Effect of aspect ratio on the unlimited flow-induced vibration of an elliptical cylinder-plate assembly

The transverse flow-induced vibration (FIV) of an elastically supported elliptical cylinder-plate assembly is investigated numerically for a laminar flow at a Reynolds number of 100. The aspect ratio (AR) of the elliptical cylinder is varied over a range of values (namely, $\text{AR}=0.5$, 0.67, 0.75, 1, 1.5, and 2). In addition, two normalized splitter-plate lengths, $L_{\text{SP}}/D=0.75$ and 2.5, are investigated (where $L_{\text{SP}}$ is the splitter-plate length, and $D$ is the equivalent diameter of the elliptical cylinder). A low mass ratio of 10 and zero structural damping are used in the numerical simulations to induce larger oscillations in the assembly. The numerical results show that all cases investigated exhibit a FIV over an unlimited range of reduced velocity. An increase in the AR promotes the vibrations of the assembly through a reduction in the reduced velocity associated with the onset of FIV and a concomitant increase in the vibration amplitude. In addition, a larger AR facilitates the transition from a pure galloping (for $\text{AR}\le 1$) to an integrated VIV-galloping response (for $\text{AR}>1$) for an assembly with $L_{\text{SP}}/D=0.75$. Moreover, a larger AR significantly decreases the onset velocity of galloping for an assembly with $L_{\text{SP}}/D=2.5$. The AR determines the nature and width of the synchronization branch in the amplitude response. In general, a larger AR leads to the inception of higher-order synchronization branches in the amplitude response and to the suppression of some branches (e.g., still and initial galloping branches) for assemblies with long splitter plates. Finally, with respect to the flow dynamics associated with an unlimited FIV, increasing AR promotes the shedding of more complex vortices in the wake of the assembly (e.g., the emergence of a tail-shaped vortex and a slender vortex)—despite this, the wake mode remains unaltered.

Figure 1

Definition of the aspect ratio for various cross-sectional body shapes.

Figure 2

An elastically mounted elliptical cylinder-plate assembly. The computational domain is partitioned into three regions, and the boundary conditions imposed in our numerical simulations are exhibited. The assembly is supported on a mass-spring-damper system that is constrained to oscillate only in the transverse (or $y$) direction.

Figure 3

Effect of the aspect ratio ($0.5\le \text{AR}\le 2.5$) on the flow-induced vibration (FIV) of an elliptical cylinder-plate assembly with two different splitter-plate lengths ($L_{\text{SP}}/D=0.75$ and 2.5). The following quantities of interest are shown: (a) the maximal transverse displacement $Y_{\text{max}}$, (b) the normalized dominant frequency $f_Y^{*}/f_n$ of the transverse displacement, (c) the root-mean-square lift coefficient $C_{L,\text{rms}}$, and (d) the mean drag coefficient $C_{D,\text{mean}}$ as a function of the reduced velocity $U_r$.

Figure 4

Branching behavior of an unlimited flow-induced vibration (FIV) response provoked on an elliptical cylinder-plate assembly with various aspect ratios (ARs) and a fixed splitter-plate length $L_{\text{SP}}/D=0.75$. The maximal transverse displacement $Y_{\text{max}}$ (left vertical axis) and the root-mean-square lift coefficient $C_{L,\text{rms}}$ (right vertical axis) are plotted as a function of the reduced velocity $U_r$ for ARs of (a) $0.75$, (b) 1, (c) $1.5$, and (d) 2. The synchronization branches are shaded in different colors. The kinks (discontinuity of the slope) in the amplitude response are delineated by the red boxes.

Figure 5

(a) The maximum transverse displacement $Y_{\text{max}}$ as a function of the reduced velocity $U_r$ for an elliptical cylinder-plate assembly with aspect ratios (ARs) of $0.75$–2 and a splitter-plate length $L_{\text{SP}}/D=0.75$. The normalized power spectral density isopleths (logarithmic scale) of $Y$ exhibited as a function of $f_Y/f_n$ and $U_r$ for ARs of (b) $0.75$, (c) 1, (d) $1.5$, and (e) 2. The vertical dashed lines delineate the boundaries of synchronization regimes; the horizontal dashed line corresponds to $f_Y/f_n=1$; the diagonal dashed lines represent the vortex-shedding frequency of the stationary assembly; and the solid lines with squares correspond to the phase difference $\varphi$ (${}^{\circ }$) between $Y$ and $C_L$.

Figure 6

(a) The maximum transverse displacement $Y_{\text{max}}$ and the normalized power spectral density isopleths of $C_L$ for an elliptical cylinder-plate assembly with a constant splitter-plate length of $L_{\text{SP}}/D=0.75$ and various aspect ratios (ARs) of (b) $0.5$, (c) 1, (d) $1.5$, and (e) 2. Other notations used here are the same as those described in the caption of Fig. 5.

Figure 7

Dynamical characteristics of the 1:1 synchronization branch (viz., lock-in) for an elliptical cylinder-plate assembly with aspect ratios (ARs) of (a) $\text{AR}=1.5$ and (b) $\text{AR}=2$ at $U_r=6$ and of the transition regime between the 1:1 and 1:2 synchronization branches for (c) $\text{AR}=2$ at $U_r=8$. The four columns of panels in (a)–(c) exhibit the time series $Y\left(t\right)$ and $C_L\left(t\right)$, the phase portraits of $Y\text{−}{Y}^{\prime }$ and $C_L\text{−}{C}_L^{\prime }$, the Lissajous figures $Y$–$C_L$, and the power spectra of $Y$ and $C_L$. In these plots, the results for $Y$ and $C_L$ are shown as the black and red curves, respectively. The evolution of the instantaneous vorticity field during one oscillation cycle with period $T$ is displayed for (d) $\text{AR}=1.5$ at $U_r=6$, (e) $\text{AR}=2$ at $U_r=6$, (f) $\text{AR}=2$ at $U_r=7$, and (g) $\text{AR}=2$ at $U_r=8$. The 2S mode is identified in the 1:1 synchronization branch and the quasi-2S mode in the transition regime.

Figure 8

Dynamical characteristics of the 1:3 synchronization branch for an elliptical cylinder-plate assembly with aspect ratios (ARs) of (a) $0.75$, (b) 1, (c) and (d) $1.5$, and (e) and (f) 2 at $U_r=16$ and 20. The notation used here is the same as that described in the caption of Fig. 7.

Figure 9

The temporal evolution of instantaneous vorticity field over one oscillation cycle with period $T$ in the 1:3 synchronization branch for an elliptical cylinder-plate assembly with aspect ratios (ARs) of (a) $0.75$, (b) 1, (c) $1.5$ and (d) and (e) 2 for $U_r=16$ and 20. The 1:3 synchronization is associated with a $3\times \left(2\mathrm{S}\right)$ mode.

Figure 10

Dynamical characteristics of the 1:5 synchronization branch for an elliptical cylinder-plate assembly with (a)–(c) aspect ratios $\text{AR}=1$, 1.5 and 2 at $U_r=29$ and 30. The notation used here is the same as that described in the caption of Fig. 7. The vortex-shedding pattern over one oscillation cycle with period $T$ is exhibited in (d) and (e) where a $5\times \left(2\mathrm{S}\right)$ wake mode is seen to be associated with the 1:5 synchronization branch.

Figure 11

Dynamical characteristics of the 1:2 synchronization branch for an elliptical cylinder-plate assembly with (a)–(c) aspect ratios $\text{AR}=1$, 1.5, and 2 at $U_r=11$. The transition regime between the 1:2 and 1:3 synchronizations for the assembly with (d) $\text{AR}=2$ at $U_r=14$. The notations used in (a)–(d) follow that given in the caption of Fig. 7. Temporal evolution of instantaneous vorticity field during one oscillation cycle for an assembly with (e)–(g) $\text{AR}=1$, 1.5, and 2 at $U_r=11$ and (h)–(i) $\text{AR}=2$ at $U_r=13$ and 14. The 1:2 synchronization branch supports a $2\times \left(2\mathrm{S}\right)$ wake mode and the transition regime supports a quasi-$2\times \left(2\mathrm{S}\right)$ wake mode.

Figure 12

Dynamical characteristics of the 1:4 synchronization branch for an elliptical cylinder-plate assembly with (a) aspect ratio $\text{AR}=1.5$ and (b) 2 at $U_r=24$. The notations used in (a) and (b) follow that given in the caption of Fig. 7. (c) and (d) Temporal evolution of instantaneous vorticity field during one oscillation cycle with a period $T$. The 1:4 synchronization branch supports a $4\times \left(2\mathrm{S}\right)$ wake mode.

Figure 13

The normalized power spectra of $C_L$ at reduced velocities in the range from 6 to 30 for an elliptical cylinder-plate assembly with (a)–(d) aspect ratios $\text{AR}=0.75$, 1, 1.5, and 2.

Figure 14

Dynamical characteristics in the nonsynchronization branches of an elliptical cylinder-plate assembly with $L_{\text{SP}}/D=0.75$ experiencing an unlimited flow-induced vibration (FIV). Each row of panels exhibits the results for the assembly for a given aspect ratio at a representative reduced velocity. The dynamical characteristics consist of (i) the time series $Y\left(t\right)$ (gray lines) and $C_L\left(t\right)$ (blue lines), (ii) phase portrait $C_L\text{−}{C}_L^{\prime }$, and (iii) Poincaré section. The red dashed lines delineate the amplitude envelope of $C_L$.

Figure 15

The instantaneous vorticity field over five consecutive oscillation cycles for an elliptical cylinder-plate assembly with $L_{\text{SP}}/D=0.75$ and aspect ratio $\text{AR}=1.5$ at $U_r=28$. This is an example of a period-5 oscillation.

Figure 16

The unlimited flow-induced vibration (FIV) response of an elliptical cylinder-plate assembly with $L_{\text{SP}}/D=2.5$ for (a) and (b) aspect ratios $\text{AR}=1$ and 1.5. The dynamical characteristics are expressed in terms of the branching behavior and the power spectral density (PSD) of $Y$ and $C_L$. Other notations used here are the same as those described in the captions of Figs. 4, 5, 6.

Figure 17

The branching behavior and wake mode for an elliptical cylinder-plate assembly with $L_{\text{SP}}/D=0.5$ in the [aspect ratio (AR), $U_r$] plane. The flow-induced vibration (FIV) response occurs over a limited reduced-velocity range.

Figure 18

The branching behavior and wake mode of an elliptical cylinder-plate assembly with $L_{\text{SP}}/D=0.75$ in the [aspect ratio (AR), $U_r$] plane. The flow-induced vibration (FIV) response occurs over an unlimited reduced-velocity range.