = 0 and Re = 170.

A marked rise in A* also characterizes the boundary between IB and LB (figure 4c). In the LB regime, A* after reaching a peak declines with increasing U_r. The LB is further characterized by f_s/f_o = 1.0 (lock-in, figure 4b), declining St and increasing f_r. These three attributes are similar to those in IB. How do IB and LB differ? Why are there jumps in St and f_r between IB and LB? The distinct phase lag ϕ between C_L and Y* $(=Y/D)$ answers these two questions: ϕ = 0° in IB and ϕ = 180° in LB (not shown), given ζ = 0. The jump in ϕ from 0° to 180° occurs at the boundary between IB and LB.

The GB can be identified from the recovery in the increase of A* (figure 4a) and jump in f_s/f_o (figure 4b) or from the plunge in f_r (figure 5b). Therefore, the critical reduced velocity U_rc for the onset of galloping is 13, 12 and 12 for m* = 5, 10 and 20, respectively. In GB, A* increases rapidly with U_r. Deviating from this trend, A* surges at 1:3 (i.e. f_s/f_o = 3) and 1:5 (i.e. f_s/f_o = 5) synchronizations (figure 4a,b). The surge is stronger at 1:3 synchronization than at 1:5 synchronization. The onset of galloping for m* = 5 coincides with 1:3 synchronization. Sen & Mittal (Reference Sen and Mittal2016), Yao & Jaiman (Reference Yao and Jaiman2017) and Daniel et al. (Reference Daniel, Todd and Yahya2021) all observed 1:3 synchronization. Here, we find that 1:5 synchronization also plays a role in the vibration responses. The vibration response is generally quasi-periodic (amplitude varying with time) but is periodic (constant amplitude) for 1:3 and 1:5 synchronizations. Sen & Mittal (Reference Sen and Mittal2016) reported that the vibration response for m* = 5 was quasi-periodic for the entire GB because they ignored the 1:5 synchronization.

Term f_s/f_o in GB generally increases linearly with increasing U_r except for the occurrence of synchronizations around f_s/f_o = 3 or 5 (figure 4b). The linear relationship between f_s/f_o and U_r can be expressed as

(3.1a)\begin{gather}{f_s}/{f_o} = 0.1638{U_r} + 0.0939\quad \textrm{for}\;{m^\ast } = 20, \end{gather}

(3.1b)\begin{gather}{f_s}/{f_o} = 0.1708{U_r} + 0.2432\quad \textrm{for}\;{m^\ast } = 10,\end{gather}

(3.1c)\begin{gather}{f_s}/{f_o} = 0.1923{U_r} + 0.4505\quad \textrm{for}\;{m^\ast } = 5.\end{gather}

The value of St is more or less independent of U_r in the galloping regime while f_r being smaller than 1.0 slightly increases with U_r (figure 5a,b).

Parameter m* plays a crucial role in A*. It is a well-accepted argument that an increase in m* reduces A* (e.g. Bahmani & Akbari Reference Bahmani and Akbari2010). In the vortex excitation regime (including IB and LB), a larger m* leads to a smaller A* and a postponement of the A* peak. On the other hand, in the GB regime, the scenario is the opposite: the larger the value of m*, the larger is A* and the smaller is U_r for galloping onset (figure 4a). That is, the vortex excitation regime shrinks, and the onset of galloping advances when m* is increased. The surge in A* at 1:3 synchronization grows with increasing m*. In addition, a larger m* in the GB regime leads to (i) reductions in f_s/f_o and gradient of f_s/f_o with U_r (figure 4b; (3.1)), (ii) a decrease of St and (iii) an augmentation of f_r.

The maximum A* of a square cylinder in the VIV regime is much smaller than that of a circular cylinder case (Sen & Mittal Reference Sen and Mittal2011, Reference Sen and Mittal2015; Li et al. Reference Li, Lyu, Kou and Zhang2019). Li et al. (Reference Li, Lyu, Kou and Zhang2019) for a square cylinder at Re = 150 demonstrated that the maximum A* is 0.17, 0.14 and 0.11 at m* = 5, 10 and 20, respectively. Zhao (Reference Zhao2015) observed the maximum A* of 0.11 at m* = 10 and Re = 200. The maximum A* in our study (Re = 170) is 0.18, 0.14 and 0.1 at m* = 5, 10 and 20, respectively. Thus, the observed influence of m* on the vibration response within the VIV regime in our study appears to be consistent with the previous findings. Experimental results, typically obtained at higher Re values, feature larger A* values in the galloping regime (Zhao et al. Reference Zhao, Leontini, Jacono and Sheridan2014) than the numerical results at lower Re values.

3.2. Effect of m* on fluid forces

Figure 5(c,d) illustrates dependencies of ${\bar{C}_D}$ and ${C^{\prime}_L}$ on U_r for different m* values. The corresponding ${\bar{C}_{D0}}$ and ${C^{\prime}_{L0}}$ for the fixed cylinder are represented by blue dashed lines. In IB, with increasing U_r, ${\bar{C}_D}$ ($< {\bar{C}_{D0}}$) declines and ${C^{\prime}_L}$ ($> C^{\prime}_{L0}$) grows for all m* values. A larger m* corresponds to a larger ${\bar{C}_D}$ but a smaller ${C^{\prime}_L}$. The scenario is the opposite in LB, i.e. ${\bar{C}_D} > {\bar{C}_{D0}}$ and ${C^{\prime}_L} < {C^{\prime}_{L0}}$, the former declining and the latter growing. On the other hand, ${\bar{C}_D}$ augments in GB, with surges at 1:3 synchronization. Although rising in the early part of GB (i.e. up to 1:3 synchronization for m* = 10 and 20, and 1:5 synchronization for m* = 5), ${C^{\prime}_L}$ does not appreciably vary in the late part of GB.

Parameter m* has distinct influences in different regimes. Increasing m* renders increased ${\bar{C}_D}$ and decreased ${C^{\prime}_L}$ in IB but the reverse correspondence in LB. In GB, relationships of ${\bar{C}_D}$ and ${C^{\prime}_L}$ with m* are straightforward, both increasing with m*. A fixed cylinder, complementing f_n = ∞, corresponds to U_r = 0. Parameters ${\bar{C}_D}$, ${C^{\prime}_L}$ and St all approach their fixed cylinder values when U_r decreases toward U_r = 0.

3.2.1. Quiescent fluid added-mass force and flow-induced force

According to Lighthill (1986), the lift force (F_L) can be decomposed into a ‘potential force’ component and a ‘vortex force’ component, where the ‘potential force’ is caused by the potential added-mass force and the ‘vortex force’ is due to the dynamics of vorticity in the flow. Given that the fluid is viscous, Alam (Reference Alam2022) proposed a quiescent-fluid added-mass force F_{La 0} (replacing the potential force) and a flow-induced force F_Lf (replacing the vortex force). Naturally, the magnitude of the quiescent-fluid added-mass force differs from that of the potential force, depending on the body shapes and orientations (Chen, Alam & Zhou Reference Chen, Alam and Zhou2020). Alam (Reference Alam2022) further proved that the quiescent-fluid added-mass force must be considered to correctly estimate the flow-induced force and the phase lag between force and displacement. The total lift force is expressed as

(3.2)\begin{equation}{F_L} = {F_{La}}_0 + {F_{Lf}}.\end{equation}

Normalizing all forces by $(1\textrm{/}2\rho U_\infty ^2D)$,

(3.3)\begin{equation}{C_L}(t) = {C_{La}}_0(t) + {C_{Lf}}(t).\end{equation}

Here, ${C_L}(t)$ is the lift force coefficient, defined as ${C_L}(t) = {F_L}(t)/(0.5\rho DU_\infty ^2)$, and ${F_L}(t)$ is the total force on the cylinder in the y direction. Coefficients C_{La 0}(t) and C_Lf(t) are the instantaneous quiescent-fluid added-mass force coefficient and flow-induced force coefficient, respectively.

Force F_{La 0}(t) is given by

(3.4)\begin{equation}{F_{La\textrm{0}}}(t) =- {m_{a{\kern 1pt} \textrm{0}}}\ddot{y}(t)\quad (\textrm{neglecting}\;\textrm{quiescent - fluid}\;\textrm{damping}\;\textrm{force}).\end{equation}

Here, ${m_{a\textrm{0}}}$ is the quiescent-fluid added mass and is expressed as ${m_{a\textrm{0}}} = m_{a\textrm{0}}^\ast{\times} {m_d}$, where $m_{a\textrm{0}}^\ast $ is the quiescent-fluid added-mass ratio and ${m_d} = \rho {D^2}$ is the fluid mass displaced by the cylinder.

The quiescent-fluid added mass and potential added mass are the added masses measured in the quiescent fluid (still and viscid fluid) and potential fluid (inviscid, incompressible fluid), respectively. The former is generally measured numerically while the latter can be measured both numerically and experimentally by plucking the cylinder in quiescent fluid. When the damping is small, the natural frequency of the cylinder system in a vacuum can be considered as

(3.5)\begin{equation}{f_n} = \frac{1}{{2{\rm \pi} }}\sqrt {\frac{k}{m}} .\end{equation}

When the cylinder is submerged in a fluid (e.g. water), the natural frequency of the cylinder oscillation changes because of the added mass m_a generated by the fluid. The modified natural frequency f_nf of the cylinder in a fluid can be expressed as

(3.6)\begin{equation}{f_{nf}} = \frac{1}{{2{\rm \pi} }}\sqrt {\frac{k}{{m + {m_a}}}} .\end{equation}

Combining (3.5) and (3.6),

(3.7)\begin{equation}{m_a} = m\left[ {{{\left( {\frac{{{f_n}}}{{{f_{nf}}}}} \right)}^2} - 1} \right].\end{equation}

When f_nf is measured directly from the decay of the cylinder oscillation, all the parameters on the right-hand side of (3.7) are known, which leads to the estimation of m_a. See Chen et al. (Reference Chen, Alam and Zhou2020) and Alam (Reference Alam2022) for details of measuring m_a. The quiescent-fluid added-mass coefficient $m_{a\textrm{0}}^\ast $ is coincidentally about 1.0 for a circular cylinder but approximately 1.5 for a square cylinder with zero incidence angle (Chen et al. Reference Chen, Alam and Zhou2020).

Following (3.3), C_L(t) is then decomposed into C_{La 0}(t) and C_Lf(t), and their corresponding fluctuating (root-mean-square) coefficients ${C^{\prime}_{La\textrm{0}}}$ and ${C^{\prime}_{Lf}}$, respectively, are obtained as presented in figure 6. Coefficient ${C^{\prime}_{La\textrm{0}}}$ gets smaller with increasing m* in all branches. In IB, the growth in ${C^{\prime}_{La\textrm{0}}}$ is very high for all m* values, largely following A* trends in IB. On the other hand, ${C^{\prime}_{Lf}}$ in IB does not change appreciably for the high m* = 10 and 20 but does increase for the low m* = 5. Both ${C^{\prime}_{La\textrm{0}}}$ and ${C^{\prime}_{Lf}}$ decline in LB for all m* values. This suggests that the increase of A* in IB is largely due to the imminent resonance effect (vortex shedding frequency approaching the cylinder natural frequency), while the decrease of A* in LB results from the decreasing ${C^{\prime}_{Lf}}$ and the retreating resonance effect (vortex shedding frequency retreating from the cylinder natural frequency). A larger m* leads to a smaller ${C^{\prime}_{Lf}}$ in LB but the opposite scenario takes place in GB. Interestingly, in GB, although A* increases with increasing U_r for all m* values, ${C^{\prime}_{Lf}}$ and ${C^{\prime}_{La\textrm{0}}}$ gently increase and decrease (U_r > 16), respectively. This indicates that flow-induced force (F_Lf) and added-mass force (F_{L 0}) in GB vary with $U_r^{2 + }$ and $U_r^{2 - }$, respectively. There are peaks in ${C^{\prime}_{Lf}}$ at U_r = 16 and 17.7 for m* = 10 and 20, respectively, where both U_r values correspond to f_s/f_o = 3. Similar peaks are also observed for ${C^{\prime}_{La\textrm{0}}}$, albeit weaker than those for ${C^{\prime}_{Lf}}$.

Figure 6. (a) Variations of fluctuating quiescent-fluid added-mass force coefficient ${C^{\prime}_{La\textrm{0}}}$ and flow-induced lift coefficient ${C^{\prime}_{Lf}}$ with U_r and m*. (b) Zoomed-in view of (a) for U_r = 1–8. Here, ζ = 0 and Re = 170.

3.3. Role of effective added mass ratio in vibration branches

When a square cylinder oscillates freely in the transverse direction, the effective added mass ratio $m_{ae}^\ast $ and added damping ratio ζ_a can be presented as

(3.8)\begin{equation}m_{ae}^\ast= \frac{{{m_{ae}}}}{{\rho {D^2}}} = \frac{{{F_{L0}}\,\textrm{cos}\,\phi }}{{\rho {{(2{\rm \pi} {f_o})}^2}{Y_0}{D^2}}}\end{equation}

and

(3.9)\begin{equation}{\zeta _a} = \frac{{{c_a}}}{{{c_c}}} =- \frac{{{F_{L0}}\,\textrm{sin}\,\phi }}{{8{{\rm \pi} ^2}m{f_o}{f_n}{Y_0}}}.\end{equation}

The different branches of VIVs have different relationships of $m_{ae}^\ast $ with U_r (figure 7e). In IB, the F_L and Y signals are in phase, cos ϕ > 0; $m_{ae}^\ast $ is therefore positive. On the contrary, the F_L and Y signals are out of phase, cos ϕ < 0, in LB, which makes $m_{ae}^\ast $ negative. The IB and LB are distinguished by $m_{ae}^\ast $ changing from positive to negative (figure 7e). In the IB and LB, the cylinder motion is sinusoidal. The natural frequency of the vibrating cylinder is then modified because of ${m_{ae}}$ generated by the surrounding fluid. The modified frequency f_nf equals the natural frequency of the cylinder vibrating in the fluid, i.e. f_nf = f_o:

(3.10)\begin{equation}{f_o} = \frac{1}{{2{\rm \pi} }}\sqrt {\frac{k}{{m + {m_{ae}}}}} .\end{equation}

Combining (3.5) and (3.10),

(3.11)\begin{equation}{m_{ae}} = m[{(\kern1pt{f_n}/{f_o})^2} - 1].\end{equation}

Parameter $m_{ae}^\ast $ can be presented as

(3.12)\begin{equation}m_{ae}^\ast= \frac{{{m_{ae}}}}{{\rho {D^2}}} = {m^\ast }[{(\kern1.5pt{f_n}/{f_o})^2} - 1] = {m^\ast }[{(1/{f_r})^2} - 1].\end{equation}

The equation demonstrates that $m_{ae}^\ast $ is proportional to $[{(1/{f_r})^2} - 1]$. Since f_r being <1.0 linearly grows with U_r in IB (figure 5b), $m_{ae}^\ast $ being >0 declines parabolically with U_r. On the other hand, again f_r grows with U_r in LB but now f_r > 1.0, hence $m_{ae}^\ast $ is negative, declining with U_r at a smaller slope than that in IB. A large m* yields a greater $m_{ae}^\ast $ magnitude in both IB and LB as $m_{ae}^\ast $ is directly proportional to m* (figure 7e; (3.12)).

Figure 7. (a) Time histories, (b) power spectral density functions, (c) time histories of low-pass-filtered and(d) time histories of high-pass-filtered Y* (black lines) and C_L (red lines) at U_r = 26 and m* = 20. (e) Variations of effective added mass $m_{ae}^\ast $ with U_r and (f) zoomed-in view of $m_{ae}^\ast $ variations in GB. Here, ζ = 0 and Re = 170.

In GB, both F_L and Y signals are composed of low- and high-frequency components corresponding to the oscillation and vortex shedding frequencies, respectively (figure 7a). These low- and high-frequency components were decomposed using the FFT-filter tool (figure 7c,d). The cutoff frequency for the decomposition was chosen as the average of the high and low frequencies. It can be observed that there are two major peaks in the power spectra (figure 7b) for the vibration in GB. The C_L and Y* associated with the cylinder oscillation are in phase (figure 7c), while those with the vortex shedding frequency are antiphase (figure 7d). In GB, the effective added mass associated with the cylinder vibration can be referred to as $m_{ae}^\ast $ while that with the vortex shedding frequency is termed as $m_{aes}^\ast $ (figure 7e). Here $m_{ae}^\ast $ jumps from negative to positive at the onset of galloping. The jump in $m_{ae}^\ast $ is caused by the drop in f_r while $m_{ae}^\ast $ in GB is positive because f_r < 1.0 (figure 5b; (3.12)). The value of $m_{ae}^\ast $ in GB slightly declines with increasing U_r and m* (figure 7f), following the increase of f_r with U_r (figures 5b and 7e; (3.12)). On the other hand, $m_{aes}^\ast $ is negative and declines when U_r is increased, and a larger m* corresponds to a smaller $m_{aes}^\ast $ (figure 7e). In IB and LB, the cylinder vortex shedding frequency (f_s) equals the vibration frequency (f_o), i.e. f_s = f_o, where only one frequency persists in both F_L and Y signals. Parameter $m_{aes}^\ast $ in IB and LB thus can be expressed as

(3.13)\begin{equation}m_{aes}^\ast= \frac{{{m_{ae}}}}{{\rho {D^2}}} = {m^\ast }[{(\,{f_n}/{f_s})^2} - 1] = {m^\ast }[{(1/{f_r})^2} - 1].\end{equation}

Therefore, $m_{aes}^\ast= m_{ae}^\ast $ in IB and LB, where the dependence of $m_{aes}^\ast $ on U_r is the same as that of $m_{ae}^\ast $. The major features of different branches are summarized in figure 8.

Figure 8. Typical response curve showing major characteristics in IB, LB and GB as well as at their borders.

3.4. Identifications of IB–LB and LB–GB boundaries

In FIV experiments and simulations, researchers largely get vibration response (A* versus U_r) and have difficulty in identifying the borders between different branches in A* versus U_r curves, particularly when the A* variations with U_r are smooth. Bhatt & Alam (Reference Bhatt and Alam2018) showed that the borders between IB and LB can be distinguished from the relationship between St and U_r as St < St ₀ for IB and St > St ₀ for LB. That is, the IB–LB boundary (i.e. boundary between IB and LB) is accompanied by a jump in St (figure 8). Without measuring St, how can the borders be determined? Here are additional methods to identify the borders. Firstly, the IB–LB boundary can be identified from the change in $m_{ae}^\ast $ from +ve to −ve, and the LB–GB boundary can be pinpointed from the change in $m_{ae}^\ast $ from −ve to +ve (figures 7e and 8). Secondly, the IB–LB boundary is characterized by a jump in Y* − C_L phase lag from <90° to >90° (Williamson & Goverdhan Reference Williamson and Goverdhan2004; Bhatt & Alam Reference Bhatt and Alam2018). Thirdly, a dramatic drop and a rise in ${C^{\prime}_L}$ mark the IB–LB and LB–GB boundaries, respectively (figures 5d and 8). Fourthly, the IB–LB and LB–GB boundaries undergo a jump and a drop in f_r, respectively. In the literature, the identification of different branches was made based on the relationship between St and U_r. Here, we explore more avenues and generalize them to expand the methodology for identifying these branches. Figure 8 encompasses all possible parameter changes (i.e. $m_{ae}^\ast $, St, ϕ, ${C^{\prime}_L}$ and f_r), which aids in discerning distinct vibration branches when one of the parameters is measured.

3.5. Beat mechanism different from the classical beat

Next, we focus on the reasons for the uneven features of the vibration amplitude in the GB, taking the case of m* = 20 and ζ = 0 as an example. Except at U_r = 17.7–17.8 and U_r = 30–30.1, the vibrations in GB do not exactly repeat from one period to another (figures 9 and 10), which indicates that there are multiple frequency components, resulting in an obvious beat-like variation in the amplitudes of Y* histories. As already discussed with figure 4(a,b), U_r = 17.7–17.8 and U_r = 30–30.1 correspond to 1:3 (i.e. f_s/f_o = 3) and 1:5 (i.e. f_s/f_o = 5) synchronizations, respectively. Interestingly, the beating amplitude in Y* (i.e. the change in the vibration amplitude due to beating) grows and declines as f_s/f_o respectively approaches and departs from f_s/f_o = 3 and/or 5, as does the beating period (figures 9 and 10). The general consensus is that a beat is characterized by amplitude modulation. It is an interference pattern between two different frequencies, perceived as a periodic variation in amplitude. As two frequencies are close to each other (i.e. small difference between them), the beating period becomes longer and it will be infinite (constant amplitude) when the two frequencies are identical.

Figure 9. (a–l) Time histories of cylinder displacement Y* for different U_r. The red curve is the envelope of Y*. Here, m* = 20, ζ = 0 and Re = 170.

Figure 10. (a–i) Time histories of cylinder displacement response Y* for different U_r values. Here, m* = 20, ζ = 0 and Re = 170.

First, we analyse the beat phenomenon for f_s/f_o = 3. The Lissajous diagrams of C_L–Y* for U_r = 15–20 presented in figure 11 also demonstrate that the vibration amplitude in GB changes from cycle to cycle except at U_r = 17.7–17.8 (i.e. f_s/f_o = 3), where the C_L–Y* diagram is an enclosed curve (figure 11e,f), and the frequency in C_L is three times that in Y* at U_r = 17.7–17.8. It can be deemed ‘1:3 lock-in’ where f_s locks in with 3f_o. Figure 12 shows power spectra (E_Y and E_CL) and envelopes of Y* and C_L at U_r = 17.5 and 19. For U_r = 17.5, f_s/f_o = 2.951 < 3, whereas for U_r = 19, f_s/f_o = 3.185 > 3. In the power spectra of E_Y (figure 12a,e), there is a minor peak ($\,{f^{\prime}_o}$) on the left-hand side of f_o for U_r = 17.5 (figure 12a) and on the right-hand side of f_o for U_r = 19 (figure 12e). The beat frequency (i.e. the frequency of the envelope) in Y* can be represented as ${f_{bY}} = 1/{T_{bY}} = |\,{f_o} - {f^{\prime}_o}|$ (figure 12c,g). The question is, what is the origin of ${f^{\prime}_o}$? Generally, such a beat frequency is generated from the interference between two frequencies. One can expect this beat frequency is due to the difference between the vortex shedding frequency and cylinder vibration frequency as these two frequencies are predominantly active during the cylinder vibrations. This is, however, not the case here as is shown below.

Figure 11. (a–i) Lissajous C_L–Y* diagrams for different U_r. Here, m* = 20, ζ = 0 and Re = 170.

Figure 12. Power spectrum of (a,e) Y* and (b,f) C_L. Envelopes of (c,g) Y* and (d,h) C_L. (a–d) U_r = 17.5, (e–h) U_r = 19. Here, m* = 20, ζ = 0 and Re = 170.

Similarly, there is a minor peak ($\,{f^{\prime}_s}$) on the right-hand side of f_s (i.e. f_s/f_o < 3) and on the left-hand side of f_s (i.e. f_s/f_o > 3) for U_r = 17.5 and 19, respectively (figure 12b,f). The corresponding beat frequency is ${f_{bL}} = 1/{T_{bL}} = |\,{f_s}-{f^{\prime}_s}|$ (figure 12d,h).

For a given U_r,

(3.14)\begin{equation}{f_{bY}} = {f_{bL}}\quad (\textrm{i}\textrm{.e}\textrm{.}\;{T_{by}} = {T_{bL}})\end{equation}

and

(3.15)\begin{equation}|\,{f_o}-{f^{\prime}_o}|= |\,{f_s}-{f^{\prime}_s}|.\end{equation}

It is found that ${f^{\prime}_s} = 3{f_o}$.

Using (3.14), it can, therefore, be written as

(3.16)\begin{equation}|\,{f_s}-{f^{\prime}_s}|= |\,{{f_s} - \textrm{ }3{f_o}} |= |\,{f_o}-{f^{\prime}_o}|.\end{equation}

This is to say that the beat frequency in both Y* and C_L is f_b = |f_s − 3f_o|. This explains that the beat frequency will be smaller as f_s/f_o gets closer to 3, and it will be zero (no beating) when f_s/f_o = 3. That is, the beating phenomenon is linked to the difference between f_s and 3f_o, tending to 1:3 synchronization, not directly linked to the difference between f_s and f_o. In GB, since f_s/f_o is proportional to U_r (figure 4b; (3.1)), the beating U_r around f_s/f_o = 3 can be predicted from (3.1) for different values of m*. The other peaks E_CL and E_Y emerge at $2{f_s} - {f^{\prime}_s}$ and $2{f_o}-{f^{\prime}_o}$, respectively. A beating phenomenon was also observed in some studies of the FIV of a circular cylinder associated with a mode change or mode competition. Voorhees et al. (Reference Voorhees, Dong, Atsavapranee, Benaroya and Wei2008), Zhang et al. (Reference Zhang, Li, Ye and Jiang2015) and Shen et al. (Reference Shen, Chan and Wei2018) indicated that this beating is due to the difference between the vibration frequency and the vortex shedding frequency (i.e. f_o and f_s), while Pan, Cui & Miao (Reference Pan, Cui and Miao2007) and Mittal (Reference Mittal2017) claimed that the beating is the modulation between the natural frequency f_n of the cylinder and the Strouhal frequency f_{s ,fixed} of the fixed cylinder. The beating phenomenon of a vibrating square cylinder in GB, however, is not the same as the cases of the circular cylinder mentioned above. It is the difference between f_s and 3f_o that gives rise to the beating phenomenon as f_s/f_o gets close to 3. This observation is made for the first time.

Apart from the 1:3 synchronization mentioned above, we also observed the 1:5 synchronization (f_s/f_o = 5) between the vibration frequency and the shedding frequency at U_r = 23.65, 27.6 and 30 for m* = 5, 10 and 20, respectively. Here the beat frequency f_b = |f_s − 5f_o| as f_s/f_o gets close to 5. The beat amplitude (figure 10) is, however, much smaller than that in the case involving 1:3 synchronization (figure 9).

3.6. Effects of m*, ζ, m*ζ, $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ and $({m^\ast } + m_{ae}^\ast )\zeta$ on U_rc for galloping onset

Next, we analyse the effects of m* and ζ on U_rc for the onset of galloping. To save time and computational resources, we simulated the responses at high U_r values only (figures 13 and 14). Parameter U_rc is identified from the abrupt increase in A* (figure 13) and/or from the sudden decrease in f_r (figure 14). For all ζ values examined, A* decreases with increasing m* before the galloping onset (i.e. for U_r < U_rc), which implies that A* in LB and/or DB (desynchronization branch) weakens with increasing m*. However, for a given m*, A* for U_r < U_rc does not change appreciably with increasing ζ. Figure 14 displays that f_r > 1.0 for U_r < U_rc but f_r < 1.0 for U_r > U_rc. In the latter U_r regime, when m* is increased, f_r increases to approach 1.0.

Figure 13. Variations of vibration amplitude A* with reduced velocity U_r for different m* and ζ values. Here, Re = 170.

Figure 14. Variations of frequency ratio f_r with reduced velocity U_r for different m* and ζ. The red dotted line represents f_r = 1. Here, Re = 170.

To further investigate the influence of m* and ζ on U_rc, more cases of m* and ζ are also considered. Table 3 summarizes U_rc for m* = 2, 3, 5, 10, 20, 30, 40 and 50 and ζ = 0, 0.001, 0.01, 0.05, 0.1, 0.2, 0.4, 0.6 and 1.0 at U_r ≤ 80. When ζ = 0, U_rc is 17, 13, 12, 12, 12, 12 and 12 for m* = 3, 5, 10, 20, 30, 40 and 50, respectively, i.e. U_rc decreases with increasing m* for 3 ≤ m* < 10 before being insensitive to m* for m* ≥ 10. Figure 15 shows the dependence of galloping occurrence on m* = 2–50 and ζ = 0–1. No galloping is observed at m* = 2 for all ζ values examined. Galloping is observed for m* = 3–10 when ζ = 0.2, while the same occurs at m* = 5 only for ζ = 0.4. For ζ ≥ 0.6, no galloping is observed, regardless of m*, until U_r = 80 examined. When m* is increased from 5 (i.e. m* ≥ 5), U_rc changes insignificantly for ζ ≤ 0.01, but significantly for ζ ≥ 0.05 (table 3, figure 16a). Parameter U_rc, however, significantly increases when m* is decreased from 5 to 3 for all ζ values (table 3, figure 16b). For all m* values examined, an increase in ζ makes U_rc higher, particularly when ζ > 0.01 (table 3, figure 16b). That is, the effect of ζ on U_rc is insignificant for ζ ≤ 0.01 but significant for ζ > 0.01, a higher ζ suppressing the cylinder vibration. Joly et al. (Reference Joly, Etienne and Pelletier2012) also found a higher U_rc value at ζ = 0.1 than that at ζ = 0.

Table 3. Effects of m* and ζ on U_rc. ‘—’ means no galloping observed for U_r ≤ 80 examined.

Figure 15. Dependence of galloping occurrence on m* = 2–50 and ζ = 0–1. Here, Re = 170.

Figure 16. Effect of (a) mass ratio m* and (b) damping ratio ζ on critical reduced velocity U_rc. Here, Re = 170.

The three-dimensional view of the dependence of U_rc on m* and ζ shown in figure 17 allows us to observe the variations of U_rc with m* and ζ more intuitively at ζ ≤ 0.2. The value of U_rc achieves its maximum at m* = 30 and ζ = 0.1. For all m* values, an increase in ζ leads to a larger U_rc. The increase is, nevertheless, larger for a large m*. Particularly for ζ > 0.01, U_rc declines for m* ≤ 5 and rapidly increases with increasing m*. The degree of the decrease and/or increase becomes larger for a higher ζ value.

Figure 17. Relationship of U_rc with m* (= 3–50) and ζ (= 0–0.2). Here, Re = 170.

Some researchers have debated whether m*ζ can serve as the FIV criterion of a circular cylinder. Griffin, Skop & Ramberg (Reference Griffin, Skop and Ramberg1975) and Griffin (Reference Griffin1980) used a Skop–Griffin parameter S_G ($= 2{{\rm \pi} ^3}{S^2}{m^\ast }\zeta $, where S is the Strouhal number of the static cylinder) to plot the maximum vibration amplitude $A_{max}^\ast $ against S_G (i.e. m*ζ). It is known as the Griffin plot. However, this plot showed a large scatter of $A_{max}^\ast $ data. Later, Khalak & Williamson (Reference Khalak and Williamson1999) and Govardhan & Williamson (Reference Govardhan and Williamson2000) introduced a combined parameter $({m^\ast } + m_{a0}^\ast )\zeta$ to modify the Griffin plot, where $m_{a\textrm{0}}^\ast $ (i.e. potential added-mass ratio C_A in their investigations) is the quiescent-fluid added-mass ratio and is about 1.0 for a circular cylinder. They demonstrated that $({m^\ast } + m_{a0}^\ast )\zeta$ collapses the $A_{max}^\ast $ data well for a wide range of $({m^\ast } + m_{a0}^\ast )\zeta > 0.06$. Sarpkaya (Reference Sarpkaya1978, Reference Sarpkaya1995) stated that the vibration response depends on m*ζ for S_G > 1 (i.e. m*ζ > 0.4), while it depends on m* and ζ separately rather than on m*ζ for m*ζ < 0.4. Blevins & Coughran (Reference Blevins and Coughran2009) for two-degrees-of-freedom vibration found that the ratio of inline to transverse amplitudes is governed by m* and ζ independently. Bahmani & Akbari (Reference Bahmani and Akbari2010) claimed that m*ζ can be used to characterize $A_{max}^\ast $ with an acceptable accuracy. For a cylinder submerged in the wake of another cylinder, Alam (Reference Alam2021) claimed that m*ζ (i.e. m*ζ = 0–8) does not serve as a unique parameter to characterize the vibration amplitude. In addition, Bearman (Reference Bearman1984) reported that low values of m* affect the frequency ratio independently, not m*ζ. Williamson & Goverdhan (Reference Williamson and Goverdhan2004) and Govardhan & Williamson (Reference Govardhan and Williamson2004) demonstrated that the VIV range depends only on m* at (m* + C_A)ζ < 0.05. Here, we will see whether m*ζ or $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ works well to collapse U_rc data. If not, what is the intrinsic parameter that dictates the vibration? Figure 18(a,b) shows the relationships of U_rc with m*ζ and $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$. As shown in figure 18(a), there are more than one U_rc data point, significantly different, for a given m*ζ (e.g. three U_rc values for m*ζ = 1.0 and four U_rc values for m*ζ = 2.0), which indicates that m*ζ cannot serve as the only criterion for galloping onset. In figure 18(b), the U_rc data scattering alleviates, which points to a linear increase in U_rc with $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ overall at m* ≥ 5. But still, there are some scattered data points, and the collapse of U_rc data on a line is still not credible. It hints that something is still missing in $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$. We find the missing parameter here.

Figure 18. Combined mass-damping parameter (a) m*ζ and (b) $({m^\ast } + m_{a0}^\ast )\zeta $ with critical reduced velocity U_rc. Here, Re = 170.

As m* decreases, f_r in GB deviates considerably from unity, i.e. the vibration frequency differs from the natural frequency (figures 5b and 14). The deviation is attributed to the flow-induced added mass $m_{af}^\ast $ that should be considered in $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$. The effective added mass is the quiescent-fluid added mass plus flow-induced added mass, i.e. $m_{ae}^\ast= m_{a0}^\ast+ m_{af}^\ast $ (Alam Reference Alam2022). While $m_{a\textrm{0}}^\ast $ is independent of U_r, $m_{af}^\ast $ is highly dependent on U_r and markedly changes at U_rc. Therefore, $m_{af}^\ast $ is crucial and should be added with $m_{a\textrm{0}}^\ast $. That is, in the mass-damping parameter, we have to consider $m_{ae}^\ast $, not $m_{a\textrm{0}}^\ast $, i.e. $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ should be replaced by $({m^\ast } + m_{ae}^\ast )\zeta$. As shown in figure 7(f), $m_{ae}^\ast $ jumps at U_rc and declines with increasing m* and U_r. We consider $m_{ae}^\ast $ at the galloping onset for $({m^\ast } + m_{ae}^\ast )\zeta$, and the dependence of U_rc on $({m^\ast } + m_{ae}^\ast )\zeta$ is presented in figure 19. For $({m^\ast } + m_{ae}^\ast )\zeta > 0.55$, as shown in figure 19(a), the U_rc data points collapse well on a line at m* ≥ 5, showing that U_rc linearly increases with $({m^\ast } + m_{ae}^\ast )\zeta$. The relationship between U_rc and $({m^\ast } + m_{ae}^\ast )\zeta$ can be represented by a curve-fitting equation:

(3.17)\begin{equation}{U_{rc}} = 17.4({m^\ast } + m_{ae}^\ast )\zeta + 2.5\quad (\textrm{for}\;({m^\ast } + m_{ae}^\ast )\zeta > 0.55\;\textrm{and}\;{m^\ast } \ge 5).\end{equation}

Figure 19. (a) Relationship between critical reduced velocity U_rc and combined mass-damping parameter $({m^\ast } + m_{ae}^\ast )\zeta $. (b) Zoomed-in view of U_rc variations at $({m^\ast } + m_{ae}^\ast )\zeta \le 0.55$.

For $({m^\ast } + m_{ae}^\ast )\zeta \le 0.55$ as shown in figure 19(b), the U_rc also increases linearly, albeit with a smaller slope, with increasing $({m^\ast } + m_{ae}^\ast )\zeta$ for m* ≥ 5. The U_rc data points for m* = 5 appear more scattered when $({m^\ast } + m_{ae}^\ast )\zeta > 0.55$. The linear relationship at $({m^\ast } + m_{ae}^\ast )\zeta \le 0.55$ and m* ≥ 5 can be represented as

(3.18)\begin{equation}{U_{rc}} = 5.6({m^\ast } + m_{ae}^\ast )\zeta + 12.3\quad (\textrm{for}\;({m^\ast } + m_{ae}^\ast )\zeta \le 0.55\;\textrm{and}\;{m^\ast } \ge 5).\end{equation}

Equation (3.18) further proves that when ζ = 0 or is very small, U_rc becomes independent (U_rc = 12.3) of m*, which is consistent with the observation made in figure 16(a).

For m* = 3, U_rc again linearly increases with $({m^\ast } + m_{ae}^\ast )\zeta$, with a larger slope than the case for m* ≥ 5 (figure 19a). The relationship between U_rc and $({m^\ast } + m_{ae}^\ast )\zeta$ can be expressed as

(3.19)\begin{equation}{U_{rc}} = 18.5({m^\ast } + m_{ae}^\ast )\zeta + 16\quad (\textrm{for}\;{m^\ast } = 3).\end{equation}

The slopes of U_rc in (3.17) and (3.19) are close to each other although the U_rc intercepts differ between the two cases.

The use of $({m^\ast } + m_{ae}^\ast )\zeta$ in (3.17)–(3.19) can predict U_rc. Term $({m^\ast } + m_{ae}^\ast )\zeta$ can, therefore, be considered a criterion for the prediction of galloping onset. Equation (3.17) also illustrates why galloping is not observed for m* = 20 and ζ = 0.2, where $({m^\ast } + m_{ae}^\ast )\zeta = 4.76$. The value of U_rc predicted by (3.17) is 85.19 > 80 examined. It can therefore be concluded that the criterion for a fluid–structure system is $({m^\ast } + m_{ae}^\ast )\zeta$, not m*ζ or (m* + C_A)ζ or $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$. For a given $({m^\ast } + m_{ae}^\ast )\zeta$, U_rc is higher for m* < 5 than for m* ≥ 5. There is a threshold of m* (= 5) after (m* ≥ 5) or before (m* < 5) which the relationships between U_rc and $({m^\ast } + m_{ae}^\ast )\zeta$ differ. We further shed light on this threshold in the next section.

3.7. Underlying physics of m* and ζ effects on galloping onset

3.7.1. Effects of m* and ζ when ζ = 0

Using the global linear stability analysis method, Meliga & Chomaz (Reference Meliga and Chomaz2011), Zhang et al. (Reference Zhang, Li, Ye and Jiang2015) and Yao & Jaiman (Reference Yao and Jaiman2017) investigated the mechanism of VIVs of a freely vibrating cylinder. They identified two modes responsible for the onset of instability in the fluid–structure system: wake mode (WM) and structure mode (SM). These two leading modes have a strong coupling for low m*, while they are decoupled for high m*. Parameter m* plays an important role in the selection of the leading mode. The essential characteristic of the flutter-induced lock-in is the competition between the two unstable modes (Zhang et al. Reference Zhang, Li, Ye and Jiang2015). Li et al. (Reference Li, Lyu, Kou and Zhang2019) utilized linear stability analysis and direct numerical simulations to investigate the underlying mechanisms of galloping of a square cylinder with zero damping. They concluded that the unstable SM leads to the low-frequency large-amplitude vibration of the cylinder, while the unstable WM results in the high-frequency vortex shedding in the wake. The instability of SM is the primary cause of the galloping phenomenon. The onset of galloping is postponed when SM and WM compete with each other. As such, the galloping phenomenon can be completely suppressed or U_rc can be delayed at relatively low-Re and low-m* (< 5) conditions. In their study, the galloping vibration completely disappeared for low m* (m* < 4, Re = 150) because of the strong competition between the enhanced SM and WM at low m* (lighter structure). They also introduced a ‘pre-galloping’ region, located between VIV and galloping, where galloping does not occur but the SM is unstable.

Here, we investigate the influence of m* on galloping by analysing the high-frequency (f_s) and low-frequency (f_o) signals of the vibration. At ζ = 0, U_rc remains constant for m* ≥ 10 while it increases with decreasing m* for 3 ≤ m* < 10. Ultimately, galloping is suppressed for m* ≤ 2, i.e. a relatively low m* inhibits galloping. Given that the maximum U_r examined in this study is 80, galloping may be observed for a smaller m* (≤ 2) if U_r is further increased (Han & Langre Reference Han and Langre2022). It has already been shown that the high-pass-filtered C_L and Y* associated with the vortex shedding are antiphase, not auspicious for the cylinder vibration. Conversely, the low-pass-filtered signals are in phase, auspicious for the cylinder vibration. First, we analyse vibration signals (Y*) at m* = 20 with U_r = 12 and 13, corresponding to pre-galloping and galloping regions, respectively (figure 20a,d). To gain insight into the evolution of the frequency content over time, the continuous wavelet transform (CWT) is performed using the Morlet wavelet (figure 20b,e). The wavelet centre frequency Fc = 3 and the wavelet scales equal 2Fc–2Fc × 2¹⁴, which allows us to measure frequency in the range of Fs/2¹⁴–Fs/2 (where Fs is the sampling frequency). Additionally, power spectra of vibration responses in different developing regimes are shown in figure 20(c,f). The cylinder vibration in each signal comprises two distinct frequencies f_o and f_s, associated with SM and WM, respectively. Consider E_Y,fo and E_{Y, fs} as the energy intensity at f_o and f_s, respectively.

Figure 20. (a,d) Time histories of Y*. (b,e) Time–frequency spectrum of Y* based on CWT. (c,f) Power spectral density functions of Y* at different regimes. (a–c) m* = 20, U_r = 12; and (d–f) m* = 20, U_r = 13. Here, ζ = 0 and Re = 170.

In the pre-galloping region, the vibration amplitude rapidly develops and reaches its maximum at t* ≈ 200 (figure 20a). At t* < 800, there is an intensified competition between f_o and f_s, where E_{Y, fo} and E_Y,fs are considerable (figure 20c); this transition is marked as regime I. In regime II, E_{Y, fo} rapidly decreases and becomes insignificant while E_{Y, fs} dominates the competition, suppressing the vibration at f_o (figure 20c). In regime III (t* > 2500), the cylinder vibration solely occurs at f_s, and the amplitude remains constant.

After galloping, E_{Y, fo} increases gradually in regimes I and II, and it remains unchanged in regime III (figure 20e). In regime I, E_{Y, fo} is lower than E_{Y, fs} (figure 20f). Parameter E_{Y, fo} increases and E_{Y, fs} remains unchanged from regimes I to II; eventually, E_{Y, fo} becomes larger than E_{Y, fs}. This indicates that SM dominates the competition and results in the occurrence of galloping. In regime III, E_{Y, fo} remains unchanged, with E_{Y, fo}/E_{Y, fs} = 7.92 (figure 20f).

The E_Y values can be used to characterize the vibration amplitude. As shown in figure 21, the cylinder vibration is decomposed into low-frequency and high-frequency vibrations after the commencement of galloping. We define the vibration amplitudes of the low- and high-pass-filtered Y* as $A_{low}^\ast $ and $A_{high}^\ast $ corresponding to E_{Y, fo} and E_{Y, fs}, respectively. In the pre-galloping region, the vibration amplitude A* in regime III can also be characterized by E_{Y, fs}, which declines gently with increasing U_r (figures 4a and 21). After the onset of galloping, $A_{high}^\ast $ remains almost unchanged. The almost unchanged E_{Y, fs} and $A_{high}^\ast $ nearly before the galloping occurrence and after the galloping occurrence indicate that vortex shedding is not involved in the occurrence of galloping. The value of $A_{low}^\ast $, however, increases markedly with the increase of U_r in the galloping regime. It is the low-frequency motion (SM) that is auspicious for the cylinder vibration, and consequently promotes the occurrence of galloping. As shown in figure 20(b,e), E_{Y, fo} at m* = 20 and 13 is significantly greater than that at U_r = 12, while E_{Y, fs} is almost the same.

Figure 21. Dependence of A* on U_r and m*. Here, ζ = 0 and Re = 170.

Since U_rc is delayed with a decrease in m* from 5 to 3 and galloping is suppressed for m* = 2 (table 3, figure 16a), we will further investigate the mode competition phenomenon at m* = 3 to evaluate to influence of m* on the competition. At m* = 3, both A* in pre-galloping region and $A_{high}^\ast $ in galloping regime increase compared to that at m* = 20 (figure 21), and the E_{Y, fs} value at m* = 3 is significantly intensified (figures 20b,e and 22b,e). Consequently, the competition between E_{Y, fo} and E_{Y, fs} becomes more intense at m* = 3, with E_{Y, fo}/E_{Y, fs} = 1.52 at the galloping onset in regime III (figure 22f). It can be inferred that if m* is further decreased, E_{Y, fs} could completely overpower E_{Y, fo} and suppress galloping. At the galloping onset, E_{Y, fo}/E_{Y, fs}, i.e. $A_{low}^\ast{/}A_{high}^\ast $, diminishes with decreasing m*. It can be reasonably inferred that if $A_{low}^\ast{/}A_{high}^\ast $ at the galloping onset drops to 1 with a further decrease in m*, then galloping will be completely suppressed. Figure 23 presents the influence of $A_{low}^\ast{/}A_{high}^\ast $ on m* at the galloping onset. It can be seen that $A_{low}^\ast{/}A_{high}^\ast $ decreases linearly with decreasing m*, and can be represented by a curve-fitting equation:

(3.20)\begin{equation}A_{low}^\ast{/}A_{high}^\ast= 0.41{m^\ast } + 0.155.\end{equation}

When $A_{low}^\ast{/}A_{high}^\ast= 1$, the critical m* predicted by (3.20) is $m_c^\ast= 2.06 \approx 2$. Equation (3.20) explains why galloping is not observed for m* ≤ 2. Additionally, we observed galloping at m* = 2.5 and U_r = 40, further verifying the rationality of the prediction of $m_c^\ast $. Figure 23 also shows the dependence of $A_{low}^\ast{/}A_{high}^\ast $ on m* at Re = 150 and 160, where $A_{low}^\ast{/}A_{high}^\ast $ declines with decreasing m*, the declining rate increasing with decreasing Re. When $A_{low}^\ast{/}A_{high}^\ast $ decreases to 1, $m_c^\ast $ is larger at a smaller Re.

Figure 22. (a,d) Time histories of Y*. (b,e) Time–frequency spectrum of Y* based on CWT. (c,f) Power spectral density functions of Y* at different regimes. (a–c) m* = 3, U_r = 17; and (d–f) m* = 3, U_r = 18. Here, ζ = 0 and Re = 170.

Figure 23. Effect of m* and Re on $A_{low}^\ast \textrm{/}A_{high}^\ast $ at the galloping onset. Here, ζ = 0.

In summary, at ζ = 0, as m* decreases, the contribution of WM becomes more significant, leading to a delay in the onset of galloping. Eventually, galloping can be completely suppressed at sufficiently low m* values. At m* = 3, the occurrence of galloping at ζ = 0 depends on the competition between SM and WM while the competition is contingent on m*. However, at m* ≥ 5, the contribution of WM becomes negligible; the cylinder vibration is thus dominated by SM, leading to a constant U_rc as m* increases.

3.7.2. Effects of m* and ζ when ζ > 0

Moving on to discussing the influence of ζ on galloping, we observe that at ζ > 0, U_rc increases with the increase of ζ. Additionally, U_rc shows a nonlinear relationship with m*, initially decreasing and then increasing as m* increases. When ζ = 0 or is small, U_rc is independent of m* (≥ 5). As previously discussed with figure 7, when the cylinder vibrates in GB without damping (ζ = 0), the phase lag between the low-pass-filtered C_L and Y* is 0° (figure 7c), while that between the high-pass-filtered C_L and Y* is 180° (figure 7d). As ζ increases from 0, taking ζ = 0.05 and m* = 20 as an example, where U_rc = 22.5, the phase lag ϕ_low (in degrees) between the low-pass-filtered C_L and Y* at U_r = 23 (beyond the galloping onset) undergoes a shift from 0 to a positive value, i.e. ϕ_low ≈ 27° (figure 24a). As discussed with figure 20(a–c), in regime I of the pre-galloping region, the cylinder vibration consists of both high- and low-frequency components. Therefore, at ζ = 0.05, ϕ_low ≈ 26.5° is achieved in regime I of the pre-galloping region (i.e. U_r = 22.5) (figure 24c). The phase lag between the high-pass-filtered C_L and Y* at ζ > 0, however, remains 180° both before and after U_rc (figure 24b,d). Therefore, the presence of damping (ζ > 0) introduces a positive phase lag between the low-pass-filtered C_L and Y*, both before and after galloping, modifying the coupling between the flow and cylinder vibration.

Figure 24. Time histories of (a,c) low-pass-filtered and (b,d) high-pass-filtered Y* (black lines) and C_L (red lines) in GB. (a,b) m* = 20, ζ = 0.05 and U_r = 23 in regime III; (c,d) m* = 20, ζ = 0.05 and U_r = 22.5 in regime I. Here, Re = 170.

Figure 25(a) demonstrates the variations of $m_{ae}^\ast,\ {\zeta}_a $ and ϕ_low with U_r at m* = 20, ζ = 0.05. In GB, $m_{ae}^\ast > 0$, declining with increasing U_r, in a similar fashion to the case of ζ = 0. The added damping ratio $\zeta_{a} $ compensates the structural damping ratio −ζ (dashed line), so that $\zeta_{a}+\zeta=0 $. Notably, ϕ_low is positive and rises with increasing U_r. Compared with ζ = 0, it is the positive phase lag that introduces a modification in the coupling between the cylinder vibration and the fluid dynamics.

Figure 25. (a) Variations of phase lag ϕ_low (deg.), added damping ζ_a and effective added mass $m_{ae}^\ast $ with U_r at m* = 20 and ζ = 0.05. The dashed line represents −ζ. (b) Dependence of U_rc on phase lag ϕ_low at the galloping onset. Here, Re = 170.

The energy W_f transferred from fluid to the cylinder for each cycle of oscillation is given by

(3.21)\begin{equation}{W_f} = \int {{F_L}\,\textrm{d}Y} = \int_t^{t + T} {{F_{L0}}\,\textrm{sin}(2{\rm \pi} {f_o}t + \phi )\,\textrm{d}({Y_0}\,\textrm{sin}\,2{\rm \pi} {f_o}t) = {\rm \pi}{F_{L0}}{Y_0}\,\textrm{sin}\,\phi } .\end{equation}

When C_L leads Y* (ϕ > 0), positive work is done on the cylinder. An increasing ϕ (<90°) provides more energy to the cylinder vibration.

The presence of damping (ζ > 0) introduces a positive phase lag, modifying energy transfer between the flow and cylinder, so the relationship between U_rc, ζ and m* becomes more complex and it is worth understanding the influence of ζ on galloping behaviour. The coupling in the presence of damping is crucial in determining the galloping onset, especially for lower m* (m* < 5) where the role of vortex shedding becomes significant.

Figure 25(b) shows the dependence of U_rc on ϕ_low. At m* ≥ 5, a larger ϕ_low corresponds to a larger U_rc overall. The value of U_rc at m* = 3, however, is obviously larger than that at m* ≥ 5, which is due to the fierce competition between SM and WM at m* = 3. As discussed before, for ζ = 0, U_rc at m* = 3 entirely depends on the competition between SM and WM. Therefore, for ζ > 0, because of the presence of non-zero ϕ_low, U_rc at m* = 3 is influenced not only by the competition but also by the magnitude of ϕ_low. On the other hand, when m* ≥ 5, the competition is weak, with no influence on the galloping onset. The value of U_rc is, therefore, mainly influenced by ϕ_low, while a larger m* and/or ζ yields a larger U_rc.

Overall, ϕ_low being associated with work done is an important factor in determining the onset of galloping, and its magnitude is influenced by both m* and ζ. Understanding the interplay between m*, ζ and ϕ_low provides valuable insights into the stability of the fluid–structure system during galloping.

Figure 26(a,b) shows the effect on ϕ_low (in degrees) of m* and ζ at the onset of galloping. The relationship between ϕ_low and m* or ζ is straightforward, where ϕ_low increases with m* (figure 26a) and/or ζ (figure 26b). In the previous sections, the relationship of U_rc with m*ζ, (m* + C_A)ζ or $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ has been made clear. Is there any relationship of ϕ_low with the same? Figure 26(c,d) shows the dependencies of ϕ_low on m*ζ and $({m^\ast } + m_{ae}^\ast )\zeta$. It is evident that there is no clear relationship between ϕ_low and m*ζ (figure 26c) or $({m^\ast } + m_{ae}^\ast )\zeta$ (figure 26d), which precludes the use of m*ζ or $({m^\ast } + m_{ae}^\ast )\zeta$ to predict ϕ_low effectively. However, when $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ is employed (figure 27), the ϕ_low data points collapse well on a line for all m* values, following a quadratic polynomial relationship between ϕ_low and $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$. This reiterates that it is $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$, not m*ζ, which intrinsically combines and reflects the mass-damping properties of the system. This relationship between ϕ_low and $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$ can be obtained as

(3.22)\begin{equation}{\phi _{low}} =- 3.1{[({m^\ast } + \textrm{ }m_{a0}^\ast )\zeta ]^2} + 28.6({m^\ast } + \textrm{ }m_{a0}^\ast )\zeta .\end{equation}

This equation allows us to predict ϕ_low using the value of $({m^\ast } + m_{a\textrm{0}}^\ast )\zeta$. Parameter $m_{a\textrm{0}}^\ast= 1.5$ for a square cylinder with zero incidence angle (Chen et al. Reference Chen, Alam and Zhou2020). As long as m* and ζ are provided, we can obtain the phase lag ϕ_low between the low-pass-filtered C_L and Y* at the onset of galloping.

Figure 26. Effect of (a) mass ratio m*, (b) damping ratio ζ, (c) m*ζ and (d) $({m^\ast } + m_{ae}^\ast )\zeta$ on phase lag ϕ_low at the galloping onset. Here, Re = 170.

Figure 27. Effect of combined mass-damping parameter $({m^\ast } + m_{a0}^\ast )\zeta $ on phase lag ϕ_low at the galloping onset. Here, Re = 170.

As discussed with figure 19, the U_rc data points collapse well on $({m^\ast } + m_{ae}^\ast )\zeta$ at m* ≥ 5, where the influence of ϕ_low is significantly greater and U_rc is mainly determined by ϕ_low. At m* = 3, however, U_rc is influenced by both the competition and the magnitude of ϕ_low. As a result, U_rc at m* = 3 deviates from the fitting curve. The competition affects U_rc at m* = 5 to some degree, where U_rc at m* = 5 is 13, that is, slightly larger than 12 at m* ≥ 10. However, the influence of ϕ_low on U_rc at m* = 5 is greater than that at m* = 3. Particularly, at $({m^\ast } + m_{ae}^\ast )\zeta > 0.55$, U_rc is largely determined by ϕ_low, so that the U_rc data points fit well on a line at m* ≥ 5 and $({m^\ast } + m_{ae}^\ast )\zeta > 0.55$ (figure 19a). At $({m^\ast } + m_{ae}^\ast )\zeta \le 0.55$, however, the U_rc data points at m* ≥ 5 all lie above the fitted curve for $({m^\ast } + m_{ae}^\ast )\zeta > 0.55$ (figure 19b), as ϕ_low at $({m^\ast } + m_{ae}^\ast )\zeta \le 0.55$ is not large enough to curb the mode competition.