Coherent structures in the wake of a long wall-mounted rectangular prism at large incident angles

The unsteady wake of a wall-mounted rectangular prism at large incident (yaw) angles is investigated numerically at Reynolds number of 250. The incident angles considered here ranged between $0^{\circ }$ and ${45}^{\circ }$, which altered the wake dynamics substantially. A symmetric steady wake at small angles transforms into an asymmetric unsteady wake with complex structures at large angles. Approaching ${40}^{\circ }$, the wake appears weakly unsteady with small amplitude fluctuations in velocity and forces. At ${45}^{\circ }$, however, the wake is dominated by unsteady formation of helical structures with a dominant Strouhal number of ${\mathrm{St}}_v=0.14$ that are formed by the interaction of tip vortices on the front and windward edges of the prism. There also exists an embedded low-frequency mechanism that corresponds to the formation and suppression of unsteady helical structures in the wake. This mechanism creates a low-frequency signature in the wake at ${\mathrm{St}}_l=0.011$. A skeleton model is developed and proposed for the wake that characterizes coherent structures at large incident angles. Furthermore, a more representative length scale is proposed based on the projected width of the prism. The range of Reynolds number based on this new scale varies between 208 and 885 with increasing incident angle.

Figure 1

The Becker et al. (2002) [8] wake model for a wall-mounted rectangular prism at different incident angles.

Figure 2

Schematic of the cylinder at ${45}^{\circ }$ incident angle. (Not to scale.)

Figure 3

Schematic of the computational domain at ${45}^{\circ }$ incident angle. (Not to scale.)

Figure 4

The spatial grid distribution for the long rectangular prism at $\mathrm{Re}=250$ and incident angle of $i={45}^{\circ }$. Top view at $z/h=0.5$ (top) and side view at $y/h=0$ (bottom).

Figure 5

Profiles of time-averaged streamwise velocity ($\overline{u}/U_0$) in the wake of a long rectangular prism using different spatial grids for incident angles (a) $0^{\circ }$ and (b) ${45}^{\circ }$.

Figure 6

The ratio of spatial grid size to the Kolmogorov length scale for the case of ${45}^{\circ }$ incidence angle at $\mathrm{Re}=250$ in the mid-height plane, zoomed in around the critical flow region.

Figure 7

Time-averaged (a) streamwise and (b) spanwise velocity distributions at $\mathrm{Re}=250$ behind a rectangular prism with $\mathrm{AR}=4$ at $z/w=3.5$ compared with DNS results of Saha (2013) [39] and Zhang et al. (2017) [34].

Figure 8

Boundary layer profiles prior to reaching the prism at $x/h=-4.15$ and $y/h=6$ compared with the Blasius boundary layer solution.

Figure 9

Isosurface of $Q=30$ at different incident (yaw) angles, $0^{\circ }\le i\le {45}^{\circ }$, at $\mathrm{Re}=250$. (a) $i=0^{\circ }$, (b) $i={15}^{\circ }$, (c) $i={25}^{\circ }$, (d) $i={35}^{\circ }$, (e) $i={40}^{\circ }$, and (f) $i={45}^{\circ }$.

Figure 10

The power spectral density diagram of the (a) lift coefficient, (b) streamwise ($x$) component of velocity vector at $x/h=2.5, y/h=0$, and $z/h=0.1$, and (c) normal ($z$) component of velocity vector at $x/h=7.5, y/h=-0.5$, and $z/h=0.5$.

Figure 11

Contours of instantaneous vorticity in the normal ($z$) direction on the mid-plane ($z/h=0.5$) of the prism at $\mathrm{Re}=250$ and $i={45}^{\circ }$. Here, $t^{*}$ is the convective time step and ${\mathrm{St}}_v$ is the Strouhal number of the vortex shedding process. (a) $t^{*}=t_0^{*}$, (b) $t^{*}=t_0^{*}+\frac{1}{3S{t}_v}$, (c) $t^{*}=t_0^{*}+\frac{2}{3S{t}_v}$, and (d) $t^{*}=t_0^{*}+\frac{1}{S{t}_v}$.

Figure 12

The variation of (a) the normal component of velocity fluctuations at ($x/h=5,y/h=-2.5,z/h=0.9$) for $i={45}^{\circ }$, and (b) coefficient of lift fluctuations.

Figure 13

Variations of Q = 30 isosurfaces for the case of $i={45}^{\circ }$ at different convective times, $t^{*}$. (a) $t_1^{*}=530$, (b) $t_2^{*}=550$, (c) $t_3^{*}=570$, (d) $t_4^{*}=580$, (e) $t_5^{*}=595$, and (f) $t_6^{*}=620$.

Figure 14

Nodal points associated with the attachment locations on leeward (right) and back (left) faces of the prism at $i=0^{\circ }, {15}^{\circ }$, and ${25}^{\circ }$.

Figure 15

Continuation of Fig. 14 for the cases of $i={35}^{\circ }, {40}^{\circ }$, and ${45}^{\circ }$.

Figure 16

2D time-averaged streamlines on the leeward and back faces of the prism at $i={35}^{\circ }$ and ${45}^{\circ }$.

Figure 17

2D time-averaged streamlines in the $xy$ plane at $z=0.5$ plane at different incident angles. (a) $i={25}^{\circ }$, (b) $i={35}^{\circ }$, (c) $i={40}^{\circ }$, and (d) $i={45}^{\circ }$.

Figure 18

2D streamlines on $yz$ plane at different streamwise locations at two incident angles. (a) $x/h=0$, (b) $x/h=0.6$, (c) $x/h=1.1$, and (d) $x/h=1.5$.

Figure 19

Skeleton model of the unsteady wake of a long rectangular prism at large incident angles.

Figure 20

The variation of lift and drag coefficients with incident angle and Reynolds number based on the projected length (${\mathrm{Re}}_{w_p}$).