Three-dimensional direct numerical simulations of vortex-induced vibrations of a circular cylinder in proximity to a stationary wall

In this paper, we present three-dimensional direct numerical simulations (3D DNS) of vortex-induced vibrations of an elastically mounted circular cylinder near a stationary wall at a subcritical Reynolds number of 500. The circular cylinder can oscillate in both the streamwise and transverse directions. A typical gap of 0.$8D$ between the cylinder surface and the stationary wall was selected to evaluate the characteristics of the responses and vortex dynamics in the presence of a stationary wall and boundary layer. We observe that the vibration response is two-branched, with two desynchronization regions lying at the ends of the simulated $U_r$ range. Because of the wall proximity, the dominant vibration frequencies in the two directions are generally identical, and the trajectories of the displacement significantly differ in each branch. Figure-eight, combined figure-eight and raindrop, raindrop, and chaotic trajectories appear successively with increasing $U_r$, dominating the first desynchronization region, initial branch, lower branch (LB), and second desynchronization region (DS-II), respectively. In addition, the phase lag between the lift and displacement jumps from 0° to 180° at the transition between the LB and DS-II. Furthermore, we evaluated the three-dimensionality of the wake through the instantaneous vortical structures, spanwise-averaged vorticity contours, and statistics of the enstrophies. We found that the three-dimensionality increases linearly with the amplitude, leading to substantial variations in the vortex dynamics. The statistics of the gap flow indicates that the mean gap flow velocity is determined by only the time-averaged gap, whereas the fluctuating gap flow velocity is governed by the amplitude. Finally, the flow physics behind the response was analyzed using the time histories of the displacement, gap flow velocity, drag and lift coefficients, and vorticity contours in one vibration period. We observed that the interactions of the vortices with the wall-generated boundary layer play significant roles in altering the cylinder vibration. In the small-amplitude case, the boundary layer merges with the freestream-side vortex of the cylinder, forcing the gap-side vortex to pair with the freestream-side vortex. In contrast, in the large-amplitude case, the gap-side shear layer collides with the boundary layer and disintegrates into small parts, resulting in negative vortices in the wake.

Figure 1

(a) Computational domain and boundary conditions, (b) mesh settings in the $x\text{−}y$ plane, and (c) mesh around the cylinder for the near-wall VIV of a circular cylinder at $\mathrm{Re}=500$. The rectangular region (red) in (b) denotes the uniform mesh region. The center of the cylinder is at $\left(x,y\right)=\left(-3.5D,-1.7D\right)$.

Figure 2

Time histories of the streamwise and transverse displacements in the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$. Note that at $U_r=6.5$, $t^{*}$ ranges from 500 to 800.

Figure 3

Power spectral density (PSD) of the displacements in two directions for the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$ for (a) the streamwise displacement and (b) the transverse displacement.

Figure 4

The modes extracted from the $z$-vorticity field using DMD analysis at $U_r=3.0$ where the beating phenomenon is observed. (a) Mode 1 at $f=0.234$ and (b) mode 2 at $f=0.253$.

Figure 5

Variations in the amplitudes, vibration frequencies, and shift of time-averaged displacements in the VIV of a circular cylinder in proximity to a stationary wall with $U_r$ at $\mathrm{Re}=500$ and $G/D=0.8$.

Figure 6

Trajectories of the displacements in both directions in the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$. The phase lags between the displacements of both directions are presented. Due to the chaotic displacement at $U_r=7.0\text{–}9.0$, no phase lags were obtained.

Figure 7

Time histories of the drag and lift coefficients in the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$. Note that at $U_r=6.5$, $t^{*}$ ranges from 500 to 800.

Figure 8

Statistics of the drag and lift coefficients in the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$: (a) for the drag coefficient, (b) for the lift coefficient, and (c) the decomposition of the lift coefficient.

Figure 9

PSD results of the drag and lift coefficients in the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$ (a) for the drag coefficient and (b) for the lift coefficient.

Figure 10

Phase lag between the lift and displacement in the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$.

Figure 11

Instantaneous vortical structures represented by the isosurface at ${\lambda }_2=-1$ for the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$. Colors denote the nondimensional $Z$-vorticity in the range of −2 to 2 with an increment of 0.2, with blue and red representing the negative and positive $z$-vorticity, respectively. The left column shows the wake viewed from the bottom, while the right column shows the wake viewed from the top. The instant that the cylinder is at the highest position is selected for each $U_r$ case.

Figure 12

Spanwise-averaged $z$-vorticity contours in the VIV of a circular cylinder in proximity to a stationary wall at $U_r=1.0\text{–}9.0$. The instants are the same as those in Fig. 11 when the cylinder is at the highest position.

Figure 13

Statistics of the enstrophies in the VIV of a circular cylinder in proximity to a stationary wall with the reduced velocity and the vibration amplitude at $\mathrm{Re}=500$ and $G/D=0.8$.

Figure 14

Spanwise-averaged secondary enstrophy (${\varepsilon }_p$) contour in the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$. For each $U_r$ case, the instant that the cylinder is at the highest position is applied.

Figure 15

Time- and spanwise-averaged gap flow velocities in the VIV of a circular cylinder at $\mathrm{Re}=500$ and $G/D=0.8$: (a) mean streamwise velocity and (b) rms streamwise velocity with $U_r$; (c) mean streamwise velocity with the displacement shift; (d) rms streamwise velocity with the amplitude.

Figure 16

Streamwise velocities in the VIV of a circular cylinder in proximity to the stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$. The left and right columns represent the mean and fluctuating velocities, respectively.

Figure 17

Transverse velocities in the VIV of a circular cylinder in proximity to a stationary wall at $\mathrm{Re}=500$ and $G/D=0.8$. The left and right columns represent the mean and fluctuating velocities, respectively.

Figure 18

Time histories of (a), (b) the transverse displacement, spanwise-averaged gap flow velocity, drag, and lift coefficients, and (c)–(g) vorticity contours of one period in the VIV of a circular cylinder in proximity to the stationary wall at $U_r=1.0$ (DS-I). Because the vorticity contour at instant $vi$ is the same as that at instant $i$, it is not presented here. The color bar for the contour plots is the same as that of Fig. 12.

Figure 19

Time histories of (a), (b) the transverse displacement, spanwise-averaged gap flow velocity, drag, and lift coefficients, and (c)–(g) vorticity contours of one period in the VIV of a circular cylinder in proximity to a stationary wall at $U_r=3.0$ (IB). Instants from $i$ to $vi$ are marked in (a), (b) from left to right. Because the vorticity contour at instant $vi$ is similar to that at instant $i$, it is not presented here. The color bar for the contour plots is the same as that of Fig. 12.

Figure 20

Time histories of (a), (b) the transverse displacement, spanwise-averaged gap flow velocity, drag, and lift coefficients, and (c)–(g) vorticity contours of one period in the VIV of a circular cylinder in proximity to the stationary wall at $U_r=4.0$ (LB). Because the vorticity contour at instant $vii$ is the same as that at instant $i$, it is not presented here. The color bar for the contour plots is the same as that of Fig. 12.

Figure 21

Time histories of the (a), (b) transverse displacement, spanwise-averaged gap flow velocity, drag, and lift coefficients, and (c)–(g) vorticity contours of one period in the VIV of a circular cylinder in proximity to the stationary wall at $U_r=8.0$ (DS-II). Because the vorticity contour at instant $vi$ is similar to that at instant $i$, it is not presented here. The color bar for the contour plots is the same as that of Fig. 12.