Parametric Reduced-Order modeling and Closed-Loop Control of Tandem-Cylinder Wakes

Abstract

The flow around two circular cylinders arranged in a tandem exhibits complex wake interactions that lead to amplified unsteady loads, particularly in the co-shedding regime where a fully developed wake forms in the gap between the cylinders. Although various control strategies have been proposed to mitigate these effects, most prior studies have focused primarily on load alleviation. Complete suppression of vortex shedding, both in the gap region and in the wake of the second cylinder, has so far only been achieved using open-loop approaches. In this work, we propose a closed-loop control framework for suppressing vortex shedding in tandem cylinder flows in the co-shedding regime. Focusing on low Reynolds numbers and sufficiently large spacings, we derive a parametric reduced-order model using a global weakly nonlinear analysis of the incompressible Navier-Stokes equations. The model is generalized to account for time dependent forcing and facilitates the real time prediction of the flow evolution. Using this model, we design a model predictive controller and apply it to the full-order system via velocity measurements and volumetric forcing. The approach is demonstrated for a cylinder spacing of eight diameters. Vortex shedding is fully suppressed in both the gap region and the downstream wake for Reynolds numbers $Re=50$, $60$, and $70$, while a significant reduction in flow unsteadiness is achieved at $Re=80$. We further show that effective control is possible with limited sensing: suppression is achieved using a single measurement point for $Re=50$ and two-point measurements for $Re=60$ and $70$.

Figures (15)

• Figure 1: Schematic of the closed-loop architecture implemented here to control the oscillating flow around tandem cylinders. (a) Output-feedback loop for the forced incompressible Navier-Stokes equations (full-order plant). (b) State-feedback loop for the reduced-order model.
• Figure 2: Schematic of the model-predictive controller. At each time step $t_j$, the future states $\mathbf{X}_{j+1},...,\mathbf{X}_{j+m}$ are predicted by the surrogate model $\mathbf{F}$ with the initial state $\mathbf{X}_j$ set equal to its estimate $\widetilde{\mathbf{X}}_j$ from the flow measurements. The inputs are determined by minimizing the cost function $J$. The first input $\mathbf{q}_j$ is applied to the system and the same process is repeated at the next time step.
• Figure 3: Schematic defining the computational domain. The streamwise and transverse coordinates $x_{-\infty}$/$x_{\infty}$ and $-y_{\infty}$/$y_{\infty}$, determine respectively the location of the inlet/outlet and lower/upper boundaries. The primary flow direction is from left to right.
• Figure 4: Velocity fields illustrating the steady base flow and the oscillatory flow on the limit cycle at different Reynolds numbers. Shown are the $x$-component $u_b$ of the base flow at $Re=50$ (a), $Re=60$ (c), $Re=70$ (e), and $Re=80$ (g), and snapshots of the $y$-component $v'$ of the velocity field oscillating on the limit cycle at $Re=50$ (b), $Re=60$ (d), $Re=70$ (f), and $Re=80$ (h).
• Figure 5: Hydrodynamic force coefficients for fully developed limit-cycle flow over two cylinders in tandem configuration with spacing $\gamma=8$, compared with the single-cylinder case, as a function of Reynolds number. (a) RMS lift coefficient $C_L'$. (b) Mean drag coefficient $\bar{C}_D$ and steady drag coefficient $C_{Db}$. (c) RMS drag coefficient $C_D'$.