Time-delayed feedback technique for suppressing instabilities in time-periodic flow

A numerical method is presented that allows to compute time-periodic flow states, even in the presence of hydrodynamic instabilities. The method is based on filtering nonharmonic components by way of delayed feedback control, as introduced by Pyragas [Phys. Lett. A 170, 421 (1992)]. Its use in flow problems is demonstrated here for the case of a periodically forced laminar jet, subject to a subharmonic instability that gives rise to vortex pairing. The optimal choice of the filter gain, which is a free parameter in the stabilization procedure, is investigated in the context of a low-dimensional model problem, and it is shown that this model predicts well the filter performance in the high-dimensional flow system. Vortex pairing in the jet is efficiently suppressed, so that the unstable periodic flow state in response to harmonic forcing is accurately retrieved. The procedure is straightforward to implement inside any standard flow solver. Memory requirements for the delayed feedback control can be significantly reduced by means of time interpolation between checkpoints. Finally, the method is extended for the treatment of periodic problems where the frequency is not known a priori. This procedure is demonstrated for a three-dimensional cubic lid-driven cavity in supercritical conditions.

Figure 1

Vorticity snapshots of paired and unpaired states, obtained without stabilization for two different parameter settings. The paired state (a) was obtained at $\mathrm{Re}=2000$ and $\mathrm{St}=0.6$ while the unpaired state (b) was obtained at $\mathrm{Re}=1300$ and $\mathrm{St}=0.6$. Reynolds and Strouhal numbers are defined in Sec. 4a.

Figure 2

Gain of the delayed feedback transfer function.

Figure 3

Real part of eigenvalues ${\alpha }_j^h$ for $-4\le j\le 4$, pertaining to fundamental oscillations, as functions of $\lambda$. It is found numerically that ${\alpha }_j^h$ and ${\alpha }_{-j}^h$ always have the same real part. The system is neutrally stable for any value of $\lambda$, with the neutral mode ${\alpha }_0^h=i$.

Figure 4

Real part of eigenvalues ${\alpha }_j^s$, pertaining to subharmonic oscillations, as functions of $\lambda$. On the left, eigenvalues for $-2\le j\le 2$; on the right, zoom on small $\lambda$ values only for $j=0$. All these eigenvalues are stable, but least so for the $j=0$ branch. Higher eigenvalues than those shown are even more stable.

Figure 5

Evolution of the residual norm $e\left(t\right)$ [Eq. (17)], in a jet simulation with $\text{Re}=2000,\text{St}=0.6$, and $\lambda =0.04432$. Dashed line: decay rate found in the model problem. Markers indicate the instances of snapshots shown in Fig. 6.

Figure 6

Vorticity (left) and nonharmonic component magnitude $\parallel \mathbf{u}\left(t\right)-\mathbf{u}(t-T)\parallel$ (right) represented at $t=T$ (a)-(b), $3T$ (c)-(d), $10T$ (e)-(f ), $23T$ (g)-(h) and $44T$ (i)-(j) for ${\lambda }_{\mathrm{opt}}=0.04432$. The vorticity colorbar is in Fig. 1.

Figure 7

Residual norm as a function of time for several values of $\lambda :\lambda \le 0.0432$ (a) and $\lambda \ge 0.0432$ (b). Curves for $\lambda =0.0425,0.0475$ are omitted for clarity. At values of $\lambda$ larger than 0.5, the convergence is increasingly ill-behaved, displaying huge oscillating behavior, and results are not reported.

Figure 8

Residual norm $e\left(t\right)$ at $t=50T,100T,$ and $250T$ as a function of $\lambda$.

Figure 9

Evolution of the residual norm $e\left(t\right)$ with and without stabilization applied. The different time steps defined in Sec. 4d are reported.

Figure 10

Vorticity (left) and nonharmonic component magnitude $\parallel \mathbf{u}\left(t\right)-\mathbf{u}(t-T)\parallel$ (right) without time-delayed feedback applied represented at $t=14T$ (a)-(b) and $200T$ (c)-(d). The vorticity colorbar is in Fig. 1.

Figure 11

Vorticity (left) and nonharmonic component magnitude $\parallel \mathbf{u}\left(t\right)-\mathbf{u}(t-T)\parallel$ (right) with time-delayed feedback applied represented at $t=4T$ (a)-(b), $15T$ (c)-(d) and $23T$ (e)-(f ). The vorticity colorbar is in Fig. 1.

Figure 12

Convergence analysis of the stabilization procedure with interpolation, for different storage requirements $N$: (a) residual norm based on the interpolated velocity ${\tilde{e}}_N\left(t\right)$ as a function of time for different $N$ and comparison to full-storage residual $e\left(t\right)$ and (b) ratio between the exact residual $e_N\left(t\right)$ and the residual obtained with full storage $e\left(t\right)$ as a function of time for different $N$. For $N=5$, 10 and 20, ${\tilde{E}}_N$ is defined as the maximum peak of ${\tilde{e}}_N\left(t\right)$ when the residual starts oscillating. For $N=50$, no oscillations of ${\tilde{e}}_N\left(t\right)$ are observed.

Figure 13

Convergence analysis of the lid-driven cavity: (a) norm of the residual component as a function of time for the different cases studied normalized with the total velocity norm at $t=50T_0$ for the fixed ${\omega }_0$ case and (b) evolution of the global frequency guess for each case as a function of time.