*5.1. Vibration Analysis*

According to Figure 7, only the CIR and EGG cross sections produce reasonable power followed by BOX with a large difference (of around 50% lower power). The drag, lift, and moment ratio versus arm ratio are shown in Figure 8 for all sections along with the calculated power on a separate axis. These three sections show more power while the pivot is on the downstream side of the section as shown in Figure 8. However, the power is nearly zero for a BOX section pivoted on the upstream side (*l* > 0) regardless of the length of the arm. The opposite behavior is observed for DIA section: the power is nearly zero for a DIA section pivoted on the downstream side (*l* < 0) regardless of the length of the arm. The two remaining sections (RAU, TRI) show the lowest power with almost no effect of the arm length and the pivot location. The difference between the BOX and the rest of the sections is more clear by analyzing the vibration response shown in Figure 9. The vibration frequency (*fv*) is far away from the natural frequency while the pivot is at the downstream of the section but it gradually goes up and close to the natural frequency. Even though aeroelastic instability is expected to be responsible for oscillation in this kind of cross section, the lock-in phenomena seem to improve the oscillation for sections with round edges. A similar change is seen for the Strouhal number St (= *fsD*/*U*), where = *fs* is the predominant vortex shedding frequency), as shown in Figure 9. The Strouhal number is very low for the BOX section while pivoted on the downstream. It eventually increases by the arm length and converges to 0.13 but for the rest of the sections, the Strouhal number is close to 0.2 which is considered in the lock-in range.

The maximum power depends largely on the natural frequency of the system which is a function of the pivot location and spring stiffness in our setup. Arionfard and Nishi [5] found that for a circular cross section the drag force assists the motion by reducing the natural frequency when the pivot is located at the downstream side of the cylinder (*l* < 0) based on Equation (7). As the moment of inertia (*It*), flow velocity (*U*), spring stiffness (*K*), and the projected area of the sections are constant, *l AD*<sup>0</sup> ∝ *l CD* is responsible for changes in the natural frequency.

Note that the mathematical analysis provided in [5] is only valid for round shapes where the drag and lift coefficients are not a function of the angle of attack. This is with agreement with the results shown in Figure 8: The calculated power changes with *l CD* for CIR, EGG, and RAU shape while the calculated power for BOX, DIA, and TRI shapes shows less dependency to the drag coefficient or arm length.

**Figure 8.** The calculated drag, lift, and moment ratios versus arm ratio for each cross-section. The calculated power is shown on a second Y-axis.

**Figure 9.** (**a**) The frequency ratio verses the arm ratio for each cross section. (**b**) The Strouhal number versus the arm ratio for each cross section. Here, negative *Arm*/*D* represents a pivot point at the downstream of the cross section and positive *Arm*/*D* represents a pivot point at the upstream side of the cross section

#### *5.2. Vorticity Analysis*

The steady-state vorticity field for the cases with the highest performance is shown in Figure 10. For sufficient oscillation amplitudes, symmetrical shedding with 2S mode is triggered in all cases as expected due to the low Reynolds number. The 2S mode is associated with the initial branch [19] where two single vortices shed per cycle, one by the top shear layer and another one by the bottom shear layer. The vorticity field animations can be found in the Supplementary Videos S1–S6.

To compare the correlation length (which is a measure of the span-wise length, that the vortices remain in phase) for each section, the three-dimensional state of the wake for each simulated case is visualized in Figures 11 and 12. The wakes are extracted by using a threshold filter the way that the pressure lies within 10 to 100 Pascal for all cases. A few factors influence the correlation length in FIV, including the amplitude of vibration, aspect ratio, surface roughness and the Reynolds number [15]. Here, the Reynolds number and surface roughness are similar for all cases while the amplitude of vibration and aspect ratio (which is a function of geometry) are changing.

The correlation length is higher when the pivot is at the upstream side (*l* > 0) for all sections except for the BOX. It is well known that body motion at a frequency close to that of the natural vortex shedding has a strong organizing effect on the shedding wake, which is manifested by a sharp increase in the spanwise correlation of the flow and forces on the body. However, the increase in three dimensionality of the flow behind the BOX section contradicts this pattern. A similar increase in three dimensionality is observed for the TRI section as well, but it is due to smaller vibration amplitude for all lengths in this section. The formation of the vortex line for the cross sections with the highest calculated power is more evident in the animations provided in the online Supplementary Videos V1–V6.

**Figure 10.** The steady state vorticity field for all cases with highest performance. The pivot is located at the origin of the black axes and the arm length is shown on top-left of each figure.

**Figure 11.** The velocity field on the wake side of the cylinders. The wakes are extracted by using a threshold filter on the pressure within 10 to 100 Pascal.

**Figure 12.** The velocity field on the wake side of the cylinders (continued).

### **6. Conclusions**

3D numerical simulation of fluid-induced vibration has been reported for a series of cylinders with different cross sections including circular, rectangle, diamond, triangle, reuleaux, and egg shape. The cylinders are pivoted at distance from the centre to study the geometrical effect of the FIV performance and to compare the results with our previous experimental study. The cross-sectional area, moment of inertia, spring stiffness, inlet velocity, and damping coefficient are set to be similar for all cases to eliminate the effect of non-geometrical parameters. According to the results, the circular and egg shape cross sections are the most efficient shapes regardless of the pivot location followed by the box, diamond, reuleaux, and triangle shapes. The vorticity field shows that the 2S mode is triggered for all cases mainly due to the low Reynolds number; thus, the vibrations are expected to be in the initial branch. Moreover, 3D visualization of the wake for each section shows that the correlation length is higher for round shapes especially when the pivot is at the upstream side while for the shapes with sharp edges, the three-dimensionality of the wake is higher.

There are two major limitations in this study that could be addressed in future research. First, the domain size: even though a grid independency study is done for the circular cylinder and there is a good agreement with the experiment, similar results are not necessarily expected for other cross sections or arm lengths. This applies to the blockage ratio as well. It is assumed that the blockage has a similar effect on all cases if kept constant for all cross sections. Second, the Reynolds number: the results are compared to the experiments done with Reynolds number of around 2800 assuming both numerical and experimental tests are in the same flow regime (1000 ≥ *Reynolds* ≥ 3000). Moreover, the Reynolds number in this study is much smaller than that of actual operating conditions. Being aware of the limitations of this numerical study, we concluded that the hydrodynamic forces, displacement and calculated power of the cross sections are still comparable with each other if not to the experiment.

**Supplementary Materials:** The following are available online at https://www.mdpi.com/1996-107 3/14/4/1128/s1, Video V1: The velocity field on the wake side of the BOX (*l* = −2*D*), Video V2: The velocity field on the wake side of the CIR (*l* = −3*D*), Video V3: The velocity field on the wake side of the DIA (*l* = +2.5*D*), Video V4: The velocity field on the wake side of the EGG (*l* = −2.5*D*), Video V5: The velocity field on the wake side of the RAU (*l* = +3*D*), Video V6: The velocity field on the wake side of the TRI (*l* = +2.5*D*), Video S1: The vorticity field for the CIR (*l* = −3*D*), Video S2: The vorticity field for the DIA (*l* = +2.5*D*), Video S3: The vorticity field for the EGG (*l* = −2.5*D*), Video S4: The vorticity field for the RAU (*l* = +3*D*), Video S5: The vorticity field for the TRI (*l* = +2.5*D*), Video S6: The vorticity field for the BOX (*l* = −2*D*).

**Author Contributions:** Formal analysis, S.M.; Investigation, H.A.; Visualization, H.A.; Writing original draft, S.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors wish to thank Fatemeh Talebi for her useful input and contribution. The calculations were carried out on supercomputer ABACUS 2.0 provided by the Southern Denmark University's eScience Center.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

