Examples

Articles using the previous versions of the toolbox

The following articles were based on previous versions of the code:

• Jussiau, W., Leclercq, C., Demourant, F., & Apkarian, P. (2022). Learning linear feedback controllers for suppressing the vortex-shedding flow past a cylinder. IEEE Control Systems Letters, 6, 3212-3217.
• Jussiau, W., Leclercq, C., Demourant, F., & Apkarian, P. (2024). Data-driven stabilization of an oscillating flow with linear time-invariant controllers. Journal of Fluid Mechanics, 999, A86.
• Jussiau, W., Demourant, F., Leclercq, C., & Apkarian, P. (2025). Control of a Class of High-Dimensional Nonlinear Oscillators: Application to Flow Stabilization. IEEE Transactions on Control Systems Technology.

Flow past a cylinder at Re=100

Illustration

Stationary solution

Periodic attractor (stable limit cycle)

Feedback configuration

The default feedback configuration (same as in Jussiau, W., Leclercq, C., Demourant, F., & Apkarian, P. (2022). Learning linear feedback controllers for suppressing the vortex-shedding flow past a cylinder. IEEE Control Systems Letters, 6, 3212-3217.) is as follows:

• Cross-stream velocity measurement $y(t)=v_2({x_s}, t)$ in the wake at ${x_s} = [3, 0]$,

• Boundary actuation at the poles of the cylinder, acting on the cross-stream velocity $v_2$. The velocity profile on the actuated boundary reads:

$${v_{act}}({x}, t) = - \dfrac{(x_1-l)(x_1+l)}{l^2} u(t)$$

where $u(t)$ is the control input, $l = \frac{1}{2} D \sin \left( \frac{\delta}{2} \right)$, $\delta=10\degree$ are tunable actuator parameters.

Flow over an open cavity at Re=7500

Illustration

Stationary solution

Quasi-periodic attractor

Contrary to the cylinder, the attractor on the cavity at Re=7500 is quasi-periodic (featuring two incommensurable frequencies in its frequency spectrum):

Feedback configuration

The default feedback configuration (same as in Leclercq et al. (2019). Linear iterative method for closed-loop control of quasiperiodic flows. Journal of Fluid Mechanics, 868, 26-65.) is as follows:

• Actuation is produced near the upstream edge of the cavity by a volume force $f({x}, t)=B({x}) u(t)$ in the momentum equation, acting on the cross-stream velocity, with:

$$B({x})=\left[ 0, \eta \exp\left( \frac{\left(x_1 - x_1^0\right)^2 + \left(x_2 - x_2^0\right)^2}{2\sigma_0^2} \right), 0 \right]^T$$

By default, the center of the actuator is $(x_1^0, x_2^0) = (-0.1, 0.02)$, just before the cavity and slightly above the wall. The amplitude $\eta\approx 8.25$ is chosen such that $\int_\Omega B({x})^T B({x}) d\Omega = 1$. The spatial extent of the actuation is set by $\sigma_0 = 0.0849$, making the force reach $50%$ of its peak value at a distance $0.1$ from its center.

• The measurement is made through wall friction on the bottom wall just downstream of the cavity:

$$y(t) = \int_{x_1=1}^{1.1} \left. \frac{\partial v_1(x, t)}{\partial x_2} \right\rvert_{x_2=0} dx_1$$