Bayesian optimization for active flow control

Abstract

A key question in flow control is that of the design of optimal controllers when the control space is high-dimensional and the experimental or computational budget is limited. We address this formidable challenge using a particular flavor of machine learning and present the first application of Bayesian optimization to the design of open-loop controllers for fluid flows. We consider a range of acquisition functions, including the recently introduced output-informed criteria of Blanchard and Sapsis (2021), and evaluate performance of the Bayesian algorithm in two iconic configurations for active flow control: computationally, with drag reduction in the fluidic pinball; and experimentally, with mixing enhancement in a turbulent jet. For these flows, we find that Bayesian optimization identifies optimal controllers at a fraction of the cost of other optimization strategies considered in previous studies. Bayesian optimization also provides, as a by-product of the optimization, a surrogate model for the latent cost function, which can be leveraged to paint a complete picture of the control landscape. The proposed methodology can be used to design open-loop controllers for virtually any complex flow and, therefore, has significant implications for active flow control at an industrial scale.

Graphic Abstract

Appendix A: Validation of the computational approach for the fluidic pinball

To determine appropriate values for the time-step size $\Delta \tau$ and polynomial order N, we consider the uncontrolled case ($\mathbf{b } = \mathbf{0 }$) at $Re=100$ with initial condition $\mathbf{v }(x, y) = [1+ 10^{-3} \sin (y)] \mathbf{e }_x$. The slight asymmetry in the initial condition allows vortex shedding to develop more rapidly than if one were to rely on small asymmetries in the numerics. The flow is evolved for 2000 time units and only the last 500 time units of that interval are retained to compute the statistics. We report the temporal mean, standard deviation (std), and peak-to-peak amplitude (amp) of $C_D$ and $C_L$, as well as the Strouhal frequency St (i.e., the dominant frequency of $C_L$). Table 2 shows that the polynomial order has a larger effect on the statistics than the time-step size. None of the computational parameters has a significant effect on the Strouhal frequency. The results show that, for the present purposes, adequate convergence is achieved by specifying $\Delta \tau =0.02$ and $N=7$. In our production runs, however, we use a smaller time-step size ($\Delta \tau =0.01$) to alleviate the possibility of solution blow-up when the actuation vector is non-zero.

Fig. 12

Long-time averages of $C_D$ and $J_T$ for the symmetric actuation configuration of Cornejo Maceda et al. [34] at $Re=100$

For $\Delta \tau =0.01$ and $N=7$, we compare our computational approach with the finite-element solver used by Deng et al. [32]. We consider the actuation configuration described in Cornejo Maceda et al. [34] in which the front cylinder is not allowed to rotate ($v_1=0$) and the aft cylinders rotate with equal and opposite angular velocities ($v_2 = -v_3$). As in Cornejo Maceda et al. [34], the Reynolds number is 100 and the flow is effected for 1000 time units starting from an initial state on the limit cycle of the uncontrolled configuration. We report the temporal mean for $C_D$ and $J_T$. Figure 12 shows good agreement between our approach and that of Cornejo Maceda et al. [34, 47].