Optimal external forces of the lock-in phenomena for flow past an inclined plate in uniform flow

We theoretically studied the optimal control, frequency lock-in, and phase lock-in phenomena due to the spatially localized periodic forcing in flow past an inclined plate. Although frequency lock-in is evident in many fluid phenomena, especially fluid-structure interactions, not many researchers have investigated it using a theoretical approach based on flow details. We obtained detailed information on the lock-in phenomena to external periodic forcing using phase reduction theory, a mathematical method for extracting the dynamics near the limit cycle. Furthermore, the optimal forces applied to the velocity field were determined under the condition of the minimum forcing energy and maximum lock-in range. The study of uniform periodic forces applied within spatially confined regions led to the conclusion that the effective lock-in position, which includes both the upstream and downstream areas of the plate, depends on the principal frequency of the force. The frequency lock-in range of these forces was analyzed and compared with theoretical predictions.

Figure 1

(a) Model configuration. An inclined plate is placed in the wind tunnel. A uniform external force was applied within a rectangular area of size $l_x\times l_y$ centered at $(x_c,y_c)$. (b) Computational grid and the plate model. Grid lines are drawn on every other line for visibility.

Figure 2

The velocity and vorticity fields. Snapshots at the phase $\varphi /2\pi =k/8(k=0,1,\cdots ,7)$ are shown.

Figure 3

The fields of the phase sensitivity vector. The contour indicates the magnitude $\left|\mathit{q}\right(\mathit{x},\varphi \left)\right|$. Snapshots at the phase $\varphi /2\pi =k/8(k=0,1,\cdots ,7)$ are shown.

Figure 4

Phase responses to periodic forcing, ${\tilde{q}}_u(\mathit{x};m)$ and ${\tilde{q}}_v(\mathit{x};m)$. Arrows and contours indicate the values (complex numbers in a Gauss plane) and their amplitudes, respectively. (a) ${\tilde{q}}_u(\mathit{x};1)$, (b) ${\tilde{q}}_v(\mathit{x};1)$, (c) ${\tilde{q}}_u(\mathit{x};2)$, (d) ${\tilde{q}}_v(\mathit{x};2)$.

Figure 5

Amplitudes of phase responses to periodic forcing $|{\tilde{q}}_u(\mathit{x};m)|$ and $|{\tilde{q}}_v(\mathit{x};m)|$ $(3\le m\le 6)$.

Figure 6

(a) The field of ${\mathrm{\Gamma }}_{\text{max},x}^{\left(1\right)}$ for $(l_x,l_y)=(1.0,0.5)$ as a function of ${\mathit{x}}_c$ $[{\mathit{x}}^{*}=(2.079,2.829)$; cf. Table 1]. (b) Same as panel (a), but for ${\mathrm{\Gamma }}_{\text{max},y}^{\left(1\right)}$ $[{\mathit{x}}^{*}=(1.553,2.645)]$. (c) Same as panel (a), but for ${\mathrm{\Gamma }}_{\text{max},x}^{\left(2\right)}$ $[{\mathit{x}}^{*}=(2.079,2.276)]$. (d) Same as panel (c), but for ${\mathrm{\Gamma }}_{\text{max},y}^{\left(2\right)}$ $[{\mathit{x}}^{*}=(1.289,2.645)]$.

Figure 7

Same as Fig. 6, but for $(l_x,l_y)=(0.5,1.0)$. The positions of the rectangles where maximum frequency lock-in is obtained are as follows (cf. Table 1): (a) ${\mathit{x}}^{*}=(2.566,1.763)$. (b) ${\mathit{x}}^{*}=(2.855,2.079)$. (c) ${\mathit{x}}^{*}=(1.987,2.079)$. (d) ${\mathit{x}}^{*}=(1.408,2.553)$.

Figure 8

Arnold tongues. Broken lines indicate theoretical prediction of the lock-in. Blue triangles and gray triangles indicate the lock-in state and no lock-in states, respectively. Relative deviation of the lift coefficient $\delta$ is indicated by colored symbols, circle with a vertical line. Vertical positions of the symbols are shifted downward slightly for visibility: (a) Case I, (b) Case II.

Figure 9

Difference between the angular frequency of the external force and that observed in the simulation, $\mathrm{\Omega }-{\omega }_{\text{sim}}$, for Case I. The points indicate the mean values, with vertical lines representing the standard deviations. Vertical broken lines indicate the theoretical boundary of the lock-in region.

Figure 10

Same as Fig. 9, but for Case II.

Figure 11

(a) Graph of the function $g\left(\psi \right)$. There are four solutions $\psi =0,{\psi }^{*},\pi$, and $2\pi -{\psi }^{*}$ for $g\left(\psi \right)=0$. (b) Graph of the function ${\mathrm{\Gamma }}_{-}\left(\psi \right)$. The lowest angular frequency for lock-in is achieved at $\psi ={\psi }_{-}(=0)$.

Figure 12

(a) The graph of the function ${\mathrm{\Gamma }}_1\left(\psi \right)$ (${\psi }_{+}=\pi$), illustrating symmetries as depicted by Eq. (26). (b) The graph of the function ${\mathrm{\Gamma }}_2\left(\psi \right)$ (${\psi }_{+}={\psi }^{*}$). There are multiple lock-in phases for smaller frequency difference, $\psi /\left(2\pi \right)=0.491649$ and 0.807 218.

Figure 13

The fields of the optimal force of the maximum lock-in region, ${\mathit{f}}_{*}$, in the case ${\psi }_{+}=\pi$. Arrows and contours indicate the force vectors and their magnitudes, respectively. Snapshots at phase $\theta /2\pi =k/8$ $(k=0,1,\cdots ,7)$ are shown.

Figure 14

The fields of the optimal force of the maximum lock-in region, ${\mathit{f}}_{*}$, in the case ${\psi }_{+}={\psi }^{*}$. Arrows and contours indicate the force vectors and their magnitudes, respectively. Snapshots at the phase $\theta /2\pi =k/8(k=0,1,\cdots ,7)$ are shown.

Figure 15

Comparison with the calculation by the spectral element method.

Figure 16

Comparison with different size of the domain.