14:50   Classical and Machine learning methods in flow control III Experimental investigation of burst-in-burst control of high-angle-of-attack separated flow over a NACA0015 aerofoil Ayano Watanabe, Naoki Takada, Satoshi Shimomura, Shuji Otomo, Akira Oyama, Hiroyuki Nishida Abstract: Flow separation over aerofoils causes stall, a loss of lift and an increase in drag, which is an unwanted phenomenon for many engineering devices. To suppress flow separation, feedback control is effective in responding to rapid changes in the flow field. Feedback control requires a highly responsive fluid control actuator, and plasma actuators (PA) have attracted much attention as such an actuator. In a previous study, a control strategy called burst-in-burst control was obtained by feedback control using PAs and deep reinforcement learning. Burst-in-burst control is the control method in which the PA is repeatedly turned on and off in a burst actuation to repeat separated and attached. This control method has obtained a higher control effect than the conventional fixed burst frequency control, but has not yet achieved continuous separation suppression. The aim of this study is to clarify the necessity for the duration when the PA drives off in burst-in-burst control, in order to obtain continuous separation suppression in high angle of attack separated flow. We conduct a predetermined control by varying the PA OFF duration to investigate its effect on the pressure time-history, and the pressure history of the fixed burst frequency case was measured and the average pressure distribution around the aerofoil was compared with between PA drive on and off. Low-order model of the fluidic pinball under time-varying control Alicia Rodríguez-Asensio, Guy Y. Cornejo Maceda, Luigi Marra, Andrea Meilán-Vila, Bernd R. Noack, Andrea Ianiro, Stefano Discetti Abstract: Introduction Flow control is at the heart of engineering applications. However, the high-dimensionality of the flow, the nonlinearities inherent to the Navier-Stokes equations, and time delays between actuation and sensing challenge control design and call for complex algorithms. Reduced-order models (ROMs) of the flow have been employed to reduce the computational or experimental load of control design algorithms and accelerate the learning process. Recently, low-dimensional representations of shear flow transients [Farzamnik et al., 2023] and under control [Marra et al., 2024] have been identified with manifold learning based on isometric feature mapping (ISOMAP) [Tenenbaum et al., 2000]. In particular, Marra et al. [2024] identified a five-dimensional manifold—referred to as the steady actuation manifold—that describes the dynamics of the fluidic pinball under steady actuation. However, extending them to scenarios with time-varying actuation remains an open challenge. In this work, we propose a low-order model of the fluidic pinball under time-varying control. Key enablers are the automatic extraction of manifold coordinates with ISOMAP, and their rescaling with the rate of change of the control. The fluidic pinball, a cluster of three rotating cylinders, is chosen as a testbed for control-oriented modeling of turbulent wakes. This work focuses on boat-tailing, where the rear cylinders rotate inward, reducing drag by vectoring the wake. First, a three-dimensional steady actuation manifold is extracted from constant actuation with ISOMAP. Similarly to Marra et al. [2024], the manifold coordinates are related to the drag, lift, and timedelayed lift coefficients. Second, a manifold that includes steady and unsteady control is identified with ISOMAP. Specifically, controls with constant rate of change of boat-tailing parameter are considered. As expected, the state trajectories of the unsteady cases depart from the steady actuation manifold. Interestingly, the lift and drag of the unsteady cases display two different response time scales related to the derivative and the squared derivative of the actuation command, respectively. The rescaling of the manifold coordinates allows for the collapse of all the trajectories onto the steady actuation manifold. Third, a control-oriented reduced-order model is built. Starting point is the model proposed by Noack et al. [2003] for the transient dynamics of the cylinder flow. Two control terms are included to account for the two time scales. The resulting model describes the dynamics of the fluidic pinball under steadily increasing control. The approach is validated at Reynolds number Re = 30. The proposed control-oriented ROM captures the transient wake response, providing a framework for extending the steady actuation manifold to time-varying actuation scenarios. Flow configuration The fluidic pinball consists of three rotating circular cylinders in an incoming flow. Figure 1 describes the fluidic pinball configuration. The centers of the cylinders are located at the vertices of an equilateral triangle pointing upstream. The radius of the cylinders is noted R and the incoming velocity U∞. The flow is manipulated with the independent rotation of the cylinders. The peripheral velocities of the cylinders are noted as b1, b2, and b3 for the front, top, and bottom cylinders, respectively. By varying b1, b2, and b3, the fluidic pinball can reproduce up to six wake stabilization mechanisms: high and low-frequency control, boat-tailing, base-bleeding, Magnus effect, and phasor control. In particular, we focus on the boat-tailing actuation, where the two rear cylinders rotate in opposite directions (i.e. b2 > 0, b3 < 0), vectoring the flow inwards. Simulations are carried out at Reynolds number Re = U∞2R/ν = 30 where ν is kinematic viscosity. The convective time units are defined as D/U∞. The computational domain is bounded by a rectangle of [−6, 20] × [−6, 6], with no-slip walls on the cylinder surfaces without forcing. An unstructured triangular mesh (4 225 elements, 8 633 vertices) ensures grid-independent results. The Navier-Stokes equations are solved with an inhouse code based on a fully implicit time integration and finite element method discretization [Noack et al., 2003]. Lift (Cl) and drag (Cd) coefficients are obtained by normalizing the total lift and drag forces, respectively, with 1/2ρU2∞D, being ρ the fluid density. Steady actuation manifold and transient response The steady actuation manifold is constructed by collecting post-transient velocity snapshots of the fluidic pinball under no rotation rate changes. Following Marra et al. [2024], the complete manifold embeds a dataset which spans combinations of (b1, b2, b3) ∈ [−3, 3]3 with dbi/dt = 0. In the present study, the actuation mechanisms are restricted to the boat-tailing subspace defined by b1 = 0, b2 = −b3, b2 ∈ [0, 1.5]. For the sake of simplicity b = b2 = −b3. It interpolates between the unforced limit cycle (b = 0) and the disappearance of coherent structures (b = 1.5). Applying a nonlinear manifold learning technique, the dataset is embedded into a fivedimensional manifold that captures the flow dynamics across all steady actuation states. Remarkably, the first three manifold coordinates correlate strongly with the lift coefficient Cl(t), drag coefficient Cd(t), and time-delayed lift coefficient Cl(t − τ ) (with τ = T0/4, where T0 is the unforced period). This equivalence implies that the steady actuation manifold can be constructed using either the first three coordinates of the full-field snapshots low-dimensional space or the time series {Cd(t), Cl(t), Cl(t − τ )}. In this abstract, the force coefficients are exploited both to represent the manifold and to build the analytical low-order model. To probe transient departures from the steady actuation manifold, additional simulations are performed in which the actuation ramps from b = 0 to b = 1.5 at constant rate ˙b. ˙b is measured in units of b per limit-cycle periods. For example, ˙b = 0.15 requires ten vortex-shedding cycles to reach b = 1.5, whereas ˙b = 0.75 completes the control in four cycles. As seen in Figure 2a, the steady actuation manifold (gray) is composed by a set of curves, each of them corresponding to one actuation case. It overlays with two transient trajectories that departs from the steady actuation manifold: a slow instance (˙b = 0.15, orange) that closely follows the manifold with only a slight phase lag in Cl, and a faster case (˙b = 0.75, red) that exhibits larger deviations in Cd and Cl before relaxing back onto the steady actuation manifold. In summary, time-varying actuation induces transient trajectories that leave the steady actuation manifold and then return once the target rotation rate is achieved and the force coefficients equilibrate. The magnitude and duration of these departures depend on the actuation change rate ˙b, with faster changes producing more pronounced deviations. Understanding these transients is critical for developing an analytical, control-oriented low-order model. Low-order analytical model By examining the transient trajectories, one can observe that they align around the steady actuation manifold, albeit with deviations caused by the rate of actuation change. This suggests that transient dynamics can be interpreted as scaled versions of the steady actuation manifold. This allows to fit the time-varying control effects into a unified model. Specifically, the force coefficients can be rescaled by factors that depend on the actuation rate (˙b). As shown in Figure 2b, both slower and faster trajectories collapse into the steady actuation manifold when appropriately scaled. To capture these dynamics analytically, we build on the three-mode model of Noack et al. [2003], which in cylindrical form reads: d dt A = (μ − w + f (b, ˙b))A, d dt ϕ = 1, d dt w = −w + A2 + g(b, ˙b) where A is the vortex-shedding amplitude, w represents the mean drag mode, and μ is the linear growth rate. In our setting, A and w are identified directly with the lift and drag coefficients, respectively. More precisely, A ∼ pCl(t)2 + Cl(t − τ )2 and w ∼ Cd(t), while μ corresponds to the unforced shedding amplitude. The terms f (b, ˙b) and g(b, ˙b) are incorporated to account for the two time delays due to control effects on the growth rate and the drag coefficient, respectively.

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