Wake interactions between two side-by-side circular cylinders with different sizes

Flows over two side-by-side circular cylinders exhibit fascinating flow physics due to complex interactions between the coupled wakes. However, their mutual interference effects have not been elucidated in a quantitative manner thus far. In this paper, we create a mismatch between the two wakes by introducing a size difference in the cylinder pair, such that the effects of one wake on the other can be distinguished. Depending on the size and gap ratios between the two cylinders, the coupled wake exhibits distinct dynamical features including the single bluff-body flows, chaotic flows, quasi-periodic flows, and synchronized flows. As with the cases of two identical cylinders, the chaotic flows are featured by the flip-flop phenomenon (random switching of the gap flow direction) even for cases with large size difference. The quasiperiodic flows are mainly composed of two primary frequencies associated with vortex shedding in the near wakes of the two cylinders. Both wakes impose their own frequencies on the other, resulting in the beating phenomenon in the lift coefficients. The phase lag associated with each of the primary frequencies between the two cylinders modulates the combined lift forces. The triad interactions between the two wakes generate the sideband frequencies, which are associated with modal structures that are mostly active in the far wake. The transition from quasiperiodicity to synchronization is dominated by the vortex shedding behind the larger cylinder, to which the wake of the smaller cylinder locks in. These results reveal new insights on the coupled wakes of two cylinders, and are pivotal for understanding more general wake interaction problems.

Figure 1

Schematic of the computational setup. The sketch is not to scale.

Figure 2

Comparison of mean drag and root-mean-squared lift coefficients between Kang (2003) [6] and the present simulations.

Figure 3

Comparison of forces coefficients and Strouhal number of a single cylinder as a function of size ratio $r$. ($a$) mean drag coefficient $\overline{C_d}$, $\left(b\right)$ maximum lift coefficient $C_{l,\text{max}}$ and ($c$) Strouhal number St. Note that $C_d$, $C_l$ and St are defined based on $D$ (corresponding to $\text{Re}=100$ in this study), as stated in Eq. (1).

Figure 4

Representative wakes for flow over two cylinders of different sizes. (a) single bluff-body wake at $(r,g)=(1.15,0.2)$, (b) chaotic wake at $(r,g)=(1.15,1.0)$, (c) synchronized wake at $(r,g)=(1.15,1.5),$ and (d) quasiperiodic wake at $(r,g)=(1.15,2.5)$. The left column shows the wake vortical structures visualized by vorticity (${\omega }_z=\mathbf{\nabla }\times \mathit{u}$) contours ranging from $-1$ (blue) to 1 (red). The middle column shows the time histories of the lift coefficients, and the right column shows their phase trajectories.

Figure 5

Classification of the flow patterns in the $r\text{−}g$ space. $⧫$: single bluff-body wakes; $▾$: chaotic wakes; $\blacksquare$: quasiperiodic wakes; $•$: synchronized wakes; $◯$: antiphase synchronized vortex shedding (for $r=1$ only).

Figure 6

The drag coefficients (top row), and flow fields at selected times for cases in the chaotic flow regimes ($g=0.7$). (a) $r=1$, (b) $r=1.15,$ and (c) $r=1.3$. $t_1$ and $t_2$ denote the time instants at which the gap flow deflects towards cylinder 1 and cylinder 2, respectively. The vorticity shown here ranges from $-1$ (blue) to 1 (red).

Figure 7

Spectral analysis of lift coefficients of (a) cylinder 1 and (b) cylinder 2 for cases with $r=1.15$ using the NAFF algorithm. $A_{i,j}$ denotes the spectral amplitude associated with the frequency component $f_j$ on cylinder $i$. (c), (d) Side views of panels (a) and (b) showing amplitudes of lift coefficients on cylinders 1 and 2, respectively. Under the dashed lines, $A_{i,n}$ denotes the amplitude of lift coefficient (i.e., $C_{l,\text{max}}$) in the single cylinder case. (e) top view of panels (a) and (b) showing primary frequencies of the lift coefficients. $f_{i,n}$ denotes the natural shedding frequency in the wake of a single cylinder. (f) Phase difference of primary frequencies $f_1$ and $f_2$ between the two cylinders. In panels (c)–(f), the shaded area represents the lock-in region. The insets show the respective quantities in $g\in [1.7,1.8]$ with an interval of $\mathrm{\Delta }g=0.01$.

Figure 8

Key parameters in the spectra of lift coefficients for $r=1–1.3$ and $g=1.5–3$. Filled and empty symbols represent synchronized and quasiperiodic cases, respectively. The dashed lines indicate the quantities for the single cylinder cases.

Figure 9

Dynamic modes for the synchronized wakes. $\text{(a)}(r,g)=(1,1.5)$, $\text{(b)}(r,g)=(1,2.5),$ and $\text{(c)}(r,g)=(1.15,1.5)$. Blue color indicates negative vorticity, and red color indicates positive vorticity.

Figure 10

Dynamic modes for the quasiperiodic flows with $r=1.15$.

Figure 11

Drag and lift coefficients of the two cylinders. $\overline{(·)}$ represents the time-averaged values and ${(·)}^{\prime }$ represents the standard deviation.

Figure 12

Spectral amplitudes of the combined lift coefficients for the case of $r=1.15$. $A_{c,1}$ and $A_{c,2}$ are the amplitudes associated with frequencies $f_1$ and $f_2$ in the combined lift coefficients.