Numerical simulation of flow over a parallel cantilevered flag in the vicinity of a rigid wall

Flow over a parallel cantilevered flag in the vicinity of a rigid wall is numerically studied using an immersed boundary–lattice Boltzmann method (IB–LBM) in two-dimensional domain, where the dynamics of the fluid and structure are, respectively, solved by the LBM and a finite-element method (FEM), with a penalty IB to handle the fluid–structure interaction (FSI). Specifically, a benchmark case considering a plate attached to the downstream of a stationary cylinder is first conducted to validate the current solver. Then, the wall effects on the flag are systemically studied, considering the effects of off-wall distance, structure-to-fluid mass ratio, bending rigidity, and Reynolds number. Three flapping modes, including symmetrical flapping, asymmetrical flapping, and chaotic flapping, along with a steady state are observed in the simulations. It is found that the flag is vibrating or stable with a mean angle inclined in the fluid when it is mounted in the vicinity of a rigid wall. The mean inclined angle first increases in the steady state and then decreases in the unsteady state with the off-wall distance. In the unsteady regime, the dependency of the inclined angle on the off-wall distance is similar to that of the gradient of the fluid velocity. In addition, the rigid wall near the flag decreases the lift and drag generation and further stabilizes the flag–fluid system. Contrarily, the flag inertia destabilizes the flag, and large flag inertia induces chaotic vibrating modes.

Figure 1

Schematic of a parallel cantilevered flag in the vicinity of a rigid wall.

Figure 2

Schematic of a flexible plate behind a stationary cylinder in a channel.

Figure 3

Instantaneous vorticity contour and stream lines of a flexible plate behind a stationary cylinder in a channel.

Figure 4

Comparison of the position histories of the trailing end with the data from Refs. [46, 48]: $\mathrm{Re}=100$, $m^{*}=2.0$, and $K_B^{*}=1.111$.

Figure 5

Time histories $C_L$ and $C_D$: $\mathrm{Re}=100$, $m^{*}=2.0$, and $K_B^{*}=1.111$.

Figure 6

Flapping patterns of the flag at $\mathrm{Re}=100$, $m^{*}=1.0$, and $K_B^{*}=0.0001$: (a) Uniform flow and (b) a rigid wall on the bottom with $d/L=0.5$.

Figure 7

Instantaneous vorticity contours and stream lines at $\mathrm{Re}=100$, $m^{*}=1.0$: (a) Far field boundary condition and (b) rigid wall on the bottom with $d/L=0.5$.

Figure 8

Instantaneous vorticity contours at $\mathrm{Re}=100$, $m^{*}=1.0$, and $K_B^{*}=0.0001$.

Figure 9

Comparison of $C_L$, $C_D$, $x$ and $y$ positions of the trailing end at $\mathrm{Re}=100$, $m^{*}=1.0$, and $K_B^{*}=0.0001$ with $d/L$ ranging from 0.5 to 2.5. Where, the gray and white regions indicate up-flapping and down-flapping, respectively. The arrow denotes the direction that the off-wall distance increases.

Figure 10

Trajectories of the trailing ends of flapping flags at $\mathrm{Re}=100$, $m^{*}=1.0$, and $K_B^{*}=0.0001$.

Figure 11

Mean inclined angles of the trailing ends of flapping flags and the gradient of the horizontal velocity at $\mathrm{Re}=100$, $m^{*}=1.0$, and $K_B^{*}=0.0001$.

Figure 12

$C_{L,\text{rms}}$, $C_{D,m}$, vibrating amplitude and St number against off-wall distance at $\mathrm{Re}=100$, $m^{*}=1.0$, and $K_B^{*}=0.0001$.

Figure 13

Flapping patterns of the flag in a period with a nondimensional interval of 0.2: $\mathrm{Re}=100$, $K_B^{*}=0.0001$, $d/\delta =0.46$ (top) and 0.68 (bottom).

Figure 14

Trajectories of the trailing ends of flapping flags at $\mathrm{Re}=100$, $m^{*}=1.0$, $K_B^{*}=0.0001$, $d/\delta =0.23$, 0.46, and 0.68 (from left to right).

Figure 15

Comparison of $C_L$, $C_D$, $x$ and $y$ positions of the trailing end at $\mathrm{Re}=100$, $K_B^{*}=0.0001$, and $d/\delta =0.46$. Where, the gray and white regions indicate up-flapping and down-flapping, respectively.

Figure 16

$C_{L,\text{rms}}$, $C_{D,m}$, $A_m$, and St number against off-wall distance at $\mathrm{Re}=100$ and $K_B^{*}=0.0001$.

Figure 17

Dynamic modes of the flag at $\mathrm{Re}=100$ and $K_B^{*}=0.0001$. Where “o,” “$\mathrm{\Delta }$,” “$\square$,” and “+” indicate steady state, asymmetrical flapping, symmetrical flapping, and chaotic flapping, respectively.

Figure 18

$C_{D,m}$ and $A_m$ against bending rigidity at $\mathrm{Re}=100$, $m^{*}=1.0$, and $d/\delta =0.46$.

Figure 19

Flapping patterns of the flag in a period with a nondimensional interval of 0.2: $\mathrm{Re}=100$, $m^{*}=1.0$, $d/\delta =0.46$, $K_B^{*}=0.0001$, 0.001, and 0.005.

Figure 20

Trajectories of the trailing ends of flapping flags at $\mathrm{Re}=100$, $m^{*}=1.0$, $d/\delta =0.46$, $K_B^{*}=0.0001$, 0.001, and 0.005. The arrow denotes the direction that the bending rigidity increases.

Figure 21

$C_{L,\text{rms}}$, $C_{D,m}$, $A_m$, and St number against local Reynolds number at $m^{*}=1.0$ and $K_B^{*}=0.0001$.