Control of flow around a circular cylinder for minimizing energy dissipation

Control of flow around a circular cylinder is studied numerically aiming at minimization of the energy dissipation. First, we derive a mathematical relationship (i.e., identity) between the energy dissipation in an infinitely large volume and the surface quantities, so that the cost function can be expressed by the surface quantities only. Subsequently a control law to minimize the energy dissipation is derived by using the suboptimal control procedure [J. Fluid Mech. 401, 123 (1999)]. The performance of the present suboptimal control law is evaluated by a parametric study by varying the value of the arbitrary parameter contained. Two Reynolds numbers, $\mathrm{Re}=100$ and 1000, are investigated by two-dimensional simulations. Although no improvement is obtained at $\mathrm{Re}=100$, the present suboptimal control shows better results at $\mathrm{Re}=1000$ than the suboptimal controls previously proposed. With the present suboptimal control, the dissipation and the drag are reduced by 58% and 44% as compared to the uncontrolled case, respectively. The suction around the front stagnation point and the blowing in the rear half are found to be weakened as compared to those in the previous suboptimal control targeting at pressure drag reduction. A predetermined control based on the control input profile obtained by the suboptimal control is also performed. The energy dissipation and the drag are found to be reduced as much as those in the present suboptimal control. It is also found that the present suboptimal and predetermined controls have better energy efficiencies than the suboptimal control previously proposed. Investigation at different control amplitudes reveals an advantage of the present control at higher amplitude. Toward its practical implementation, a localized version of the predetermined control is also examined, and it is found to work as effectively as the continuous case. Finally, the present predetermined control is confirmed to work well in a three-dimensional flow too.

Figure 1

Flow around a circular cylinder.

Figure 2

Control volume.

Figure 3

Parametric study: (a) $\mathrm{Re}=100$; (b) $\mathrm{Re}=1000$. The dissipation rate $\varepsilon$ is made dimensionless by $\rho ,U_{\infty }$, and $D$.

Figure 4

Velocity distribution on the wall $\left(\mathrm{Re}=1000\right)$. The horizontal axis is the angle from the front stagnation point, ${\theta }^{\prime }={180}^{\circ }-\theta$. The velocity $u_r$ is made dimensionless by $U_{\infty }$.

Figure 5

Time traces $\left(\mathrm{Re}=1000\right)$: (a) energy dissipation, $\varepsilon$; (b) drag coefficient, $C_D$; (c) lift coefficient, $C_L$. The dissipation rate $\varepsilon$ is made dimensionless by $\rho ,U_{\infty }$, and $D$.

Figure 6

Pressure and friction coefficients on the cylinder surface $\left(\mathrm{Re}=1000\right)$: (a) pressure coefficient, $C_p$; (b) friction coefficient, $C_f$. The horizontal axis is the angle from the front stagnation point, ${\theta }^{\prime }={180}^{\circ }-\theta$.

Figure 7

Instantaneous energy dissipation field $\left(\mathrm{Re}=1000\right)$: (a) no control; (b) $J_1$ control; (c) present suboptimal control $\left(\Delta t_c/T=0.75\right)$. The local dissipation rate is made dimensionless by $\rho ,U_{\infty }$, and $D$.

Figure 8

Time traces in the predetermined control case $\left(\mathrm{Re}=1000\right)$: (a) energy dissipation, $\varepsilon$; (b) drag coefficient, $C_D$. ${\theta }^{{}^{\prime }}={180}^{\circ }-\theta$. The dissipation rate $\varepsilon$ is made dimensionless by $\rho ,U_{\infty }$, and $D$.

Figure 9

Surface pressure coefficient in the predetermined control case $\left(\mathrm{Re}=1000\right)$. The horizontal axis is the angle from the front stagnation point, ${\theta }^{\prime }={180}^{\circ }-\theta$.

Figure 10

Dependency on the control amplitude ${\varphi }_{max} \left(\mathrm{Re}=1000\right)$: (a) energy dissipation $\overline{\varepsilon }$; (b) lowest possible efficiency ${\eta }_a$. The dissipation rate $\varepsilon$ is made dimensionless by $\rho ,U_{\infty }$, and $D$.

Figure 11

Velocity distribution on the wall (localized control, $\mathrm{Re}=1000$). The horizontal axis is the angle from the front stagnation point, i.e., ${\theta }^{\prime }={180}^{\circ }-\theta$. The velocity $u_r$ is made dimensionless by $U_{\infty }$.

Figure 12

Instantaneous energy dissipation field (localized control, $\mathrm{Re}=1000$). The local dissipation rate is made dimensionless by $\rho ,U_{\infty }$, and $D$.

Figure 13

Instantaneous energy dissipation field ($\mathrm{Re}=1000$, three-dimensional): (a) no control; (b) $J_1$ control; (c) present predetermined control. The local dissipation rate is made dimensionless by $\rho ,U_{\infty }$, and $D$.