Stabilization of unsteady flows by reduced-order control with optimally time-dependent modes

In dynamical systems theory, suppression of instabilities around a fixed point is generally achieved by controlling the linearized dynamics of infinitesimal perturbations, because considering small-amplitude disturbances allows for application of a range of celebrated techniques from linear control theory. In this paper, we consider the problem of design and implementation of a controller for fully nonlinear, high-dimensional, dynamical systems with the goal of steering trajectories to an unstable fixed point of the governing equations. Our control strategy is based on our previous work [A. Blanchard, S. Mowlavi, and T. P. Sapsis, Nonlinear Dynam. 95, 2745 (2019)] and takes advantage of the unique properties of the optimally time-dependent (OTD) modes, a set of global, time-evolving, orthonormal modes that track directions in phase space associated with transient growth and persistent instabilities. We show that the OTD control strategy introduced previously is robust with respect to perturbation amplitude even in cases in which the trajectory initially evolves on an attractor that lies far away from the target fixed point. In recognition of the fact that actuation capabilities are generally limited in practice, we also formulate a localized control strategy in which the OTD modes are computed in a spatially localized subdomain of the physical domain of interest. We suggest a strategy for selecting the optimal control domain based on a quantitative criterion derived from the OTD modes. We show that even when the range of the controller is reduced, OTD control is able to steer trajectories toward the target fixed point.

Figure 1

For the $2\times 2$ non-normal system (17a) and (17b), norm of trajectories subject to (a) no control, (b) OTD control with $r=1$ and $\zeta =0.1$, and (c) OTD control with $r=1$ and $\zeta =0.6$. In (b) and (c), control is activated at $t=100$. Initial conditions for the trajectories are ${(0,c)}^{\mathsf{T}}$, where $c={10}^{-3}$, $2.5\times {10}^{-3}$, ${10}^{-2}$, and $5\times {10}^{-2}$, from darker to lighter.

Figure 2

For flow past a cylinder at $\text{Re}=50$, (a) spanwise vorticity distribution of the steady symmetric solution ${\mathbf{w}}_e$ and (b) snapshot of the spanwise vorticity distribution of the solution on the limit cycle at an instant for which $C_L$ is maximum.

Figure 3

For flow past a cylinder at $\text{Re}=50$ on the limit-cycle attractor, vorticity distributions of (a), (c), (e), and (g) the first OTD mode, and (b), (d), (f), and (h) the second OTD mode, shown at time (a) and (b) $t_0$, (c) and (d) $t_0+T/4$, (e) and (f) $t_0+T/2$, and (g) and (h) $t_0+3T/4$, where $C_L$ reaches its maximum amplitude at $t_0$.

Figure 4

For the steady (unstable) flow past a cylinder at $\text{Re}=50$, vorticity distributions of the (a) first and (b) second OTD modes.

Figure 5

For flow past a cylinder at $\text{Re}=50$ subject to OTD control with $r=2$ and $\zeta =0.1$, (a) time series of $\parallel \mathbf{w}-{\mathbf{w}}_e\parallel$ and (b) eigenvalues of the open-loop reduced operator $({\mathbf{L}}_r+{\mathbf{L}}_r^{\mathsf{T}})/2$. Control is idle in the interval $0\le t<100$ and active for $t\ge 100$.

Figure 6

For flow past a NACA 0012 airfoil with $\alpha ={10}^{\circ }$ and $\text{Re}=1000$, (a) spanwise vorticity distribution of the steady solution, (b) snapshot of the spanwise vorticity distribution of the solution on the limit cycle at an instant for which $C_L$ is maximum, and (c) 21 most unstable eigenvalues of the linear operator ${\mathbf{L}}_e$ visualized in the complex plane.

Figure 7

For flow past a NACA 0012 airfoil with $\alpha ={10}^{\circ }$ and $\text{Re}=1000$, (a) time series of $|C_L-C_{L,e}|$ for trajectories subject to infinitesimal asymmetric inlet perturbation (26) with OTD control law (14) and $\zeta =0.1$ and (b) time series of $\parallel \mathbf{w}-{\mathbf{w}}_e\parallel$ for trajectories initialized on the limit cycle subject to OTD control law (25) with $r=2$ and $\zeta =1.8$, 2.4, and 3.2. In (a), OTD control is active for all $t\ge 0$. In (b), OTD control is idle in the interval $0\le t<100$ and active for $t\ge 100$.

Figure 8

For flow past a NACA 0012 airfoil with $\alpha ={10}^{\circ }$ and $\mathrm{Re}=1000$ subject to OTD control (25) with $r=2$ and $\zeta =1.8$, spanwise vorticity distributions of (a) the solution $\mathbf{w}$ and (b) the control force ${\mathbf{f}}_{c,\mathrm{all}}$ at $t=120$ [cf. Fig. 7].

Figure 9

For Kolmogorov flow with $\text{Re}=40$ and $k_f=4$, (a) spanwise vorticity distribution of the initial condition used in the computations and (b) energy dissipation for trajectories with OTD control (with $\zeta =0.1$) and without control. Control is idle in the interval $0\le t<50$ and active for $t\ge 50$.

Figure 10

For flow past a cylinder at $\text{Re}=50$, (a) spectrum of ${\mathbf{L}}_e$ computed with inflow-outflow-symmetry boundary conditions (black squares) and homogeneous Dirichlet boundary conditions (red triangles) and (b) location of control subdomains ${\overline{\mathrm{\Omega }}}_1$–${\overline{\mathrm{\Omega }}}_4$ with the spanwise vorticity distribution of ${\mathbf{w}}_e$ shown in the background.

Figure 11

For flow past a cylinder at $\text{Re}=50$ subject to localized OTD control with $r=8$, $\zeta =0.4$, and the control subdomains ${\overline{\mathrm{\Omega }}}_k$ shown in Fig. 10, (a) time series of $\parallel \mathbf{w}-{\mathbf{w}}_e\parallel$ and (b) detail of the time series for $C_L$. Control is idle in the interval $0\le t<100$ and active for $t\ge 100$.

Figure 12

For flow past a cylinder at $\text{Re}=50$, leading time-averaged OTD eigenvalue ${\overline{\mu }}_1$ computed with $r=8$ on various subdomains extending $2D$ in the streamwise and cross-stream directions (black bounding boxes). The spanwise vorticity distribution of ${\mathbf{w}}_e$ is shown in the background.

Figure 13

For flow past a NACA 0012 airfoil with $\alpha ={10}^{\circ }$ at $\text{Re}=1000$, leading time-averaged OTD eigenvalue ${\overline{\mu }}_1$ computed with $r=8$ on various subdomains extending $0.5L_c$ in the streamwise and cross-stream directions (black bounding boxes). The spanwise vorticity distribution of ${\mathbf{w}}_e$ is shown in the background.

Figure 14

For flow past a NACA 0012 airfoil with $\alpha ={10}^{\circ }$ at $\text{Re}=1000$, detail of the time series of $C_L$ for a trajectory initialized on the limit cycle and subject to localized OTD control with $r=8$ and $\zeta =3.4$ acting in the optimal subdomain identified in Fig. 13 ($x_c=1.5L_c$ and $y_c=0.05L_c$). Control is idle for $t<100$ and active for $t\ge 100$.