Vortex-induced vibration in a cylinder with an azimuthal degree of freedom

We present a numerical study on the motion of a rigid cylinder promoted by its interaction with an incoming flow when the cylinder is restricted to move along the arc of a circle and thus has only one degree of freedom. The analysis also includes the description of the flow around the cylinder, which is influenced by the motion of the cylinder. The Reynolds number, based on the diameter of the cylinder and the far-away incoming flow velocity, is 180. It is considered that the diameter of the circle is three times the diameter of the cylinder and that the ratio of the cylinder to fluid densities is 1.6. The mass and momentum conservation equations in two dimensions are solved in a Cartesian grid, and the presence and motion of the cylinder are resolved using the immersed boundary method. The motion of the cylinder and the flow are two-way coupled in the sense that the aerodynamic forces that drive the displacement of the cylinder result from the fluid-solid interaction, and in turn, the flow around the cylinder is modified by its motion. We analyze the initial transient dynamics and long-time behavior for two different cases. The present results illustrate that the motion of the cylinder and the torque are quasiperiodic with cycles composed of three oscillations with different amplitudes. The stagnation point and the boundary layers are displaced periodically around the rim of the cylinder according to the incoming direction of its relative velocity with respect to the fluid. The upper and lower separation points undergo similar periodic angular displacements. This effect is superposed to the alternating vortex shedding mechanism that occurs in fixed cylinders. Two rows of alternating vortices similar to von Karman vortex street are formed downstream of the cylinder, but their centers are farther apart from the symmetry line than those generated in the wake of a fixed cylinder; this effect is closely related to the coupling between the oscillatory motion of the cylinder and the vortex shedding process. The pressure field and the instantaneous streamlines are also presented and related to the dynamical features of the cylinder motion. It is found that the pressure is the dominant effect on the dynamics of the cylinder and its magnitude and the direction of the total force on the cylinder is related to the motion of the cylinder and the genesis and emission of vortices.

Figure 1

Sketch of the physical situation analyzed. The cylinder diameter is $d$, and the diameter of the circle where the cylinder moves is $D=3d$. The incoming flow moves from left to right, and upstream of the cylinder, the fluid has a uniform velocity $\mathbf{U}$.

Figure 2

(a) Definition of the relative velocity of the cylinder ($\widehat{\mathbf{u}}$) for a given azimuthal position ($\theta$) and angular velocity ($\dot{\theta }$) and the angle between the relative velocity and the direction of the incoming flow (${\theta }^{*}$). (b) Definition of the tangential force ($\widehat{\mathbf{F}}$) and the drag and lift forces, ${\mathbf{F}}_{\mathbf{D}}$ and ${\mathbf{F}}_{\mathbf{L}}$, respectively.

Figure 3

Position of the center of the cylinder ($\theta$, black trace) and nondimensional torque ($T^{*}$, red trace) as functions of nondimensional time ($t^{*}$) for different initial conditions. (a) Case I; (b) Case II, $t_r^{*}=100t^{*}$. The two last columns contain amplifications of the initial and long-term behavior, respectively.

Figure 4

Fourier spectrum of the angular position, $H_{\theta }$, black line; torque, $H_{T^{*}}$, red line. The initial conditions used correspond to Case I.

Figure 5

Total ($F^{*}$), streamwise ($F_x^{*}$), and transverse ($F_y^{*}$) forces and torque ($T^{*}$) on the cylinder as functions of time.

Figure 6

Vibration analysis. Upper panel, azimuthal position of the cylinder $\theta$ (thin gray line) and its periodic reconstruction $\mathrm{\Theta }$ (thick blue line), Eq. (18). Lower panel, individual waves in Eq. (18).

Figure 7

Upper figures: Three-dimensional attractor in space ${\theta }_n,{\theta }_{n+n_0},{\theta }_{n+2n_0}$ with delay time $n_o=1/f_{\theta 3}^{*}$ for Case I. Lower figures: Poincare maps for Cases I and II. The black thick lines in the upper figures and the black dots in the lower figures represent the simplified expression given in Eq. (18).

Figure 8

Upper panel: Velocity components $u^{*}$ and $v^{*}$ of the fluid for Case I at a position 4.5 diameters downstream from the center of the cylinder and $\theta =0$ as functions of time. Lower panel: Fourier spectra of $u^{*}$ and $v^{*}$ signals.

Figure 9

Upper panel, angular position, and torque as functions of time. Lower panel, snapshots of vorticity field. The times at which the vorticity fields were obtained are indicated in the upper panel.

Figure 10

Vorticity field in the surrounding area of the fixed cylinder. Upper panel: velocity components and torque as functions of time. Lower panel: vorticity fields at times indicated in the upper panel.

Figure 11

Vorticity field in the surrounding area of the cylinder for $955<t^{*}<995$. For reference, see Figs. 3 and 9.

Figure 12

Accumulated vorticity (${\mathrm{\Omega }}_{⊺}^{*}$) as defined in Eq. (19) in a $0.1d$ width ring around the cylinder as a function of time. The azimuthal position of the cylinder and the torque are also shown for reference.

Figure 13

Trajectories of the vortices. The black line represents the trajectory of the first negative vortex that corresponds to the positive displacement of $\theta$ in the large swing (see point $a_1$ of top panel in Fig. 11 and Supplemental Material [17]). The red line corresponds to the trajectory of the next vortex (first positive). The blue, yellow, green, and cyan lines represent, respectively, the third to sixth vortices. The empty black dots labeled with capital $V^{\prime }\mathrm{s}$ denote the position of the center of the vortices at $t^{*}=980$ (see panel $i$ Fig. 9). The cylinder's central and extreme positions are indicated by black and gray large circles on the left of the figure.

Figure 14

Snapshots of pressure field. The cylinder's angular position and times where the fields were obtained are indicated in the upper panel.

Figure 15

Instantaneous streamlines near (${\mathrm{t}}^{*}=980$) when the cylinder is at a maximum displacement. Point $i$ in the upper panel. The decimal numbers in the labels are nondimensional time units.