Wake behind a concave curved cylinder

The wake behind a quarter-of-ring concave curved cylinder is investigated in this paper by means of direct numerical simulations. The plane of curvature is aligned with the uniform incoming flow. We have appended straight extensions to both ends of the curved part of the cylinder, such that free ends are eliminated from the simulations. The effect of the vertical extension, i.e., the straight extension with its axis normal to the inflow, is carefully studied and turns out to be significant. The results from several different Reynolds numbers ($\mathrm{Re}=100–500$) are presented, from which a clear picture of the wake transition behind this configuration could be sketched. The concave curved cylinder wake consists of different flow regimes along the span. Oblique shedding, vortex dislocations, and various shedding frequencies are captured in different flow regimes. At $\mathrm{Re}\le 300$, the flow regimes change abruptly, but at $\mathrm{Re}=400$ and 500, the changes are continuous, so the boundaries between them are difficult to observe. A frequency band, instead of one single dominating frequency, manifests itself in the three-dimensional (3D) energy spectrum.

Figure 1

(a) Three-dimensional computational domain for the concave cylinder configuration. Notice the domain is not to scale. (b) Projection sketch of the geometry in the symmetry plane, the $\left(x,z\right)$ plane at $y/D=0$. The origin of the configuration is marked as $O$.

Figure 2

An example (case VE-12D) of the multilevel grids. The slice of the grid box distribution in the symmetry plane, i.e., the $\left(x,z\right)$ plane at $y/D=0$, is shown. Each square represents a 3D grid box, while each grid box, regardless of its size, contains the same amount ($N\times N\times N$) of grid cells. Therefore, the square size also indicates the different levels of grid resolution. One can find five levels of grids, among which the first four levels are indicated with numbers.

Figure 3

(a) The mean axial velocity $u_{\mathrm{ax}}/U_0$ plotted along a concentric arc of the curved cylinder whose radius is $14D$, i.e., $1D$ behind the leeward face of the curved cylinder. Results from cases GI-coarse, GI-medium, GI-fine, and GI-fine2 are plotted. (b) The curved dashed line indicates the concentric arc along which the mean axial velocities in (a) are plotted. $\mathrm{Re}=500$. The physical interpretation of the peculiar variation of the axial velocity is deferred to the discussion concerning Fig. 7 in Sec. 3.

Figure 4

The wake structure of different VE-simulations, visualized by the iso-surface of ${\lambda }_2=-0.01$. All snapshots are viewed in the $+y$ direction. The coordinate system is sketched in (a). Note that $O^{\prime }$ is not the true origin of the domain. Since each snapshot has a different scale, we indicate $z/D=0$, i.e., the interface between the curved part and the vertical extension, with a short horizontal solid line in each subplot. The dashed lines in (e) separate the different flow regimes. $\mathrm{Re}=200$.

Figure 5

Comparison of the axial velocity $u_{\mathrm{ax}}/U_0$ distribution along a concentric arc of the curved cylinder, whose radius is $13.6D$, i.e., $0.6D$ behind the cylinder. Results from case VE-18D, VE-21D, and VE-24D are plotted. The abscissa is divided into two parts: the curved part is scaled by the angle $\theta$, while the vertical part is scaled by height $z/D$. $\mathrm{Re}=200$.

Figure 6

Results of the time-averaged vertical velocity $\langle w\rangle /U_0$ contours in the symmetry plane for three cases (VE-24D, VE-12D, VE-6D) are plotted together for comparison. For each case, the contours of $\langle w\rangle /U_0=0.2,0.5,0.8$ are plotted. $\mathrm{Re}=200$.

Figure 7

The time-averaged streamwise velocity $\langle u\rangle /U_0$, vertical velocity $\langle w\rangle /U_0$, and the axial velocity $u_{\mathrm{ax}}/U_0$ along the same concentric arc for the curved part of the cylinder, similarly as in Fig. 5. Results taken from simulation VE-24D, at $\mathrm{Re}=200$.

Figure 8

The location of the boundaries between adjacent regimes. Left ordinate for the boundary between regime 1 and 2, right ordinate for that between regimes 2 and 3.

Figure 9

Power spectrum density distribution of the cross-flow velocity $v$ along a sampling line $3D$ behind the vertical extension in the symmetry plan ($y/D=0$), as sketched in (c). (a) VE-12D, (b) VE-24D. The main frequencies are indicated. $\mathrm{Re}=200$.

Figure 10

The rendering of ${\lambda }_2$, showing instantaneous wake snapshots at (a) $\mathrm{Re}=100$, (b) $\mathrm{Re}=200$, (c) $\mathrm{Re}=300$, and (d) $\mathrm{Re}=500$, respectively. All snapshots are viewed in the $+y$ direction, and $L_v=24D$. Notice that (a) and (b) use the same color map scale indicated in (b), while (c) and (d) use the different color map scale indicated in (d).

Figure 11

Locations of the boundaries between different flow regimes at different Re. To read the boundary location between regimes 1 and 2, one uses the left vertical axis, while the right vertical axis is used to read the boundary location between regimes 2 and 3.

Figure 12

The axial velocity distribution $u_{\mathrm{ax}}/U_0$ following the axial direction of the cylinder, both the curved part and the vertical extension part, $0.6D$ behind the leeward side of the cylinder surface. The abscissa is arranged in the same way as in Fig. 5. Results for all five Re are plotted. The arrows show the tendency when Re increases. $L_v/D=24$.

Figure 13

The cross-flow velocity energy spectrum plotted over a vertical line at $x/D=16, y/D=0$, and over a vertical span from $z/D=0$ to 24, i.e., a vertical line $3D$ behind the vertical extension and in the symmetry plane. Results at (a) $\mathrm{Re}=100$, (b) $\mathrm{Re}=300$, (c) $\mathrm{Re}=400$, and (d) $\mathrm{Re}=500$. The frequency values are marked in each subplot.

Figure 14

The wake structure at $\mathrm{Re}=300$ is described by the iso-surface of ${\lambda }_2=-0.2$. The four flow regimes are separated by the black dashed lines. The red dashed lines are drawn based on the calculated oblique angles (${\alpha }_1, {\alpha }_2$, and ${\alpha }_3$) indicated.