2.1. Numerical methodology

The fluid–structure interaction is simulated using the immersed boundary (IB) method which was first introduced by Peskin (Reference Peskin1972) to simulate the blood flow around the flexible leaflet of a human heart. In the framework of the IB method (see Appendix A), the flow governing equations are discretized on a fixed Cartesian grid, which generally does not conform to the geometry of moving solids. As a result, the boundary conditions on the fluid–cylinder interface, manifesting the interaction between the fluid and the structure, cannot be imposed directly. Instead, an extra body force is added to the momentum equation using interpolation and distribution functions to take such interaction into account. Compared with conventional numerical methods, the IB method has significant advantages, particularly in fluid–structure interaction simulations with topological changes. Another merit of the IB method lies in its parameterized and fast implementation for a large number of simulations with different geometric configurations compared with conventional methods using body-conformal grids.

The dynamics of the elastically supported D-section prism is simplified as a mass–damper–spring system. In this study, the prism is free to oscillate only in the transverse direction and the governing equation of prism motion is as follows:

(2.1)\begin{equation} m \ddot{y}+c \dot{y}+k y=F_{y}, \end{equation}

where $m$ is the prism mass, $c$ is the structural damping, $k$ is the stiffness coefficient, $y$ is the transverse displacement and $F_y$ is the transverse fluid force. The non-dimensional form of the governing equation is as follows:

(2.2)\begin{equation} \ddot{Y}+4 {\rm \pi}F_{n}\zeta \dot{Y}+\left(2 {\rm \pi}F_{n}\right)^{2} Y=\frac{4 C_{y}}{{\rm \pi} m^{*}}, \end{equation}

where $m^*$ is the mass ratio, $\zeta$ is the structural damping ratio, $F_n = f_nD/U_\infty$ is the reduced natural frequency of the prism, $Y = y/D$ is the normalized transverse displacement and $C_y = 2F_{y} / \rho U_{\infty }^{2}D$ is the dimensionless transverse fluid force. The governing equation of prism motion is based on Newton's second law and is solved by applying the standard Newmark-$\beta$ method – a method of numerical integration used to solve differential equations and widely used in numerical evaluation of the dynamic response of structures. Details of the methodology can be found in our previous works (Ji, Munjiza & Williams Reference Ji, Munjiza and Williams2012; Chen et al. Reference Chen, Ji, Xu, Liu and Campbell2015; Chen, Ji & Xu Reference Chen, Ji and Xu2019a).

In the present two-dimensional simulations, the streamwise length $L$ ($= L_u+L_d$) and transverse length $H$ of the computational domain are both 100$D$, as shown in figure 1(a), with $L_u$ and $L_d$ denoting the distances of the prism's centre to the inlet and outlet boundaries, respectively. The domain is discretized by a non-uniform Cartesian grid with a resolution of $768\times 1024$. To improve the accuracy of numerical results, a rectangular region of $8D\times 12D$, covering possible motions of the prism, is discretized by using a uniform mesh with a non-dimensional grid spacing of $1/64$ in both directions. A stretched mesh is adopted out of the region to keep the total grid number within an affordable range. This mesh configuration was applied in our previous simulations (Chen et al. Reference Chen, Ji, Xu, Liu and Campbell2015, Reference Chen, Ji, Williams, Xu, Yang and Cui2018, Reference Chen, Ji, Alam, Williams and Xu2020a), demonstrating high accuracy and fidelity. As for boundary conditions, a Dirichlet-type boundary is adopted at the inflow whereas a Neumann-type boundary is employed at the outflow. The top and bottom boundaries are set as free-slip boundaries. The number of IB points for the prism is selected as $506$ to ensure at least one IB point is allocated in each grid cell. The simulations start with a uniform flow velocity and zero pressure.

Figure 1.

(a) The computation domain for the FIV of a D-section prism at ${\textit {Re}} = 100$ and (b) a sketch for the AoA ($\alpha$).

In the present simulations, the D-section prism is allowed to vibrate only in the transverse direction and $\alpha$ varies in the range $0^\circ \text {--}180^\circ$, with $\alpha = 0^\circ$ ($180^\circ$) corresponding to the configuration with the flat surface facing downstream (upstream); see figure 1(b). The reduced velocity normalized by the prism diameter is $U^* = 2.0$–$20.0$, the mass ratio is $m^* = 2.0$ and the structural damping ratio ($\zeta$) is set to zero to excite larger vibrations. In this paper, we aim to explore the effects of $\alpha$ and $U^*$ on the responses of the D-section prism. The Reynolds number, based on the effective diameter $D_e$, is fixed at ${\textit {Re}} = U_\infty D_e/\nu = 100$, where $D_e=0.5(1+|\cos \alpha |) D$, varying from $0.5D$ to $D$, and $\nu$ is the fluid kinematic viscosity. In addition, parameter intervals as small as $\Delta \alpha =5^{\circ }$ and $\Delta U^{*}=0.1$ are applied, leading to more than $1000$ simulation cases.

The drag and lift coefficients are defined as $C_{D}=2 F_{D} / \rho U_{\infty }^{2} D$ and $C_{L}=2 F_{L} / \rho U_{\infty }^{2} D$, respectively. Forces $F_D$ and $F_L$ are the drag and lift, respectively. Parameters $\bar {C}_{D}$ or $\bar {C}_{L}$ and $C_{D}^{\prime }$ or $C_{L}^{\prime }$ are the time-mean and root-mean-square (r.m.s.) values for the corresponding coefficients, respectively. The non-dimensional vibration amplitude is defined as $A^{*}=\sqrt {2} y_{rms} / D$, where $y_{r m s}$ is the r.m.s. value of the displacement in the transverse direction. The lift, drag and vibration frequencies are obtained through the fast Fourier transform, and the phase lag between the lift and displacement is obtained through the Hilbert transform.

2.2. Convergence analysis and validation

The convergence of the grid spacing for FIV of a D-section prism is given in table 1. The difference in the results between the grid spacings of 1/64 and 1/128 is insignificant, with the largest value being $1.3\,\%$ for $C_{L}^{\prime }$. Thus, it can be concluded that the grid spacing of 1/64 is acceptable for the simulations. Further, the computational domain sizes and non-dimensional time step $(\Delta t U_{\infty }/ D)$ are also checked. Five cases are adopted, with Case 3 being the reference case. As shown in table 2, with $L_d = 50D$ to $100D$ (Case 1), with $L_u = 50D$ to $100D$ (Case 2) and with $H = 100D$ to $200D$ (Case 4), the results show a negligible difference from those in Case 3, with the most significant difference being $1.9\,\%$ for $C_{L}^{\prime }$. In addition, when decreasing $\Delta t U_{\infty } / D=0.005$ to $0.0025$ (Case 5), the difference is acceptable, with the largest value being $5.1\,\%$ for $C_{L}^{\prime }$. Thus, in subsequent simulations, $L_u = 50D$, $L_d = 50D$, $H = 100D$ and $\Delta t U_{\infty } / D=0.005$ are adopted.

Table 1.

Results of FIV of a D-section prism at ${\textit {Re}} = 100$, $m^* = 2.0$, $\zeta = 0$, $\alpha = 180^\circ$ and $U^* = 8.0$ with different grid spacings. The values in parentheses represent the difference from those of the grid spacing of $1/128$.

Table 2.

Results of FIV of a D-section prism at ${\textit {Re}} = 100$, $m^* = 2.0$, $\zeta = 0$, $\alpha = 180^\circ$ and $U^* = 8.0$ with varying computational domain sizes and non-dimensional time step.

The validation is conducted with VIV results of a circular cylinder and flow past a stationary D-section prism at ${\textit {Re}} = 100$. As shown in table 3, the present results are consistent with published data, which suggests the present method has high fidelity and accuracy. Besides, as shown in table 4, the present results agree well with published data, further validating the reasonably high accuracy of the present method. Note that the same computational domain as that of Bhinder, Sarkar & Dalal (Reference Bhinder, Sarkar and Dalal2012) was adopted for the flow past the D-section prism.

Table 3.

Comparison of results of VIV of a circular cylinder at ${\textit {Re}} = 100$, $U^* = 4.46$, $m^* = 1.59$ and $\zeta = 0$. The cylinder is constrained to vibrate only in the transverse direction. Note that $y_{max}$ denotes the maximum value of the transverse displacement.

Table 4.

Comparison of present results with published data for flow past a D-section prism at ${\textit {Re}} = 100$. The same computational domain as Bhinder et al. (Reference Bhinder, Sarkar and Dalal2012) is applied, i.e. $L_u = 8D$, $L_d = 26D$ and $H = 16D$.

3.1. Regime i: typical VIV

As shown in figure 3(a), in the typical VIV regime, $A^*$ increases significantly with increasing $U^*$ ($= 2.0\text {--}5.0)$. The maximum $A^* = 0.57$ is obtained at $U^* = 5.0$, independent of $\alpha$. This peak $A^*$ is very close to that of the VIV of a circular cylinder at the same set of parameters (${\textit {Re}}$ = 100, $m^*$ = 2.0), as indicated by the stars in figure 3(a). Beyond $U^*$ = 5.0, $A^*$ decreases gradually with increasing $U^*$. The amplitude difference becomes obvious when $U^* > 8.0$, i.e. $A^*$ is slightly decreased for a higher $\alpha$ but is larger than that of a circular cylinder. However, we can see that if only the behaviour of the amplitudes with $U^*$ is considered, it is not easy to differentiate response branches because of the smooth transitions. Instead, the normalized vibration frequency ($f_v/f_n$) should be a better choice to identify the response branches and transitions (Kumar et al. Reference Kumar, Singh and Sen2018). As shown in the inset of figure 3(b), according to the $f_v/f_n$ variation with $U^*$, two branches, following the terminology in the VIV of an isolated circular cylinder (Williamson & Govardhan Reference Williamson and Govardhan2004), are identified. At $U^* < 3.5$, the vibration amplitude is minimal and $f_v/f_n$ closely follows the $St$ line for $\alpha = 0^\circ$, suggesting the desynchronization regime. While at $3.5 \le U^* < 5.0$, $f_v/f_n$ deviates from the $St$ line and becomes constant at a value smaller than 1.0 – a feature of the initial branch of the VIV of a circular cylinder (Singh & Mittal Reference Singh and Mittal2005). It is thus named as the initial branch, followed by the lower branch when $U^* \ge 5.0$. However, different from the VIV of a circular cylinder, there is no demarcation via an amplitude/frequency jump between the lower branch and the desynchronization regime for the D-section prism. Moreover, the normalized vibration frequency shows a linear increment with increasing $U^*$ and significantly deviates from the natural frequency, even when the vibration amplitude is significant. This is distinct from the frequency variation of a circular cylinder undergoing VIV at the same mass ratio $m^* =$ 2.0, that is, the vibration frequency is close to its natural frequency in the lower branch (see the stars in figure 3b). It is worth noting that the excitation of large-amplitude vibration is not necessarily accompanied by the match of the vibration frequency and the natural frequency of a body in vacuum due to the added mass effects (Alam Reference Alam2022). Williamson & Govardhan (Reference Williamson and Govardhan2004) stated that light bodies can conceivably vibrate with large amplitudes at hundreds of times the natural frequency. Also, the VIV response of a very low-$m^*$ circular cylinder in the laminar flow regime (Navrose & Mittal Reference Navrose and Mittal2016) shows that the frequency ratio $f_v/f_n$ linearly increases in the lock-in region characterized by the large-amplitude vibration when $m^* =$ 0.5. However, it is argued that the linearly increasing trend in $f_v/f_n$ for the D-section prism is not a fallout of the mass ratio because the adopted $m^*$ in this study is significantly larger than those in Navrose & Mittal (Reference Navrose and Mittal2016). Instead, we attribute this phenomenon to the stationary flow separation from the prism and the lacking afterbody. Detailed discussion is presented in § 7.

The significant vibration amplitude of the D-section prism at a small $\alpha$ is unexpected, considering the absence of the afterbody. It is conventionally believed that the vibration of a D-section prism at $\alpha = 0^\circ$ is very weak, or even ceases to exist, due to the lack of the vortex–afterbody interaction. The early wind-tunnel tests proved this at a high mass ratio $m^* \sim$ O($10^2$), including Brooks (Reference Brooks1960) and Parkinson (Reference Parkinson1963). However, in the present simulations with $m^* = 2.0$, the amplitude of the D-section prism closely follows that of an isolated circular cylinder at $4.0 < U^* < 7.0$ (indicated by the star markers in figure 3a). Similar results were also reported in the experimental study of Zhao et al. (Reference Zhao, Hourigan and Thompson2018) at $m^* = 6.0$. The above difference indicates the significant influence of $m^*$ on the response of the D-section prism, which is discussed in § 7.

The spectral features of the displacement shown in figure 4(a) indicate that the vibration of the D-section prism in the typical VIV regime is dominated by one frequency, with an exception at $U^* = 3.5$ where a secondary frequency appears, resulting in a beating-like displacement (figure 5a). Moreover, the spectral regularity suggests that the vortex shedding frequency at each $U^*$ equals the vibration frequency. The time history of the displacement in figure 5(b) clearly shows a perfect periodicity.

Figure 4.

Power spectral density of the displacement of prism for $U^* = 2$–$20$. (a) Typical VIV at $\alpha = 10^\circ$, (b) extended VIV at $\alpha = 50^\circ$, (c) combined VIV and galloping at $\alpha = 80^\circ$, (d) narrowed VIV at $\alpha = 125^\circ$, (e) transition response at $\alpha = 155^\circ$ and ( f) pure galloping at $\alpha = 170^\circ$, representing the six types of responses marked in figure 2. The power spectral contours are logarithmically scaled. The dash-dotted line in each panel denotes the vortex shedding frequency ($St$) of a stationary D-section prism at the same AoA.

Figure 5.

Time histories of the normalized displacements at different $U^*$ and $\alpha$ values: (a) $U^* = 3.5$ and $\alpha = 10^\circ$, (b) $U^* = 6.0$ and $\alpha = 10^\circ$, (c) $U^* = 9.0$ and $\alpha = 50^\circ$, (d) $U^* = 17.0$ and $\alpha = 50^\circ$, (e) $U^* = 3.0$ and $\alpha = 80^\circ$, ( f) $U^* = 18.0$ and $\alpha = 80^\circ$, (g) $U^* = 7.0$ and $\alpha = 125^\circ$, (h) $U^* = 7.5$ and $\alpha = 155^\circ$, (i) $U^* = 11.0$ and $\alpha = 155^\circ$, (j) $U^* = 17.0$ and $\alpha = 160^\circ$, (k) $U^* = 18.0$ and $\alpha = 160^\circ$, (l) $U^* = 7.0$ and $\alpha = 170^\circ$, (m) $U^* = 13.0$ and $\alpha = 170^\circ$ and (n) $U^* = 18.0$ and $\alpha = 170^\circ$. (a,b) Typical VIV, (c,d) extended VIV, (e,f) combined VIV and galloping, (g) narrowed VIV, (h–k) transition response and (l–n) pure galloping.

3.2. Regime ii: extended VIV

As shown in figure 3(c), in the extended VIV regime ($\alpha = 40^\circ \text {--}65^\circ$), the large-amplitude vibration region, following the definition in Bourguet & Lo Jacono (Reference Bourguet and Lo Jacono2014), becomes wider with increasing $\alpha$. Depending on $\alpha$, the response of the prism consists of two or three branches. As shown in figure 3(d), at $\alpha = 40^\circ \text {--}45^\circ$, two branches are observed, with the initial branch at $3.0 < U^* \le 4.0$ and the lower branch at $4.0 < U^* < 7.5$. A desynchronization region exists at $7.5 \le U^* \le 20.0$, characterized by linearly increasing $f_v/f_n$ and smaller vibration amplitudes. The jump between the lower branch and the desynchronization region is rather sharp. At $\alpha = 50^\circ \text {--}55^\circ$, the lower branch exists in a much wider $U^*$ region. However, different from that at $\alpha = 40^\circ \text {--}45^\circ$, the amplitude at $\alpha = 50^\circ \text {--}55^\circ$ decreases gradually in the lower branch with increasing $U^*$. At $\alpha = 60^\circ \text {--}65^\circ$, the third branch is observed between the lower branch and the desynchronization region. For the $\alpha = 60^\circ$ case, the initial branch is at $3.0 < U^* \le 4.0$ and the lower branch is in the range $4.0 < U^* \le 9.0$, with the maximum amplitude ($A^* = 0.72$) being achieved at $U^* = 8.0$. Significant amplitude fluctuations with $U^*$ are observed in the third branch with the largest $A^* (= 0.67)$ occurring at $U^* = 12.5$, indicating the variety of wake–prism interactions. For the $\alpha = 65^\circ$ case, the third branch appears in a much wider region, i.e. $10.5 < U^* < 18.8$. Similar to the third branch at $\alpha = 60^\circ$, $A^*$ also varies significantly with increasing $U^*$, with the largest amplitude ($A^* = 1.0$) appearing at $U^* = 14.0$. The desynchronization region appears once $U^*$ is larger than a critical value, $U^* = 14.0$ and 18.8 for $\alpha = 60^\circ$ and $65^\circ$, respectively. As shown in figure 3(d), the peak $A^*$ of the two-branched VIV of the D-section prism is similar to that of a circular cylinder in low-$Re$ flow, while the peak $A^*$ of the three-branched VIV is comparable to that of a circular cylinder in high-$Re$ flow (Khalak & Williamson Reference Khalak and Williamson1999; Zhao et al. Reference Zhao, Leontini, Lo Jacono and Sheridan2014). Although the largest amplitudes are comparable, the vibration region of the D-section prism is much wider than that of the VIV of a circular cylinder at high-$Re$ flow (see the results in figure 3c).

As shown in figure 3(d), with increasing $\alpha$, the resonance $f_v/f_n = 1.0$ appears at a larger $U^*$. For $\alpha = 45^\circ \text {--}65^\circ$, $f_v/f_n$ slightly deviates from the corresponding $St$ line, similar to the initial branch in the typical VIV regime. However, in the lock-in region consisting of the lower and third (if existing) branches, $f_v/f_n$ significantly deviates from the corresponding $St$ line and approaches a constant value (figure 3d). For example, at $\alpha = 65^\circ$, $f_v/f_n$ remains at 0.75 in the range of $U^* = 4.0\text {--}18.6$, suggesting a vast lock-in region compared with that in the VIV of a circular cylinder.

The spectral features of the displacement shown in figure 4(b) indicate that at $\alpha = 50^\circ$, the prism vibration is dominated by the fundamental frequency at $2.0 \le U^* < 13.0$ with the second-harmonic frequency dominating after $U^* \ge 13.0$. In the lower branch ($5.5 \le U^* \le 13.0$), the two frequencies coexist. However, because of the low amplitude of the second harmonic, the displacements are relatively regular in the whole examined $U^*$ range (figure 5c,d).

3.3. Regime iii: combined VIV and galloping

As shown in figure 3(e), the amplitude variation at $\alpha = 70^\circ \text {--}80^\circ$ consists of two distinct types of response. The first is VIV with a small amplitude. With increasing $\alpha$, the width of the VIV region grows significantly. At $\alpha = 70^\circ$ and $75^\circ$, the VIV region ends at $U^* = 3.5$ and $4.0$, respectively, while at $\alpha = 80^\circ$, VIV terminates at $U^* = 7.0$. It is expected that as $\alpha$ further increases, VIV will dominate the whole $U^*$ range, suggesting the presence of pure VIV (see figure 3g). The second is galloping with a much larger amplitude, occurring at a larger $U^*$ (figure 3e). In this region, $A^*$ increases significantly with increasing $U^*$, despite the significant kinks at $\alpha = 70^\circ \text {--}75^\circ$ which are associated with the local occurrences of superharmonic lock-in and wake mode changes. Similar features were reported in the experiments on a square prism by Bearman et al. (Reference Bearman, Gartshore, Maull and Parkinson1987), Nemes et al. (Reference Nemes, Zhao, Lo Jacono and Sheridan2012) and Zhao et al. (Reference Zhao, Leontini, Lo Jacono and Sheridan2014), and in the numerical simulations of a D-section prism by Sharma, Garg & Bhardwaj (Reference Sharma, Garg and Bhardwaj2022). Similar to the vibration response of a square prism in Sen & Mittal (Reference Sen and Mittal2011) and Bhatt & Alam (Reference Bhatt and Alam2018), the response of the D-section prism at $\alpha = 70^\circ \text {--}80^\circ$ shows a combined feature of both VIV and galloping and is thus named combined VIV and galloping (Cui et al. Reference Cui, Zhao, Teng and Cheng2015). As shown in figure 3(e), for $\alpha = 70^\circ$, $A^*$ is larger than that for $\alpha = 75^\circ$ when $U^* < 9.5$. At a larger $U^*$, $A^*$ of the two cases becomes comparable. However, $A^*$ at $\alpha = 80^\circ$ is much smaller than that at $\alpha = 70^\circ \text {--}75^\circ$, which is caused by the postponed emergence of the galloping response.

As shown in figure 3( f), $f_v/f_n$ generally follows the $St$ line in the VIV region. In the galloping region, $f_v/f_n$ for all three $\alpha$ cases is almost constant, having a value lower than $1.0$ due to the significant added mass effects when a small $m^*$ is adopted. With increasing $\alpha$, the constant $f_v/f_n$ decreases slightly in the galloping region. However, when $m^*$ is large, $f_v/f_n$ approaches unity. This feature is also observed in galloping of other non-circular prisms (Zhao et al. Reference Zhao, Leontini, Lo Jacono and Sheridan2014; Seyed-Aghazadeh et al. Reference Seyed-Aghazadeh, Carlson and Modarres-Sadeghi2017).

The spectral features of the displacements shown in figure 4(c) indicate that the frequencies in the VIV and galloping regions are different. The displacement has one dominant frequency in the VIV region, which linearly increases with increasing $U^*$. While in the galloping region, the dominant frequency is independent of $U^*$, with a second harmonic being observed. The amplitude of the second harmonic grows gradually with increasing $U^*$. Furthermore, time histories of the displacement in the VIV region are periodic (figure 5e). In contrast, in the galloping region, the vibration is not sinusoidal with the appearance of the second harmonic (figure 5f). The vibration period in the galloping region is much more extended than that in the VIV region.

3.4. Regime iv: narrowed VIV

As shown in figure 3(g), at $\alpha = 85^\circ \text {--}150^\circ$, the amplitude of the D-section prism with increasing $U^*$ shows the same behaviour, i.e. first increasing and then decreasing. Similar to typical VIV, two branches, i.e. the initial and lower branches, can be distinguished based on the variations of $f_v/f_n$ and $A^*$ while the transition from the lower branch to the desynchronization region is smooth. With increasing $\alpha$, the largest amplitude grows and appears at a larger $U^*$. At $\alpha = 85^\circ \text {--}95^\circ$, where the flat surface is almost parallel to the incoming flow, the peak amplitude is insignificant, i.e. $A^* = 0.09$. As shown later, the vortices in this configuration are shed further from the prism and give rise to a more minor stimulation to the vibration. At $\alpha = 150^\circ$, the largest amplitude is $A^* = 0.68$, which is even more significant than the VIV of an isolated circular cylinder (Prasanth & Mittal Reference Prasanth and Mittal2008; Chen et al. Reference Chen, Ji, Xu, Liu and Campbell2015). However, compared with typical VIV, the lock-in regime appears in a much narrower $U^*$ range, especially when $\alpha$ is small (see figure 3a,g). As a result, this regime is named narrowed VIV. In the desynchronization region as $U^*$ is large, the amplitude for each $\alpha$ case is constant while the value increases slightly with increasing $\alpha$ (figure 3g).

As shown in figure 3(h), $f_v/f_n$ for each $\alpha$ case shows a linear increase with increasing $U^*$ in the lower branch and the desynchronization regime. Besides, we can see that the $f_v/f_n$ slope significantly decreases with increasing $\alpha$, which is distinct from the scenario in the typical VIV regime that the $f_v/f_n$ lines at different $\alpha$ collapse. This can be contributed by the variation of the effective diameter $D_e$ with $\alpha$ because a smaller prism diameter leads to a higher vortex shedding frequency. Note that, in the narrowed VIV, the $f_v/f_n$ lines at different $\alpha$ collapse in general if $U^*$ is normalized using $D_e$ as the reference length scale except at $\alpha =85^\circ \text {--}95^\circ$ (results are not shown here). The insufficient scaling with $D_e$ indicates that there are other influential factors. For example, it is seen in figure 3(h) that the largest $f_v/f_n$ slope is attained at $\alpha =95^\circ$, instead of $\alpha =90^\circ$ at which $D_e$ is the smallest. Moreover, the $f_v/f_n$ slope at $\alpha =95^\circ$ is slightly larger than that at $\alpha =85^\circ$ even though the values of $D_e$ are the same in the two cases. It is argued that the flow separation also contributes to the variations in $f_v/f_n$, and the $f_v/f_n$ slope may depend on the transverse distance between separation points. This is other evidence of the key role played by the flow separation in determining the VIV response of the D-section prism.

As shown in figure 4(d), the spectral features of the displacement in the narrowed VIV are similar to those in the typical VIV, i.e. one dominant frequency which linearly increases with $U^*$. Furthermore, the time history of the displacement shown in figure 5(g) indicates that the vibration is perfectly periodic, which is also similar to that in the typical VIV regime.

3.5. Regime v: transition response between narrowed VIV and pure galloping

Figure 3(i) shows that the response at $\alpha = 155^\circ \text {--}160^\circ$ can be divided into regions. At $\alpha = 155^\circ$, VIV is observed at $2.0 \le U^* \le 9.0$ and $11.5 \le U^* \le 20.0$, while the galloping-like response dominates at $9.0 < U^* < 11.5$. In the first VIV-like region, consisting of the initial branch ($2.0 \le U^* < 4.0$) and the lower branch ($4.0 \le U^* \le 9.0$), the amplitude first increases and subsequently decreases with increasing $U^*$, peaking ($A^* = 0.81$) at $U^* = 4.5$. While in the second VIV-like region, the amplitude is constant at a smaller value ($A^* \approx 0.18$), signifying the desynchronized response. As shown in figure 3(j), $f_v/f_n$ in the VIV-like region linearly increases, following the $St$ line, with increasing $U^*$. Lying between the two VIV-like regions, the galloping-like region has a contrasting amplitude. The amplitude increases sharply to the largest $A^*$ ($= 0.94$) and afterwards plummets. In addition, $f_v/f_n$ of this region falls back to a value smaller than unity, a frequency feature similar to that in the galloping response. However, we can see that the increasing amplitude cannot persist to a higher $U^*$; the galloping is confined. Regarding the $\alpha = 160^\circ$ case, the VIV-like region only appears at $2.0 \le U^* \le 5.0$ while the galloping-like region dominates the remaining $U^*$ range. However, the amplitude does not show a divergent behaviour with increasing $U^*$. It is worth noting that for the $\alpha = 160^\circ$ case, the amplitudes at $U^* = 17.0\text {--}17.5$ are much larger than those of the neighbouring $U^*$, a similar behaviour occurring at $\alpha = 60^\circ \text {--}65^\circ$ (see figure 3c,i). This is caused by the variations of the separation points. As shown in figure 5(j,k), at $U^* = 17.0$, the vibration of the prism is periodic, suggesting a stable vortex–prism interaction, while at $U^* = 18.0$, the vibration amplitude changes between small and large values, corresponding a beating response.

The above suggests that the prism response at $\alpha = 155^\circ \text {--}160^\circ$ has combined VIV and galloping characteristics. However, the amplitude with $U^*$ significantly deviates from the typical galloping response. Therefore, the response at $\alpha = 155^\circ \text {--}160^\circ$ can be taken as a transition response from narrowed VIV to pure galloping.

The spectral features of the displacement shown in figure 4(e) suggest that the vibration in the VIV-like regions has only one frequency. In contrast, in the galloping-like region several frequencies are observed, with the lowest being dominant. Furthermore, the time history of the displacement shown in figure 5(h,i) indicates that the displacements are regular, especially in the VIV-like region.

3.6. Regime vi: pure galloping

As shown in figure 3(k), at $\alpha = 165^\circ \text {--}180^\circ$, $A^*$ almost monotonically increases with increasing $U^*$. This type of response is referred to as pure galloping. Besides, we can see that the vibration for all the $\alpha$ cases begins at the examined smallest $U^*$ ($= 2.0$). The amplitudes of the four $\alpha$ cases are almost identical with $U^* < 6.0$, and afterwards, the amplitude with a larger $\alpha$ is slightly larger (figure 3k). Furthermore, kinks are observed for all the four $\alpha$ cases, caused by the lock-in on higher-harmonic frequencies (Bearman et al. Reference Bearman, Gartshore, Maull and Parkinson1987; Nemes et al. Reference Nemes, Zhao, Lo Jacono and Sheridan2012; Zhao et al. Reference Zhao, Leontini, Lo Jacono and Sheridan2014; Sharma et al. Reference Sharma, Garg and Bhardwaj2022). In figure 3(k), the galloping response of a D-section prism at ${\textit {Re}} = 1080\text {--}9000$ reported in the experimental study of Zhao et al. (Reference Zhao, Hourigan and Thompson2018) is superimposed. We can see that the amplitudes obtained from the present investigation and those of Zhao et al. (Reference Zhao, Hourigan and Thompson2018) are close to each other when $U^* \le 12.5$, which indicates that for the galloping response the Reynolds number has insignificant effects if the reduced velocity is small. After $U^* > 12.5$, the amplitude deviation between the present study and that of Zhao et al. (Reference Zhao, Hourigan and Thompson2018) becomes obvious and enlarges with increasing $U^*$.

As shown in figure 3(l), $f_v/f_n$ of the galloping response is close to unity after $U^* > 5.5$ while following the $St$ line for $U^* \le 5.5$. In addition, we can see that $f_v/f_n$ in Zhao et al. (Reference Zhao, Hourigan and Thompson2018) is closer to 1.0 as a consequence of a higher $m^*$ ($= 6.0$) adopted. When compared with the galloping branch at $\alpha = 70^\circ \text {--}80^\circ$, $f_v/f_n$ is much closer to unity, which is mainly related to the variation of the added mass with the change in the cross-sectional shape with respect to the incoming flow (Sarpkaya Reference Sarpkaya2004).

The spectral features of the displacement shown in figure 4( f) indicate that the galloping response has several frequencies, suggesting complex vortex–prism interactions. At small $U^*$ ($= 2.0\text {--}9.5$), the displacement of each $U^*$ has one or two frequencies. At $U^*> 9.5$ the displacement spectrum of each $U^*$ can be irregular, dominated by several frequencies, or chaotic, such as that for $U^* = 17.0\text {--}19.0$. As discussed later, the multiple frequencies are related to the multiple vortices shed from the prism in each vibration period.

4.1. An overview of the wake modes

Figure 6 shows the partition of the wake modes in the $U^*$–$\alpha$ parametric plane. We can see that the 2S mode dominates the typical and narrowed VIVs where the vibration amplitude is relatively small. Leontini, Thompson & Hourigan (Reference Leontini, Thompson and Hourigan2006b) found that the wake mode is closely related to the cylinder amplitude, and when the amplitude is smaller than $0.85$, the $2$S mode dominates, which is consistent with the present results. In the extended VIV, the wake shows the 2S mode when the amplitude is low but displays the ${\rm P}+{\rm S}$ mode or the 2P mode when the amplitude is high. There is also another mode, ${\rm T}+{\rm S}$, observed only at $\alpha = 50^\circ \text {--}55^\circ$. Four vortices are shed from the prism in one vibration period, with three vortices forming a triplet shed at one side of the prism while the remaining one is at the other side. The P mode – two vortices shed from the prism and immediately pair up, generating an inclined trace – exists at $\alpha = 45^\circ$ and $U^* = 7.5$. The paired vortices distinguish P mode from 2S mode with separated vortices. The wake modes with more than four vortices shed in a vibration cycle are only observed in the galloping regime, suggesting that a large amplitude is required to trigger the multi-vortex mode – the $m{\rm S}+n{\rm S}$ mode, where $m$ and $n$ are integers. That is, when the prism moves in one direction, $m$ vortices alternately shed from the prism; when the prism moves the other way, $n$ vortices alternately shed from the prism. The vortices arrange in a V-shaped pattern and the vortex number on each wing increases with increasing $U^*$ owing to the amplified vibration amplitude. For example, at $\alpha = 75^\circ$ and $U^* =20$, fifteen vortices shed from the prism in a vibration cycle. Although the amplitude is higher at $\alpha = 165^\circ \text {--}180^\circ$, fewer vortices shed from the prism in each vibration cycle. This can be attributed to the higher vibration frequency at $\alpha = 165^\circ \text {--}180^\circ$ than at $\alpha = 75^\circ$; see figure 3( f,l). However, it should be pointed out here that, in some $U^*$ cases, the galloping responses are not regular, and the displacement can vary from one period to another, thus leading to the varying number of vortices shed from the prism. Although $m$ and $n$ reflect the wake mode, they may change from period to period. In the transition response, the 2S mode dominates the $\alpha = 155^\circ$ case, while both ${\rm P}+{\rm S}$ and 2P modes dominate the $\alpha = 160^\circ$ case, with the latter appearing at higher $U^*$.

Figure 6.

Wake modes in the ($U^*$, $\alpha$) parameter plane for the FIV of a D-section prism at ${\textit {Re}} = 100$ and $m^* = 2.0$.

4.2. Excitation/inhibition mechanisms of the prism vibration

In this section, the time evolutions of the wake modes are explored with the aid of time-dependent displacement $y$ and lift coefficient $C_L$, with a view to elucidating the underlying excitation/inhibition mechanisms of the prism vibration. As the transverse vibration of the prism is sustained by the lift force, special attention is paid to the lift coefficient in phase with the prism velocity, which signifies the energy transfer between the fluid and prism. However, it should be pointed out here that the time-averaged lift $\bar {C}_{L}$ is induced by the asymmetry of the prism to the incoming flow, which results in the non-zero time-averaged position $\bar {y}$ at $0^\circ < \alpha < 180^\circ$. The prism oscillation with respect to the time-averaged position, i.e. $y^{\prime }$ ($=y-\bar {y}$), depends on the energy exchange between the flow and the prism. Following Bourguet & Lo Jacono (Reference Bourguet and Lo Jacono2014), the lift coefficient in phase with the prism velocity is defined as $C_{L, V}=({\sqrt {2}(C_{L}-\bar {C}_{L}) V})/{\sqrt {\overline {V^{2}}}}$, with $V=\dot {y}$ being the prism velocity. Positive $C_{L,V}$ indicates the flow excites the oscillation while negative $C_{L,V}$ means the oscillation is damped by the fluid. Therefore, flow physics for the excitation/inhibition caused by the fluid flow can be analysed through the variations of the vorticity contours and the lift component in phase with the prism velocity.

For a stably vibrating cylinder undergoing VIV, it is generally recognized that the net energy transfer over a vibration cycle is zero when $\zeta = 0$. Specifically, in the initial branch where the total lift force $C_L$ is in phase with the vibration displacement, energy is transferred from the cylinder to the fluid flow when the cylinder moves from the top/bottom to the balanced position, leading to a negative $C_{L,V}$. On the other hand, the same amount of energy is transferred back to the cylinder when it moves from the balanced position to the top/bottom. The opposite scenario happens when the total lift is in anti-phase with the displacement in the lower branch (Ji et al. Reference Ji, Xiao, Wang and Wang2011). This simple mechanism also applies to the VIV of the D-section prism at $\alpha = 0^\circ$, as is shown in § 7. However, when the D-section prism is asymmetric to the incoming flow, the energy-in and -out mechanisms are fundamentally changed. As shown in figure 7(a,b), the wake at ($\alpha$, $U^*$) = ($10^\circ$, 6.0) is in the 2S mode with two vortices alternately shed from the prism in one period. The $C_L$ is dominated by the second-harmonic frequency due to the asymmetric prism configuration. Now, let us focus on half a vibration cycle (instant $i$ $\rightarrow$ $v$) where the prism moves from the top to the bottom. When the prism is at the top (instant $i$), the lower side shear layer of the prism grows and rolls closer to the prism base than the other side shear layer, which leads to a negative $C_L$. With the downward moving of the prism, the negative $C_L$ inputs energy into the vibration system and accelerates the prism, as indicated by the positive ${C}_{L,V}$. Further downward motion of the prism alters $C_L$ from negative to positive, and correspondingly, $C_{L,V}$ changes from positive to negative, indicating that the vibration of the prism is damped. The largest $C_{L,V}$ (negative) is achieved near the balanced position, followed by a small positive peak before the bottom position. Now, we can see that there are two separated power-in regions near the top and bottom positions in this half-cycle. Between them, there exists a power-out region where the prism has a large velocity. As stated in Qin, Alam & Zhou (Reference Qin, Alam and Zhou2017), the dimension of a vortex is proportional to the amount of energy transferred into the wake. From figure 7(b), it is observed that at instant $ii$, the anticlockwise vortex (red) rapidly develops in size and strength when the prism passes the balanced position, extracting energy from the vibrating prism. The power-out process terminates and the vortex starts to shed before the prism reaches the bottom at instant $iii$.

Figure 7.

Time histories of the lift coefficient ($C_L$ and $C_{L,V}$), the displacement ($y/D$ and $y^\prime /D$) and vorticity contours. (a,b) The 2S mode at ($\alpha$, $U^*$) = ($10^\circ$, 6.0), (c,d) 2S mode at ($\alpha$, $U^*$) = ($35^\circ$, 5.5), (e,f) P mode at ($\alpha$, $U^*$) = ($45^\circ$, 7.5) and (g,h) ${\rm P}+{\rm S}$ mode at ($\alpha$, $U^*$) = ($160^\circ$, 5.0). The normalized Z-vorticity is in the range [$-$1, 1] with an increment of 0.2. Here, $C_L$ is the total lift coefficient, $C_{L,V}$ is the lift coefficient in phase with the prism velocity, $y/D$ is the displacement and $y^\prime /D$ is the displacement with non-zero time-averaged position removed.

However, when the prism moves upward in the second half-cycle (instant $iii$ $\rightarrow$ $v$), there exists only one power-out (bottom $\rightarrow$ balanced position) region and one power-in (balanced position $\rightarrow$ top) region. This is similar to the power-in and -out mechanisms of the circular cylinder and the D-section prism at $\alpha =0^\circ$. However, there is a discrepancy in that the amounts of energy transferred in and out are unequal, resulting in a positive net energy input into the vibrating prism at a non-zero $\alpha$.

From the aspect of the entire vibration cycle, it is observed that the net energy transferred into the vibrating prism is negative in the first half-cycle while positive in the second one. As a result, the vortices (red) shed from the lower side in the first half-cycle are larger and stronger than those shed from the upper side, leading to a slightly asymmetric vortex shedding pattern in the wake (see figure 8a).

Figure 8.

Instantaneous vorticity contours. (a) The 2S mode at ($\alpha$, $U^*$) = ($10^\circ$, 6.0), (b) 2S mode at ($\alpha$, $U^*$) = ($35^\circ$, 5.5), (c) P mode at ($\alpha$, $U^*$) = ($45^\circ$, 7.5) and (d) ${\rm P}+{\rm S}$ mode at ($\alpha$, $U^*$) = ($160^\circ$, 5.0). The normalized Z-vorticity is in the range [$-$1, 1] with an increment of 0.2.

At $\alpha =25^\circ \text {--}35^\circ$, as shown in figure 7(c), the variation of $C_{L,V}$ indicates that the energy is transferred from the prism to the fluid flow when the prism moves from the top/bottom to the balanced position, and vice versa. This is similar to the scenario for the VIV of a circular cylinder and the D-section prism at $\alpha =0^\circ$, except that there exist significant fluctuations in the variations of $C_L$ and $C_{L,V}$. The vortices can be nearly parallel in the wake because the top separation point appears upstream of the bottom one (figure 7d). Figure 8(b) shows the wake mode in a wider view. In the wake, the vortices of the same row coalesce at $x/D > 16$ because of the formation of the secondary vortex street (Cimbala, Nagib & Roshko Reference Cimbala, Nagib and Roshko1988; Inoue & Yamazaki Reference Inoue and Yamazaki1999; Kumar & Mittal Reference Kumar and Mittal2012; Thompson et al. Reference Thompson, Radi, Rao, Sheridan and Hourigan2014; Jiang & Cheng Reference Jiang and Cheng2019; Zheng & Alam Reference Zheng and Alam2019; Shi, Alam & Bai Reference Shi, Alam and Bai2020). With the variations of $U^*$ and $\alpha$, the secondary wake appears at different positions. It should be mentioned here that the secondary street exists much closer to the prism than that ($x/D > 100$) in the wake of a circular cylinder (Kumar & Mittal Reference Kumar and Mittal2012; Jiang & Cheng Reference Jiang and Cheng2019). This indicates that the D-section shape may increase the wake instability.

As shown in figure 7( f), at $\alpha =45^\circ$ and $U^*=7.5$, the vortices shed from the prism are paired in a biased wake due to the AoA of the prism (see the wide view in figure 8c). This is named the P mode, with P denoting vortex pairing. This unique and asymmetric wake pattern can be explained by the significant difference in the variation of $C_{L,V}$ in each half vibration cycle. As shown in figure 7(e), the energy is mainly transferred from the prism to the fluid when the prism moves from the top towards the bottom (instant $i$ $\rightarrow$ $iii$) where the vortex pair right behind the prism develops fast in dimension and strength. In the second half (instant $iii$ $\rightarrow$ $v$), a smaller amount of energy is transferred towards the vibrating prism, compared with that transferred towards the fluid in the first half, and the vortex pair begins to shed at instant $iv$. It seems that the vortices from both sides develop in a half cycle and shed in the other one, which results in the asymmetric vortex shedding pattern. Over the entire vibration cycle, the net energy transfer is negative, leading to the reduced vibration amplitude at instant $v$.

As shown in figure 7(h), in the ${\rm P}+{\rm S}$ mode, a vortex pair sheds from the upper side while only a single vortex sheds from the lower side of the prism. The counter-rotating vortex pair have different intensities, with the counterclockwise (red) vortex being much weaker. Besides, the secondary vortex street is also observed in the wake owing to the vortex coalescence (see figure 8d). As shown in figure 7(g), the variation of $C_{L,V}$ in one vibration period is similar to that of the 2S mode at $\alpha =35^\circ$ and $U^*=5.5$, i.e. negative $\rightarrow$ positive $C_{L,V}$ when the prism moves from one side to the other, suggesting a similar role of the vortices in the prism vibration.

A hybrid mode where the constituent patterns appear is observed in the transition response. Figure 9 shows the time histories of the lift coefficient ($C_L$ and $C_{L,V}$), the displacement ($y/D$ and $y^\prime /D$) and vorticity contours at different instants with $\alpha =160^\circ$ and $U^*=9.0$. We can see that when the prism moves from the top to the bottom (instant $i$ $\rightarrow$ $iii$), because of the cutoff effects of the lower side shear layer, one small vortex (vortex 1) is shed from the upper side of the prism (figure 9b–d). Positive net energy is transferred from the fluid to the prism (figure 9a), resulting in the augmented vibration amplitude at instant $iii$. However, when the prism moves back from the bottom to the top (figure 9d–f), two vortices (vortices 2 and 3), one from the upper side and the other from the lower side of the prism, are formed at $x/D \approx 5.0$. As the vortices move downstream, the vortex from the lower side separates into two parts (vortices $3_1$ and $3_2$) with different intensities. Thus, in this period, the wake is ${\rm P}+{\rm S}$ mode in the near wake but changes to ${\rm T}+{\rm S}$ mode farther downstream. From instant $iii$ to $v$, the prism continuously extracts energy from the fluid, and the vibration amplitude is further increased.

Figure 9.

Time histories of the lift coefficient ($C_L$ and $C_{L,V}$), the displacement ($y/D$ and $y^\prime /D$) and vorticity contours at different instants marked in (a) for the hybrid mode (P + S/P) at ($\alpha$, $U^*$) = ($160^\circ$, 9.0). The normalized Z-vorticity is in the range [$-$1, 1] with an increment of 0.1.

In the following period, the vortex shedding becomes different. When the prism moves from the top to the bottom (figure 9f–h), two vortices (vortices 4 and 5), one from the top and the other from the bottom, are shed from the prism, resulting in a vortex pair in the wake. When the prism moves back from the bottom to the top (figure 9h–j), no vortex sheds. Therefore, the P mode appears. In this period (instant $v$ $\rightarrow$ $ix$), the variation of $C_{L,V}$ in the first half-cycle (instant $v \rightarrow vii$) is different from that in the second half-cycle, which is consistent with different shapes and intensities of the vortices. In the first half-cycle, the energy is positive around the top and bottom while negative near the balanced position. However, in the second half-cycle, the energy is approximately negative when the prism is below the balanced position while positive when the prism is above. This is similar to the energy-in and -out mechanisms of the D-section prism at ($\alpha$, $U^*$) = ($10^\circ$, 6.0) where the total lift has a frequency twice the displacement, as happened from instant $v$ to $ix$. However, the vortex shedding pattern is distinct owing to the varying vibration amplitude and the vortex shedding history. When the prism moves from the top to the bottom in the subsequent period (figure 9j–l), another vortex pair forms (vortices 6 and 7). The energy transfer within the time span $ix \rightarrow xi$ is similar to that within the time span $v \rightarrow vii$, despite that the second positive $C_{L,V}$ peak is tiny.

Now, we see that the difference of the vortex shedding modes (instant $i \rightarrow xi$) is attributed to the amplitude variation. In the first period, due to a smaller amplitude, the ${\rm P}+{\rm S}$ mode exists from instant $i$ to $v$. In the following period, the amplitude increases, and the P mode dominates.

Figure 10 shows the time evolution of the 2P mode at ($\alpha, U^*$) = ($50^\circ, 7.0$) and the ${\rm T}+{\rm S}$ mode at ($\alpha, U^*$) = ($55^\circ, 5.5$). Figure 10(a–e) shows that when the prism moves upward from the bottom, a paired vortex is shed from the lower and upper sides, respectively. Similarly, when the prism moves downward from the top, another vortex pair forms in the wake. In each vortex pair, the strength of one vortex is weaker than that of the other. In addition, because of the asymmetric shape of the D-section prism, the 2P mode also becomes asymmetric to the incoming flow. Interestingly, the 2P mode changes to the ${\rm T}+{\rm S}$ mode if $\alpha$ and $U^*$ are slightly altered. As shown in figure 10( f–j), in the ${\rm T}+{\rm S}$ mode, a vortex triplet (vortices 1, 2 and 3) and a single vortex (vortex 4) are shed from each side of the prism when the prism moves upward and downward, respectively. This is caused by the altered vortex shedding timing when $\alpha$ and $U^*$ are changed, resulting in different vortex pairings. Of note is that although the vibration amplitudes corresponding to the two modes are smaller than the critical value for the 2P mode of a circular cylinder (Williamson & Roshko Reference Williamson and Roshko1988; Williamson & Govardhan Reference Williamson and Govardhan2004), two vortices are shed from each side of the prism. This is mainly caused by the difference in the cross-sectional shape. Comparing figures 10(a) and 10( f), we can see that although four vortices are shed in one vibration period, the variations of $C_{L,V}$ are significantly different in the two cases. In the 2P mode, because of the appearance of higher harmonics, the $C_{L,V}$ variation is rather irregular while in the ${\rm T}+{\rm S}$ mode the $C_{L,V}$ variation is approximately sinusoidal.

Figure 10.

Time histories of the lift coefficient ($C_L$ and $C_{L,V}$), the displacement ($y/D$ and $y^\prime /D$) and vorticity contours at different instants for (a–e) the 2P mode at ($\alpha$, $U^*$) = ($50^\circ$, 7.0) and ( f–j) the ${\rm T}+{\rm S}$ mode at ($\alpha$, $U^*$) = ($55^\circ$, 5.5). The normalized Z-vorticity is in the range [$-$1, 1] with an increment of 0.1.

For elucidating the flow physics of the $m{\rm S}+n{\rm S}$ mode ($m + n \ge 5$, $m$ and $n$ are integers) that appears in the galloping region, the $4{\rm S}+4{\rm S}$ vortex evolutions at $\alpha = 175^\circ$ and $U^* = 18.0$ are explored. As shown in figure 11(a), the displacement of the prism has a much longer period than the lift, indicating that galloping response and wake unsteadiness are not synchronized. Upon closer inspection, we can see that the displacement is not symmetric with respect to the mean position, and at the same time, the descending half-cycle has a longer period than the ascending half. Figure 11(b) shows that although there are four frequencies in the displacement spectrum, the dominant one, i.e. $f_vD/U_\infty = 0.0516$, has a much larger amplitude than the others. In the lift spectrum, the dominant lift frequency is $f_LD/U_\infty = 0.2064$, which is four times larger than the dominant vibration frequency.

Figure 11.

(a) Time histories of the lift coefficient ($C_L$ and $C_{L,V}$) and the displacement ($y/D$ and $y^\prime /D$), (b) the lift and displacement spectra (PSD, power spectral density) and (c–j) vorticity contours of different instants for the $4{\rm S}+4{\rm S}$ mode at ($\alpha$, $U^*$) = ($175^\circ$, 18.0). The normalized Z-vorticity is in the range [$-$1, 1] with an increment of 0.1. The green lines in ($i$) indicate the V-like arrangement of the vortices.

As shown in figure 11(c–j), when the prism moves from the top to the bottom, four vortices (vortices 1, 2, 3 and 4), with two from each side, are shed from the prism in the first half (instant $i \rightarrow v$). Similarly, when the prism moves upward, another four vortices (vortices $1'$, $2'$, $3'$ and $4'$) are shed in the wake (figure 11g–j). Associated with the vortex shedding, $C_{L,V}$ varies between positive and negative values alternately. As shown in figure 11(a), the energy is transferred towards the prism when the vortices develop at the lower side of the prism (instants $ii$ and $iv$) as the prism moves from the top to the bottom, and vice versa (instant $iii$). This suggests that the prism vibration is maintained by the pressure difference caused by the attached vortex/shear layer. That is, as the prism moves downward, the development of the lower side vortex leads to a lower pressure at the lower side, promoting the prism vibration. However, when the upper side vortex saturates, the pressure on the upper side becomes lower than that on the lower side, thus dampening the prism vibration. As the prism moves from the bottom to the top, less significant fluctuations are found in $C_L$ and $C_{L,V}$.

The vortices shed in a vibration period are arranged in a ‘V’ shape (see figure 11i). According to the number of vortices shed from the prism in one vibration period, this mode is named the $4{\rm S}+4{\rm S}$ mode. Note that the terminology adopted for the galloping wake mode differs from that in Seyed-Aghazadeh et al. (Reference Seyed-Aghazadeh, Carlson and Modarres-Sadeghi2017). In the present study, the number of vortices in each half-period are collected as $m$ and $n$, while in Seyed-Aghazadeh et al. (Reference Seyed-Aghazadeh, Carlson and Modarres-Sadeghi2017), $m$ and $n$ are defined as the numbers of vortices shed from each side of the prism in one period. It is of note that when the prism passes through the mean position, the vortices (vortices 3, 4, $3'$ and $4'$) shed from the prism are weaker than those (vortices 1, 2, $1'$ and $2'$) shed at the highest/lowest position. When the prism passes through the mean position, the prism motion is faster, and the time for vortex development is thus shorter. Moreover, compared with the four vortices shed in the descending half-cycle, the four vortices shed in the ascending half are slightly smaller and have lower intensities, which is consistent with the shorter period of the ascending motion and the lower $C_L$ and $C_{L,V}$ amplitude in the ascending half-period (figure 11a).

For the $U^*$ cases where the vibration is aperiodic, the number of the vortices shed from the prism in each half-period may vary from period to period. To elucidate this, the case of $\alpha = 170^\circ$ and $U^* = 19.0$ is selected for further investigation. As shown in figure 12(a), the number of vortices in each half-period varies. Generally, we can see that the number in the descending half-period is larger than that in the ascending half-period as a consequence of the longer descending half-period. Furthermore, from the displacement and lift spectra shown in figure 12(b), there is only one dominant vibration frequency, i.e. $f_vD/U_\infty = 0.0488$, corresponding to the natural frequency ($f_n$) of the prism, while multiple lift frequencies appear, with the smallest peak ($f_LD/U_\infty = 0.0076$) being a beating frequency ($f_b$) generated due to nonlinear interaction. Other lift frequencies are the linear superpositions of the two frequencies ($f_n = 0.0488$ and $f_b = 0.0076$), such as $0.0891 \approx 2f_n-f_b$, $0.0966 \approx 2f_n$, $0.1379 \approx 3f_n-f_b$, $0.1455 \approx 3f_n$ and $0.1866 \approx 4f_n-f_b$. The combination of these incommensurate spectral frequencies is closely related to the irregular vortex structures (Bourguet Reference Bourguet2020).

Figure 12.

(a) Time histories of the prism displacement and (b) the displacement and lift spectra (PSD, power spectral density) at ($\alpha$, $U^*$) = ($170^\circ$, 19.0). The number of vortices shed in a half-period is given in (a).

4.3. Dependence of wake modes on $A^*$ and $f_v/f_n$

It is generally believed that the vortex shedding pattern strongly depends on the vibration amplitude. Leontini et al. (Reference Leontini, Stewart, Thompson and Hourigan2006a) stated that the 2S mode dominates when $A^*$ is smaller than 0.85. Beyond that, ${\rm P}+{\rm S}$ and 2P modes appear. However, from the above discussions, we can see that the wake mode is also closely related to the vibration frequency $f_v$. For example, more vortices are shed from the prism in a vibration cycle at ($\alpha$, $U^*$) $=$ ($75^\circ$, 20) with a smaller $A^*$ and a lower $f_v$ than at ($\alpha$, $U^*$) $=$ ($180^\circ$, 20) with a larger $A^*$ and a higher $f_v$. In this section, the dependence of the wake modes on $A^*$ and $f_v/f_n$ is further explored.

As shown in figure 13(a), the 2S mode appears when $A^* < 0.85$, which is consistent with the observation in Leontini et al. (Reference Leontini, Stewart, Thompson and Hourigan2006a). The variation of the envelope generally follows the $U^*$–$A^*$ curves in the typical and narrowed VIV regimes with the maximum $A^*$ at $f_v/f_n = 0.8$. The P mode is observed at $f_v/f_n \approx 1.0$ with a small $A^*$ ($= 0.30$), as shown in figure 13(b). Actually, the P mode occurring at ($\alpha$, $U^*$) = ($45^\circ$, 7.5) can be taken as a highly deformed 2S mode due to the asymmetric configuration of the prism. As shown in figure 13(c), the ${\rm P}+{\rm S}$ mode is distributed along two $f_v/f_n$ lines: $A^*$ increases from 0.2 to 0.6 at $f_v/f_n \approx 0.5$ while from 0.5 to 1.6 at $f_v/f_n \approx 1.0$, which indicates the significant effects of the vibration frequency in determining the vortex shedding pattern. Moreover, it is considered that the ${\rm P}+{\rm S}$ mode cannot be provoked at a vibration amplitude as low as $A^* = 0.2$ for the VIV of a circular cylinder. However, due to the asymmetric vibration of the prism, the vortex shedding timing can be markedly changed which results in the ${\rm P}+{\rm S}$ mode. Compared with the ${\rm P}+{\rm S}$ mode, the 2P mode appears at a larger $A^*$, as shown in figure 13(d). When $A^* < 0.9$, the 2P mode generally appears at two lines, approaching each other with increasing $A^*$. However, beyond $A^* = 0.9$, the 2P mode occurs only around $f_v/f_n \approx 1.0$ with the largest $A^* = 1.85$. A further increase in the vibration amplitude will change the 2P mode into the $m{\rm S} + n{\rm S}$ mode, with more vortices shed in a period. This is consistent with the experimental results of Jauvtis & Williamson (Reference Jauvtis and Williamson2004) that the 2T mode (defined as the $3{\rm S}+3{\rm S}$ mode in this study) was found in the super-upper branch ($A^* \sim 1.5$) for a two-degree-of-freedom circular cylinder. Note that there exists an overlapped region ($1.55 < A^* < 1.85$) between the 2P and $m{\rm S} + n{\rm S}$ modes, corresponding to the critical $A^*$ for the mode transition. Obviously, the critical $A^*$ shows apparent dependency on $\alpha$. Apart from the linear region at $f_v/f_n \approx 1.0$, the $m{\rm S} + n{\rm S}$ mode is also found at $f_v/f_n \approx 0.6$ with a smaller $A^*$. This further demonstrated the key roles of the vibration frequency. As shown in figure 13(e), ${\rm T}+{\rm S}$ mode is observed at $A^* \approx 0.52\text {--}0.7$ and $f_v/f_n \approx 0.7$, and it can be taken as a deformed 2P mode due to the asymmetric configuration.

Figure 13.

Dependence of the wake modes on $A^*$ and $f_v/f_n$. Here, $m{\rm S}+n{\rm S}$ mode ($m$ + $n$ $\ge$ 5, $m$ and $n$ are integers) represents the wake mode where $m$ and $n$ vortices are shed from the prism in each half-period.

5.1. The statistics of fluid forces

The statistics of fluid forces varying with $\alpha$ and $U^*$ are shown in figure 14. Generally speaking, the time-mean drag ($\bar {C}_{D}$) and lift ($\bar {C}_{L}$) coefficients are significantly affected by $\alpha$ but less by $U^*$, judging from the horizontal contour lines in the plots. However, two regions at a similar $U^*$ range ($\approx 4.0\text {--}8.0$) but with different $\alpha$ values show apparent dependency with the reduced velocity. The one with a lower $\alpha$ ($\approx 0^\circ \text {--}35^\circ$) is mainly within the typical VIV region while the other with a higher $\alpha$ ($\approx 150^\circ \text {--}180^\circ$) extends over the transition and galloping regions and even part of the narrowed VIV region for $\bar {C}_{L}$. The smallest $\bar {C}_{D}$ $(\approx 0.7)$ exists within a horizontal stripe near $\alpha = 90^\circ$, owing to the smallest projection width of the prism. Moreover, the variation of $\bar {C}_{D}$ in this stripe is negligible, which coincides with the very small vibration amplitudes near $\alpha = 90^\circ$ (figure 3g). The mean drag coefficient achieves a local peak $\bar {C}_{D} \approx 2.4$ at $U^* = 5.5$ and $\alpha = 0^\circ$, due to the drag amplification effects of the enlarged vibration amplitude in the typical VIV regime (figure 3a). However, the local peak $\bar {C}_{D} \approx 3.2$ at $U^* = 5.0$ and $\alpha = 180^\circ$ suggests that other factors influence the mean drag in the galloping regime. The distribution of $\bar {C}_{L}$ shows two regions, i.e. the positive $\bar {C}_{L}$ region at $\alpha \le 95^\circ$ and the negative region at $\alpha > 95^\circ$, with a sharp transition in the middle. The sign of $\bar {C}_{L}$ is mainly determined by the AoA of the prism. That is, when $\alpha \le 95^\circ$ the shear layer separating at the upstream corner imposes an upward lift to the prism. Also, the pressure on the upstream part of the curved surface pushes the prism in the positive $y$ direction. However, when $\alpha > 95^\circ$ the shear layer separates at the downstream corner and the pressure on the flat surface generates a large force in the negative $y$ direction. The non-vanishing $\bar {C}_{L}$ is clearly reflected by the asymmetric wake modes, such as the P mode and the ${\rm P}+{\rm S}$ mode, which introduce unbalanced lift through biased vortex shedding (see figure 8c) or non-identical vortex strengths at the two sides of the prism (see figure 8d). The largest negative $\bar {C}_{L}$ is obtained at $\alpha \approx 120^\circ \text {--}150^\circ$, while the largest positive $\bar {C}_{L}$ exists in two regions, one at $\alpha \approx 80^\circ \text {--}90^\circ$ and the other at $U^* = 5.5$ and $\alpha = 30^\circ$. Comparing the absolute values of $\bar {C}_{L}$ in the parametric plane, we found that the largest negative $\bar {C}_{L}$ is much larger (approximately four times) than the largest positive one.

Figure 14.

Dependence of force coefficients on $U^*$ ($= 2\text {--}20$) and $\alpha$ ($= 0^\circ \text {--}180^\circ$): (a) $\bar {C}_{D}$, (b) $C_{D}^{\prime }$, (c) $\bar {C}_{L}$ and (d) $C_{L}^{\prime }$. The same as in figure 2, different regimes are divided by the dashed (horizontal) lines.

The distributions of the r.m.s. drag ($C_{D}^{\prime }$) and lift ($C_{L}^{\prime }$) coefficients show a clear dependency on both $\alpha$ and $U^*$. Small values are observed in the majority of the parametric plane while large values are found in two separated regions for both $C_{D}^{\prime }$ and $C_{L}^{\prime }$. These two regions match well with those where the vibration amplitude of the prism shows an ascending trend in figure 2(a). However, where the vibration amplitude shows a descending trend or keeps invariant, the values of $C_{D}^{\prime }$ and $C_{L}^{\prime }$ are small. For example, in the narrowed VIV regime, the vibration amplitude peaks at $U^*$ ($\approx 2.5\text {--}4.25$), which linearly increases with increasing $\alpha$, as shown in figure 3(g). Similarly, the distribution of $C_{D}^{\prime }$ and $C_{L}^{\prime }$ shows a linear ridge in the region, as shown in figure 14(b,d). The largest $C_{D}^{\prime }$ $(\approx 1.3)$ is achieved at $U^* = 5.0$ and $\alpha = 180^\circ$ while the largest $C_{L}^{\prime }$ is achieved at $U^* = 3.4$ and $\alpha = 180^\circ$.

5.2. The spectral analysis of fluid forces

The dominant drag and lift frequencies at each $\alpha$ and $U^*$ case are shown in figure 15. Clearly, in the typical and narrowed VIV regimes, the normalized lift frequency ($f_LD/U_\infty$) is approximately identical to the non-dimensional vortex shedding frequency ($St$) of the stationary prism but shows less dependency on $U^*$. It is of note that the normalized lift frequency is higher in the narrowed VIV regime than in the typical VIV regime, and gradually decreases with increasing $\alpha$ in the range $90^\circ < \alpha < 150^\circ$, because the frequencies are normalized using the diameter $D$ instead of the effective diameter $D_e$. Similar to the vibration frequencies, the lift frequencies for the VIV-like regimes at different $\alpha$ collapse in general if $D_e$ is taken as the reference length scale. However, the corresponding results are not shown for the sake of the conciseness of the discussion. In the majority of the typical VIV regime, the normalized drag frequency is two times the normalized lift frequency, as in the VIV of an isolated circular cylinder. However, in the narrowed VIV regime, the normalized drag frequency equals the normalized lift frequency. This is because in the narrowed VIV regime, the asymmetry of the prism to the incoming flow is more obvious, and the resultant asymmetric vortex shedding strength leads to the dominant drag frequency being halved (Chen et al. Reference Chen, Ji, Xu and Williams2019b, Reference Chen, Ji, Alam, Xu and ZHANG2022). For the extended VIV, combined VIV and galloping, and pure galloping regimes, the normalized force frequencies are dependent on $U^*$. In the lock-in $U^*$ range of the extended VIV, the normalized force frequencies decrease with increasing $U^*$ and jump back to $St$ out of the lock-in region. The normalized drag and lift frequencies are virtually identical due to the reason discussed above. In the combined VIV and galloping regime, the normalized frequencies are smaller than those in the pure galloping regime due to the adopted reference length scale; while in the transition regime, the variations of the normalized drag and lift frequencies are irregular due to the occurrence of multiple frequency components.

Figure 15.

Dependence of (a) drag ($f_{D} D / U_{\infty }$) and (b) lift ($f_{L} D / U_{\infty }$) frequencies on $U^*$ ($= 2\text {--}20$) and $\alpha$ ($= 0^\circ \text {--}180^\circ$). The same as in figure 2, different regimes are divided by the dashed (horizontal) lines.

Figures 16 and 17 show the drag and lift spectra for each regime at selected $\alpha$ values. In the typical VIV regime, for most $U^*$ cases at $\alpha = 10^\circ$, the drag has three dominant frequencies, i.e. the fundamental frequency and its second and third harmonics, with the second harmonic having a larger amplitude than the fundamental (figure 16a). At $U^* = 3.5$, three comparable but incommensurate frequencies are observed, consistent with the presence of a beating-type vibration, as shown in figure 5(a). For the lift spectra, except for $U^* = 3.5$ and 6.0, one conspicuous frequency emerges (figure 17a). Upon closer inspection, the second and third harmonics coexist at $U^* = 6.0$, with the amplitude of the former being larger than that of the fundamental frequency. The dominance of the higher harmonic contributes to the phase between the lift and displacement jumping from $0^\circ$ to $180^\circ$ (Prasanth & Mittal Reference Prasanth and Mittal2008; Chen et al. Reference Chen, Ji, Xu and Williams2019b). Comparing figures 16(a) and 17(a), the dominant drag frequency doubles the dominant lift frequency, except at $U^* = 3.5$ and 6.0. The figure-of-eight phase portrait of the drag and lift coefficients shown in figure 18(a–d) sustains this spectral feature.

Figure 16.

The drag spectra of FIV of a D-section prism. (a) Typical VIV at $\alpha = 10^\circ$, (b) extended VIV at $\alpha = 50^\circ$, (c) combined VIV and galloping at $\alpha = 80^\circ$, (d) narrowed VIV at $\alpha = 125^\circ$, (e) transition response at $\alpha = 155^\circ$ and ( f) pure galloping at $\alpha = 170^\circ$, representing the six types of response. The dash-dotted line in each panel denotes the vortex shedding frequency ($St$) of a stationary D-section prism at the same AoA.

Figure 17.

The lift spectra of FIV of a D-section prism. (a) Typical VIV at $\alpha = 10^\circ$, (b) extended VIV at $\alpha = 50^\circ$, (c) combined VIV and galloping at $\alpha = 80^\circ$, (d) narrowed VIV at $\alpha = 125^\circ$, (e) transition response at $\alpha = 155^\circ$ and ( f) pure galloping at $\alpha = 170^\circ$, representing the six types of response. The dash-dotted line in each panel denotes the vortex shedding frequency ($St$) of a stationary D-section prism at the same AoA. In ( f), the horizontal dashed lines indicate the fundamental lift frequency and its second and third harmonics, respectively, while the broken dashed line indicates the vortex shedding frequency.

Figure 18.

The phase portraits of the drag and lift coefficients. (a–d) Typical VIV at $\alpha = 10^\circ$, (e–h) extended VIV at $\alpha = 50^\circ$, (i–l) combined VIV and galloping at $\alpha = 80^\circ$, (m–p) narrowed VIV at $\alpha = 125^\circ$, (q–t) transition response at $\alpha = 155^\circ$ and (u–x) pure galloping at $\alpha = 170^\circ$, representing the six types of response.

In the extended VIV regime, the drag coefficient has multiple frequencies at $\alpha = 50^\circ$; see figure 16(b). Generally, with increasing $U^*$, the second and the third harmonics are intensified compared to the fundamental one. When $U^* > 11.0$, the second harmonic becomes dominant, while the fundamental and third harmonic disappear gradually with increasing $U^*$. Similar to the drag coefficient, the lift coefficient also consists of multiple components, with the fundamental one dominating until $U^* > 7.5$ (figure 17b). Consistent with the identical dominant drag and lift frequencies, periodic ‘o’-shaped or like-‘o’-shaped trajectories appear for most $U^*$ cases (figure 18e,f,h); an intersected shape is observed at $U^* = 8.5$ due to the dominance of the second harmonic in the drag and the fundamental one in the lift (figure 18g).

In the combined VIV and galloping regime, the drag coefficient at $\alpha = 80^\circ$ is dominated by the fundamental frequency at $U^* \le 7.0$ and by a much lower frequency once $U^* > 7.0$ (see figure 16c), corresponding to the VIV and galloping branches, respectively. With increasing $U^*$, the dominant galloping frequency decreases slightly due to the amplified vibration amplitude, while the VIV frequency remains invariant. As shown in figure 17(c), the lift spectra show similar features to the drag spectra. The phase portrait is ‘o’-shaped in the VIV branch (figure 18i,j) while more like a knitted fabric in the galloping branch owing to multiple frequencies in the drag and lift (figure 18k,l).

In the narrowed VIV regime, the drag spectra show two significant frequencies in most $U^*$ cases at $\alpha = 125^\circ$, with the fundamental frequency having a larger amplitude than the second harmonic. In contrast, the lift spectra only show the fundamental frequency (figures 16d and 17d). The phase portrait shows a slanted figure-of-eight shape (figure 18o,p). At $U^* = 3.0$, a very low frequency dominates the drag and the phase portrait is not closed (figure 18n); at $U^* = 2.0\text {--}2.5$, the second harmonic becomes insignificant and the ‘o’-shaped trajectory emerges (figure 18m).

In the transition response regime, the features of the drag and lift spectra are different in the VIV and galloping-like branches. As shown in figures 16(e) and 17(e), in the first VIV branch ($2.0 \le U^* \le 9.0$), both the drag and lift have multiple frequencies, while in the second VIV branch ($11.5 \le U^* \le 20.0$), the drag has two frequencies and the lift has one. The trajectory in the first VIV branch is complex but periodic (figure 18q,r), while being figure-of-eight-shaped in the second VIV branch (figure 18t). In the galloping branch ($9.0 < U^* < 11.5$), because of the dominance of a low frequency in both the drag and lift, the trajectory becomes complex (figure 18s).

In the pure galloping regime, the drag has multiple frequencies, with the second harmonic being dominant at $U^* = 2.0$–$9.5$ (figure 16f). However, the lift is dominated by the fundamental frequency at $U^* = 2.0\text {--}6.0$ but by the second harmonic at $U^* = 6.5\text {--}9.5$ (figure 17f). After $U^* > 9.5$, the drag and lift spectra become rather irregular. As shown in figure 3(k), the vibration amplitude at $\alpha = 170^\circ$ shows a large kink at $U^* = 14.5\text {--}16.0$, which may be associated with the local occurrences of lock-in with harmonics (Bearman et al. Reference Bearman, Gartshore, Maull and Parkinson1987; Nemes et al. Reference Nemes, Zhao, Lo Jacono and Sheridan2012; Zhao et al. Reference Zhao, Leontini, Lo Jacono and Sheridan2014; Sharma et al. Reference Sharma, Garg and Bhardwaj2022). To clarify this, the fundamental lift or vibration frequency, and its second and third harmonics are shown by the horizontal dashed lines in figure 17( f). The broken dashed line in figure 17( f) indicates the vortex shedding frequency. It is seen that the vortex shedding frequency overlaps the third harmonic of the vibration frequency within the $U^*$ range where the kink exists. This suggests lock-in of vortex shedding with the third-harmonic frequency. As shown in figure 18(u–x), a figure-of-eight-shaped trajectory occurs at small $U^*$ and becomes complex but still periodic at intermediate $U^*$; at large $U^*$, the trajectory becomes chaotic.

5.3. The phase lag between lift and displacement

The phase lag between lift and displacement indicates vortex shedding timing relative to cylinder motion. Bourguet & Lo Jacono (Reference Bourguet and Lo Jacono2014) observed that the phase lag jumps from $0^\circ$ to $180^\circ$ when resonance ($f_v = f_n$) happens. In some situations, the phase jump divides vibration branches, such as the upper and lower branches of VIV of a circular cylinder in high-$Re$ flows (Williamson & Roshko Reference Williamson and Roshko1988; Williamson & Govardhan Reference Williamson and Govardhan2004). However, in low-$Re$ VIV, the vibration branch exhibits a continuous variation as the vibration frequency crosses the natural frequency, as reported in Leontini et al. (Reference Leontini, Thompson and Hourigan2006b) and Prasanth & Mittal (Reference Prasanth and Mittal2008). The phase jump does not happen for pure galloping because the vibration frequency is always smaller than the natural frequency.

Shown in figure 19 is the phase lag between the lift and the transverse displacement in the studied parametric space. It should be mentioned that in the cases with multiple frequencies in the lift, the phase lags are calculated using the lift component with the same displacement frequency. The phase lag can be either $0^\circ$ or $180^\circ$ depending on the combination of $\alpha$ and $U^*$, and a sharp transition from $0^\circ$ to $180^\circ$, matching $f_v = f_n$, happens. We can see that in the typical VIV regime, the phase jump occurs at a constant reduced velocity $U^* = 6.1$ approximately, while in the narrowed VIV regime, a mild increment from $U^* = 4.0$ to 6.0 is observed. The critical reduced velocity is close to the value of $1/St = 6.0$, with $St = 0.167$ being the non-dimensional vortex shedding frequency of a stationary circular cylinder at ${\textit {Re}} = 100$. The discrepancy, especially in the narrowed VIV regime, is related to the varying AoA, which alters the transverse width of the D-section prism. The insignificant variation of the critical $U^*$ with $\alpha$ is in accordance with the linearly increasing vibration frequency in the typical VIV and narrowed VIV regimes. However, in the extended VIV regime, the critical $U^*$ significantly increases from 4.5 to 18.6 with increasing $\alpha$, signifying that the vortex shedding of a vibrating D-section prism synchronizes to the vibration responses. Of note is that the phase jump at $\alpha = 55^\circ$ is slightly earlier than that at $\alpha = 50^\circ$, resulting in a small fluctuation in the curve of critical $U^*$. In the combined VIV and galloping regime, the phase lag remains at $0^\circ$ until $U^* = 20$, indicating that the lift is always in phase with the displacement in galloping. This is also true in the pure galloping regime where $f_v$ is always smaller than $f_n$. In the transition regime, the critical $U^*$ for the phase jump shows a sharp increase and after $\alpha \ge 160^\circ$, the phase jump disappears.

Figure 19.

Phase lags between lift and displacement in the FIV of a D-section prism at different $U^*$ and $\alpha$. The grey and lime regions denote that the phase lag is $0^\circ$ and $180^\circ$, respectively. Different types of responses are divided by the dashed lines. Regimes (i)–(vi) denote typical VIV, extended VIV, combined VIV and galloping, narrowed VIV, transition response and pure galloping, respectively.