Three-dimensional flow past a fixed or freely vibrating cylinder in the early turbulent regime

The three-dimensional structure of the flow downstream of a circular cylinder, either fixed or subjected to vortex-induced vibrations, is investigated by means of numerical simulation, at Reynolds number 3900, based on the cylinder diameter and current velocity. The flow exhibits pronounced fluctuations distributed along the span in all studied cases. Qualitatively, it is characterized by spanwise undulations of the shear layers separating from the body and the development of vortices elongated in the plane normal to its axis (planar vortices). A quantitative analysis of crossflow vorticity fluctuations in the spanwise direction reveals a peak of fluctuation amplitude in the near region (i.e., area of formation of the spanwise wake vortices) and opposite trends of the spanwise wavelength in the shear layer and wake regions; the wavelength tends to decrease as a function of the streamwise distance in the shear layers down to a minimum value close to 0.5 body diameters and then slowly increases further in the wake. The spanwise structure of the flow is differently altered in these two regions, once the cylinder vibrates. In the shear layer region, body motion is associated with an enhancement of planar vortex formation. The amplification of vorticity spanwise fluctuations in this region is accompanied by a reduction of the spanwise wavelength; it is found to decrease as a function of the instantaneous Reynolds number based on the instantaneous flow velocity seen by the moving body, following the global trend of the wavelength versus Reynolds number previously reported for fixed cylinders. In the wake region, the flow spanwise structure is essentially unaltered compared to the fixed body case, in spite of the major distortions of the streamwise and crossflow length scales.

Figure 1

Visualization of the three-dimensional flow downstream of a circular cylinder at $\text{Re}=3900$: instantaneous isosurface of the $Q$ criterion [31] $\left(Q=0.1\right)$ colored by isocontours of the streamwise vorticity nondimensionalized by the oncoming flow velocity and cylinder diameter $\left({\omega }_x\in [-1,1]\right)$ in the (a) fixed and (b) freely oscillating $\left(U^{*}=6\right)$ body cases. Arrows indicate the direction of the oncoming flow. In (b) the crescent-shaped trajectory of the oscillating cylinder is represented by a line at the end of the body.

Figure 2

Structural responses of an elastically mounted circular cylinder subjected to vortex-induced vibrations at $\text{Re}=3900$: (a) in-line and (b) crossflow oscillation amplitudes (nondimensionalized by the cylinder diameter) and (c) crossflow oscillation frequency normalized by the Strouhal frequency (vortex shedding frequency in the fixed body case), as functions of the reduced velocity. Circles indicate the three typical cases examined in the present work, in addition to the fixed body case.

Figure 3

Sketch of the physical configuration.

Figure 4

Schematic view of the spanwise lines $L_1$ and $L_2$ along which flow quantities are analyzed.

Figure 5

Flow patterns in the $(x,y)$ plane: instantaneous isocontours of the span-averaged, spanwise vorticity $\left({\omega }_z\in [-2,2]\right)$ in the (a) fixed and (b)–(d) oscillating body cases, for (b) $U^{*}=3$, (c) $U^{*}=6$, and (d) $U^{*}=9$. In each case, red dots indicate the positions of lines $L_2^{+}$ (defined in Sec. 2b). In (b)–(d), ${\zeta }_y=0$ and the body is moving downward.

Figure 6

Flow patterns in the $(x,y)$ plane: instantaneous isocontours of the span-averaged, planar vorticity magnitude $\left(|{\mathbf{\omega }}_{\mathit{p}}|\in [0,2]\right)$ in the (a) fixed and (b)–(d) oscillating body cases, for (b) $U^{*}=3$, (c) $U^{*}=6$, and (d) $U^{*}=9$. In (b)–(d), ${\zeta }_y=0$ and the body is moving downward.

Figure 7

Global visualization of the three-dimensional flow: instantaneous isosurface of the $Q$ criterion $\left(Q=0.1\right)$ colored by isocontours of the crossflow vorticity $\left({\omega }_y\in [-1,1]\right)$ in the (a) fixed and (b)–(d) oscillating body cases, for (b) $U^{*}=3$, (c) $U^{*}=6$, and (d) $U^{*}=9$. In (b)–(d), ${\zeta }_y=0$ and the body is moving downward.

Figure 8

Visualization of the three-dimensional flow close to the body: instantaneous isocontours of the spanwise vorticity $\left({\omega }_z\in [-10,10]\right)$ in the plane $z=0$ and isosurface of the $Q$ criterion $\left(Q=10\right)$ colored by isocontours of the crossflow vorticity $\left({\omega }_y\in [-1,1]\right)$ in the (a) fixed and (b)–(d) oscillating body cases, for (b) $U^{*}=3$, (c) $U^{*}=6$, and (d) $U^{*}=9$.

Figure 9

Visualization of the shear layers separating from the body, for $U^{*}=6$: (a), (c), and (e) isocontours of the span-averaged, spanwise vorticity $\left({\omega }_z\in [-2,2]\right)$ and (b), (d), and (f) three-dimensional upper shear layer colored by isocontours of surface elevation $\left(\xi \in [0.6,1.2]\right)$, at three instants during an oscillation cycle. In (a), (c), and (e), the trajectory, position, and velocity of the body are shown and red dots indicate the span-averaged location of the surface captured by the shear layer tracking method (described in Sec. 2b).

Figure 10

Three-dimensional shear layer separating from the body: upper shear layer colored by instantaneous isocontours of the crossflow vorticity $\left({\omega }_{\xi }\in [-1,1]\right)$ in the (a) fixed and (b)–(d) oscillating body cases, for (b) $U^{*}=3$, (c) $U^{*}=6$, and (d) $U^{*}=9$. In the oscillating body cases, ${\zeta }_y=0$ and the body is moving downward.

Figure 11

Time series of the spanwise patterns in the shear layer and wake regions: (a), (c), (e), and (g) ${\omega }_{\xi }$ on line $L_1^{+}(\eta =1)$ and (b), (d), (f), and (h) ${\omega }_y$ on line $L_2^{+}(x=5)$, in the (a) and (b) fixed and (c)–(h) oscillating body cases, for (c) and (d) $U^{*}=3$, (e) and (f) $U^{*}=6$, and (g) and (h) $U^{*}=9$. The isocontours are distributed in ranges (a) ${\omega }_{\xi }\in [-2,2]$, (c), (e), and (g) ${\omega }_{\xi }\in [-10,10]$, and (b), (d), (f), and (h) ${\omega }_y\in [-2,2]$. In (a), a black rectangle indicates an instance of shear layer vortices. In (c) and (e), striped areas denote discarded samples (low-vorticity level).

Figure 12

Streamwise evolution of the spanwise fluctuation amplitude, in the fixed and oscillating body cases. In the oscillating body cases, the amplitudes $A_{z1}$ are determined for $\{{\zeta }_y=0,\dot{{\zeta }_y}<0\}$ (Sec. 2b).

Figure 13

Time series of the PDF of the local spanwise wavelength in the shear layer and wake regions: (a), (c), (e), and (g) on line $L_1^{+}(\eta =1)$ and (b), (d), (f), and (h) on line $L_2^{+}(x=5)$, in the (a) and (b) fixed and (c)–(h) oscillating body cases, for (c) and (d) $U^{*}=3$, (e) and (f) $U^{*}=6$, and (g) and (h) $U^{*}=9$. The isocontours of the PDF are distributed in the range $[0,2]$. In (b), (d), (f), and (h), gray dots indicate the samples considered in the selective time-averaging procedure $({\langle \rangle }_{t,L_2}^{*}$, defined in Sec. 2b). In (c) and (e), striped areas denote discarded samples (low-vorticity level).

Figure 14

Streamwise evolution of the spanwise wavelength in the fixed body case: comparison between the present results and the experimental measurements of Mansy et al. [5] $\left(\text{Re}=600\right)$ and Chyu and Rockwell [9] $\left(\text{Re}={10}^4\right)$. For the present results, error bars indicate the standard deviation of the wavelength around the averaged value [defined by (5)]. The wavelengths obtained at $y=1$ are also plotted for comparison.

Figure 15

Streamwise evolution of the spanwise wavelength, in the fixed and oscillating body cases. In the oscillating body cases, the wavelengths ${\lambda }_{z1}$ are determined for $\{{\zeta }_y=0,\dot{{\zeta }_y}<0\}$ (Sec. 2b). Error bars indicate the standard deviation of the wavelength around the averaged value [defined by (5)] for $U^{*}=3$, i.e., the case of maximum variability.

Figure 16

Schematic view of the boundary layer separating from the cylinder.

Figure 17

Scaling of the shear layer spanwise wavelength with the momentum thickness of the boundary layer at separation: (a) streamwise evolution of ${\lambda }_{z1}$ normalized by ${\delta }_s$ and (b) evolution of ${\delta }_s$ as a function of the Reynolds number $(\text{Re}$ in the fixed body case and ${\text{Re}}_{ip}$ in the oscillating body cases). In the oscillating body cases, the phase-averaged values are determined for $\{{\zeta }_y=0,\dot{{\zeta }_y}<0\}$ in (a) and different phases are considered in (b). In (a), a dash-dotted line $\left({\lambda }_{z1}/{\delta }_s=90\right)$ indicates the typical value of the normalized wavelength in the plateau region; an error bar shows the typical variation (standard deviation) of ${\lambda }_{z1}/{\delta }_s$ around this plateau value at different phases, for $U^{*}=3$ and $\eta /{\delta }_s\approx 150$. In (b), the momentum thickness values issued from the experimental studies of Thom [47], Green [48], and Fage [49] are also reported for comparison.

Figure 18

Evolution of the spanwise wavelength measured close to the body as a function of the Reynolds number $(\text{Re}$ in the fixed body case and ${\text{Re}}_{ip}$ in the oscillating body cases): comparison between previous works (fixed body case) and present results (fixed and oscillating body cases). In previous works, the wavelength was measured at $x=3$ [5], $x\approx 4$ [7], and $x=0.5$ [9]. In the present cases, the values of ${\lambda }_{z1}$ at $\eta =150{\delta }_s$ are reported and various phases are considered in the oscillating body cases.

Figure 19

Typical length scales of the flow structure in the wake region: spanwise wavelength as a function of the (a) streamwise length scale and (b) crossflow width of the wake, in the four cases studied. In each case, seven streamwise positions are considered in the range $x\in [4,10]$. Error bars indicate the standard deviation of the wavelength around the averaged value [defined by (5)].

Figure 20

Influence of the computational mesh on the spanwise wavelengths downstream of the body: evolutions of ${\lambda }_{z1}(\eta =1)$ and ${\lambda }_{z2}(x=10)$ as functions of the number of cells, in the oscillating body case, for $U^{*}=6$ (peak oscillation amplitudes). Error bars indicate the standard deviation of the wavelength around the averaged value [defined by (5)].

Figure 21

Visualization of the computational mesh: isocontours of the cell length $\mathrm{\Delta }l$ in the $(x,y)$ plane: (a) an overview and (b) a closer view.

Figure 22

Power spectral densities of the time series of (a) and (b) the streamwise component of flow velocity at (a) $(x,y)=(3,0)$ and (b) $(x,y)=(5,0)$ and (c) the crossflow force coefficient, in the fixed body case. The frequencies are normalized by the shedding frequency of the primary wake vortices $\left(f_{\text{St}}\right)$. In (a) and (b), a line indicates the $-5/3$ slope.

Figure 23

Span-averaged PSD of the time series of spanwise vorticity in the shear layer, in the fixed body case. The vorticity was sampled at $(x,y)=(0.32,0.53)$. The frequencies are normalized by the shedding frequency of the primary wake vortices $\left(f_{\text{St}}\right)$. The frequency ratio predicted by (C1) is indicated by a dashed line.