\chapter{Preliminary Results and Discussion} \label{Chap: result}

Having established the theoretical background, problem formulation, and the DRL methodology, this chapter presents the empirical validation of our framework for active hydrodynamic control. The following sections provide concrete evidence of the DRL agent's ability to achieve both robust hydrodynamic cloaking and targeted wake illusion using the fluidic pinball platform. The results demonstrate not only the success of the control strategies but also explore the boundaries of their effectiveness, providing critical insights into the capabilities and limitations of this approach.

This chapter systematically presents the results, organized into four key sections that progressively build in complexity from foundational proof-of-concept to advanced applications and preliminary experimental work.

First, the investigation commences with the foundational challenge of hydrodynamic cloaking in a steady, parabolic inflow. In this baseline scenario, the primary objective is to actively manipulate the flow to eliminate the pinball's characteristic wake, thereby restoring the downstream velocity profile and nullifying the net hydrodynamic forces exerted on the body. This section will demonstrate the DRL agent's ability to rapidly learn an effective control policy that minimizes the pinball's hydrodynamic signature.

Building upon this foundation, the second section addresses the more demanding task of hydrodynamic cloaking in complex, unsteady background flows. Here, the robustness and adaptability of the DRL controller are rigorously tested against a variety of dynamic inflow conditions. We will present results for cloaking in the presence of a periodic Kármán vortex street generated by an upstream cylinder, as well as for transient events corresponding to a passing dipole vortex and an isolated monopole vortex. Furthermore, this section will analyze the influence of key physical parameters by examining the cloaking performance across a range of Reynolds numbers and for different disturbance-generating cylinder diameters, discussing the implications for the controller's generalization capabilities.

The third section transitions from the concept of invisibility to that of active deception, presenting our findings on hydrodynamic illusion. Operating in a steady background flow, we demonstrate the DRL agent's ability to command the fluidic pinball to generate downstream wakes that convincingly mimic the distinct signatures of isolated bluff bodies. We will show that the controller can successfully replicate the wake structures and, crucially, the characteristic vortex shedding frequencies of target cylinders of various sizes and at different Reynolds numbers, effectively creating a hydrodynamic disguise.

Finally, to bridge the gap between numerical simulation and physical reality, the fourth section details our preliminary experimental design and validation efforts. This section will introduce the design of a custom towing test platform for a water tunnel, developed to replicate the numerical setup. We will discuss the integration of control hardware, sensors, and the DRL inference module, as well as our initial attempts at flow field measurement using Particle Image Velocimetry (PIV). This part will candidly address the significant engineering challenges encountered, particularly with sensor noise, and outline the ongoing work to refine the experimental apparatus for future validation.

\section{Cloaking in Steady Flow}

We begin by establishing a baseline for the controller's performance in the most fundamental scenario: cloaking the fluidic pinball in a steady, uniform background flow. While previous studies have achieved similar outcomes with a focus on drag reduction, this section serves as a crucial proof-of-concept for our DRL framework, demonstrating its ability to learn a control policy aimed specifically at wake cancellation and force nullification.

The objective in this scenario is straightforward, as illustrated in the conceptual framework in \fref{fig:schematic_cloak_steady}. The DRL agent must learn to actuate the three rotating cylinders of the pinball in such a way that the steady inflow passes through the system's domain as if the pinball were not present, resulting in a restored steady outflow.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/schematic_cloak_steady.pdf}
     \caption{Conceptual framework for hydrodynamic cloaking in a steady background flow. The DRL agent controls the fluidic pinball to transform the disturbed wake back into the original uniform inflow state, as measured by downstream sensors.}
     \label{fig:schematic_cloak_steady}
\end{figure}

The efficacy of the DRL-derived control policy is immediately apparent in the vorticity fields shown in \fref{fig:on&off_cloak_steady}. The top panel depicts the uncontrolled case, where the stationary pinball generates a classic, unsteady von Kármán vortex street. This shedding process creates a significant wake, fundamentally altering the downstream flow. The velocity profile at the downstream sensor plane is visibly distorted compared to the clean parabolic profile at the inlet. In stark contrast, the bottom panel shows the result once the DRL control is activated. The agent successfully learns to rotate the cylinders to completely suppress vortex shedding, resulting in a smooth, steady flow field that appears to pass through the pinball region unperturbed. The downstream velocity profile is visually restored to match the inlet condition, providing the first qualitative evidence of successful cloaking.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.75\textwidth]{Graphs/on&off_cloak_steady.png}
     \caption{Comparison of vorticity contours with and without DRL-based control in a steady flow. Velocity profiles at the inlet and downstream sensor planes are indicated by arrows. (Top) The uncontrolled pinball generates a distinct vortex street, disrupting the downstream flow. (Bottom) The controlled pinball suppresses vortex shedding, maintaining a steady wake and restoring the downstream velocity profile.}
     \label{fig:on&off_cloak_steady}
\end{figure}

To quantify this observation, \fref{fig:velo_profile} presents a detailed comparison of the velocity profiles at the upstream inlet and the downstream sensor plane. The top panels, corresponding to the uncontrolled case, reveal a significant disruption in the downstream velocity profile, with the relative error in the x-component of velocity exceeding 40\% in the center. Conversely, the bottom panels demonstrate the remarkable performance of the controlled system. The downstream velocity profile closely tracks the original inflow profile, with the relative error contained entirely within a 10\% margin across the cross-section. This confirms that the controller is not merely stabilizing the wake but is actively restoring the flow field.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/velo_profile.pdf}
     \caption{Quantitative analysis of downstream velocity profile restoration. The plots compare the x-component of velocity and the relative error against the inflow profile for the uncontrolled (top) and controlled (bottom) cases. The DRL controller significantly reduces the downstream velocity deficit.}
     \label{fig:velo_profile}
\end{figure}

The mechanism behind this successful cloaking is revealed by analyzing the control actions and the resulting hydrodynamic forces, shown in \fref{fig:action&force_cloak_steady}. The left panel displays the time series of the tangential velocities for the three cylinders. The right panel shows the net forces acting on the pinball assembly. During the initial uncontrolled period (highlighted in blue), the pinball experiences significant oscillating lift forces due to vortex shedding and a substantial mean drag force. Upon activation of the DRL controller, both the lift and drag forces are rapidly driven to near-zero values. Intriguingly, the controller learns to generate a small amount of net thrust (negative drag). We hypothesize that this is a necessary component of perfect cloaking; the agent discovered that to make the flow downstream identical to the flow upstream, it must not only suppress its own wake but also inject a small amount of momentum into the flow to compensate for the viscous drag inherent to its physical presence.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/action&force_cloak_steady.pdf}
     \caption{Control actuation signals and resulting hydrodynamic forces. (Left) Time series of the tangential velocities for the cylinders. (Right) Time series of the net lift and drag coefficients. The controller rapidly nullifies both lift and drag, even generating a slight thrust to achieve a zero-drag state.}
     \label{fig:action&force_cloak_steady}
\end{figure}

In summary, these results conclusively demonstrate that our DRL framework can successfully learn a control policy to achieve hydrodynamic cloaking in a steady flow. However, as this represents a foundational case, the true measure of the system's intelligence and robustness lies in its ability to handle more complex and realistic scenarios. We therefore now transition to the far more challenging problem of cloaking within dynamic, unsteady background flows.

\section{Cloaking in Unsteady Flow}

This section elevates the complexity of the control task by subjecting the fluidic pinball to various unsteady inflow conditions. The objective is no longer to maintain a steady outflow but to ensure that pre-existing, complex flow structures pass by the pinball as if it were transparent.

We first investigate the case where the inflow is a periodic Kármán vortex street, generated by a stationary cylinder placed upstream with a diameter double to that of a single pinball cylinder. The goal, depicted in \fref{fig:schematic_cloak_vortex}, is for the DRL agent to manipulate the pinball's rotation such that the incoming vortex street passes through and is reconstituted downstream with its structure and frequency intact.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/schematic_cloak_vortex.pdf}
     \caption{Conceptual framework for cloaking in an unsteady background flow. The objective is to control the pinball such that the incoming vortex street passes through the control region and is restored downstream, maintaining its original structure.}
     \label{fig:schematic_cloak_vortex}
\end{figure}

\fref{fig:vort_cloak_vortex} provides a visual comparison of the system's result. The top panel shows the target flow field: an undisturbed vortex street. The bottom panel shows the uncontrolled case, where the pinball completely shreds the incoming vortices, creating a chaotic and unrecognizable downstream wake. The middle panel, however, showcases the remarkable success of the DRL controller. The incoming vortices appear to pass directly through the pinball assembly, with their shape, size, and spacing preserved almost perfectly. The contour lines of zero vorticity flow smoothly through the control region, visually demonstrating the cloaking effect. This qualitative observation is supported by quantitative metrics: similarity score between the controlled and target sensor signals reaches 89.6\%, a dramatic improvement from the 0.56\% similarity of the uncontrolled case. Furthermore, the Strouhal number of the wake, which is distorted to 0.086 in the uncontrolled case, is restored to 0.138, perfectly matching the target and correcting a 37.7\% error.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/vort_cloak_vortex.pdf}
     \caption{Vorticity contours demonstrating cloaking performance in an incoming vortex street. Panels show the target flow (top), the DRL-controlled flow (middle), and the uncontrolled flow (bottom). The controlled pinball preserves the structure of the passing vortices, which are otherwise destroyed.}
     \label{fig:vort_cloak_vortex}
\end{figure}

The error fields, shown in \fref{fig:error_cloak_vortex}, quantify the difference between the achieved flow fields and the target. For the uncontrolled pinball (bottom), massive errors exceeding 50\% and even 100\% contaminate the entire downstream domain. In contrast, the DRL-controlled case (top) successfully confines the significant error to the immediate vicinity of the cylinders themselves, with the downstream error remaining largely below 30\%.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/error_cloak_vortex.pdf}
     \caption{Instantaneous error fields relative to the target flow. (Top) The DRL-controlled case confines significant error to the pinball's surroundings. (Bottom) The uncontrolled case results in large, persistent errors throughout the downstream wake.}
     \label{fig:error_cloak_vortex}
\end{figure}

Analysis of the sensor signals provides further insight into the system dynamics. The phase portraits in \fref{fig:sensor_cloak_vortex} plot the velocity components at the three sensor locations. The uncontrolled signals (right) are chaotic and non-repeating, bearing no resemblance to the stable cycles of the target flow (top). The DRL-controlled signals (left), while not perfectly matching the target's shape and position, form stable, periodic cycles, indicating that the controller has successfully stabilized the downstream dynamics into the correct periodic state.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/sensor_cloak_vortex.pdf}
     \caption{Phase portraits of sensor velocity signals ($u_y$ vs. $u_x$) for the target flow (top), the DRL-controlled flow (left), and the uncontrolled flow (right). The controller restores the periodic cycle behavior of the downstream wake.}
     \label{fig:sensor_cloak_vortex}
\end{figure}

The frequency content of these signals, analyzed via Fast Fourier Transform (FFT) in \fref{fig:freq_cloak_vortex}, confirms this conclusion. The uncontrolled signal has a noisy, broadband spectrum. The controlled signal, however, exhibits clear spectral peaks that align precisely with the dominant frequencies of the target vortex street. While a discrepancy in the amplitude of the primary peak remains, suggesting room for further refinement, the controller has successfully captured the fundamental frequency content of the target flow.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/freq_cloak_vortex.pdf}
     \caption{Frequency analysis (FFT) of the y-velocity component from a downstream sensor. The DRL controller successfully reproduces the dominant spectral peaks of the target flow, unlike the noisy spectrum of the uncontrolled case.}
     \label{fig:freq_cloak_vortex}
\end{figure}

The control policy discovered by the DRL agent is itself complex, as shown in \fref{fig:action_cloak_vortex}. The time series of the cylinder rotations (left) are not simple sinusoids. The probability distributions show significant time spent near the mean rotation speed, indicating a more nuanced control strategy. The FFT of the control signals (right) reveals that while the dominant frequency matches the target vortex shedding frequency, several other harmonics are present. A crucial test was performed by simplifying this learned policy to a pure sinusoidal rotation at the main frequency and amplitude. This simplified control still provided a cloaking effect but resulted in a significant drop in similarity from 89.6\% to approximately 70\%, proving that the multi-frequency, non-sinusoidal nature of the DRL-derived policy is essential for high-fidelity cloaking. This suggests a promising direction for future work in extracting simplified, physics-interpretable "white-box" models from the learned DRL agent. An interesting observation is that the mean rotational speeds required for unsteady cloaking are about 20\% lower than those for the steady case, a phenomenon that warrants further investigation.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/action_cloak_vortex.pdf}
     \caption{Analysis of the DRL-derived control policy for unsteady cloaking. (Left) Time series and probability distributions of the cylinder rotational velocities. (Right) FFT of the control signals, revealing a dominant frequency with multiple harmonics.}
     \label{fig:action_cloak_vortex}
\end{figure}

The DRL agent's ability to adapt is highlighted by the transfer learning process shown in \fref{fig:train_cloak}. After being pre-trained on the steady-flow case, the agent was exposed to the new vortex-street environment. The reward curve (left) shows a rapid learning process, achieving satisfactory performance in just 500 episodes. The corresponding flow snapshots at different stages of learning (right) provide a fascinating glimpse into the agent's strategy: it first learns to restore the basic periodic structure of the vortex street (Stage A to B), and then refines its control to more accurately match the strength and shape of the individual vortices (Stage B to C), increasing the similarity from 55.3\% to a final 89.6\%.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/train_cloak.pdf}
     \caption{Transfer learning process for adapting the agent from a steady-flow task to the unsteady vortex-street task. (Left) Reward curve showing rapid learning over 500 episodes. (Right) Vorticity snapshots at stages A, B, and C, illustrating the progressive refinement of the cloaking performance.}
     \label{fig:train_cloak}
\end{figure}

We also tested the robustness of the learned policy across different Reynolds numbers, as shown in \fref{fig:re_cloak_vortex}. While the policy performs well at $Re=50$ (87.1\% similarity) and $Re=100$ (89.6\% similarity), its effectiveness diminishes significantly at higher Reynolds numbers, dropping to 65.3\% at $Re=200$ and 29.5\% at $Re=400$. The flow fields clearly show that the control task becomes progressively more difficult as the flow becomes more inertial and complex, marking a key limitation and an area for future improvement.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/re_cloak_vortex.pdf}
     \caption{Effect of Reynolds number on cloaking performance in a vortex street. The similarity score degrades as the Reynolds number increases, highlighting the increasing difficulty of the control problem.}
     \label{fig:re_cloak_vortex}
\end{figure}

To test the controller against non-periodic disturbances, we designed a scenario where an isolated monopole vortex is generated upstream and travels towards the pinball, as shown in the framework in \fref{fig:schematic_tylor}. The goal is for the controller to execute a transient maneuver to allow the vortex to pass with minimal distortion.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/schematic_tylor.pdf}
     \caption{Conceptual framework for cloaking a transient, aperiodic disturbance, represented by a single monopole vortex.}
     \label{fig:schematic_tylor}
\end{figure}

\fref{fig:cloak_tylor} presents a time-lapsed sequence of this interaction. Without control, the vortex is completely shattered upon impact with the pinball. The DRL-controlled pinball, however, performs a remarkable maneuver: it effectively "catches" the vortex and re-emits a new one downstream. While a comparison with the target flow reveals that the re-emitted vortex has undergone some changes in shape and size, the fundamental structure is preserved.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/cloak_tylor.pdf}
     \caption{Time-lapsed vorticity snapshots of the interaction with a monopole vortex. The sequence shows the vortex approaching, contacting, and leaving the pinball region, compared to the target (undisturbed) flow. The DRL agent enables the re-creation of the vortex downstream.}
     \label{fig:cloak_tylor}
\end{figure}

The control actions for this transient event are, as expected, pulse-like rather than periodic (\fref{fig:action&sensor_tylor}, top). The resulting sensor signals (bottom left) show that while the amplitude is not perfectly matched, the shape of the signal is strikingly similar to the target signal (bottom right), indicating successful replication of the transient event's signature.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/action&sensor_tylor.pdf}
     \caption{Control and sensor signals for monopole vortex cloaking. (Top) The transient control actions of the three cylinders. (Bottom) Comparison of the sensor signals for the cloaked case (left) and the target case (right), showing good shape similarity.}
     \label{fig:action&sensor_tylor}
\end{figure}

Finally, we explored the effect of the vortex's impact position, as shown in \fref{fig:pos_tylor}. We tested scenarios where the vortex's trajectory was offset vertically by up to four cylinder diameters. The results show that the cloaking performance is significantly better for off-center impacts, which is intuitive as the direct interaction with the pinball is less intense. The plot of similarity scores across nine different offset positions confirms this trend. Interestingly, the performance is not symmetric for positive and negative offsets, a result of the vortex's polarity and its differential interaction with the top and bottom cylinders of the pinball.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/pos_tylor.pdf}
     \caption{Influence of monopole vortex impact position on cloaking performance. (Left) Vorticity fields for impacts at $y=-4D$, target, and $y=+4D$. (Right) Similarity score as a variation of the vertical offset, showing improved performance for off-center impacts and an asymmetric response.}
     \label{fig:pos_tylor}
\end{figure}

In conclusion, this section has demonstrated the DRL agent's remarkable ability to achieve hydrodynamic cloaking in a variety of complex, unsteady flows, including both periodic and aperiodic disturbances. The agent learned non-trivial, multi-frequency control policies and showed impressive adaptability. Having established the ability to effectively erase a hydrodynamic wake, we now proceed to the next level of complexity: actively sculpting a new, deceptive wake to achieve hydrodynamic illusion.

\section{Illusion in Steady Flow}

Having demonstrated the ability to erase a wake, we now explore the more ambitious objective of actively sculpting one. This section presents the results for hydrodynamic illusion, where the DRL agent's goal is to command the fluidic pinball to generate a wake that convincingly mimics the signature of a different object—in this case, a single, larger stationary cylinder. The conceptual framework, shown in \fref{fig:schematic_illusion}, illustrates this objective: a steady, uniform inflow is transformed by the controlled pinball into a periodic vortex street characteristic of the target cylinder.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/schematic_illusion.pdf}
     \caption{Conceptual framework for hydrodynamic illusion. The DRL agent controls the pinball in a steady inflow to generate a downstream wake that mimics the periodic vortex street of a target bluff body.}
     \label{fig:schematic_illusion}
\end{figure}

The success of this illusion is strikingly visualized in \fref{fig:vort_illusion}. The top panel shows the target flow field: the classic von Kármán street shed by the target cylinder. The bottom panel shows the wake of the uncontrolled pinball, which, while periodic, has a distinctly different structure and shedding frequency. The middle panel displays the result of the DRL-controlled illusion. The generated wake is a remarkably high-fidelity replica of the target flow field, matching the vortex size, spacing, and structure with great precision. This is quantitatively confirmed by the Strouhal numbers: the target St is 0.133, and the DRL-generated illusion achieves an St of 0.134. In contrast, the uncontrolled pinball sheds at a much lower frequency, with an St of 0.117. The DTW-based similarity score reflects this dramatic improvement, soaring from a mere 13.6\% for the uncontrolled case to an exceptional 97.5\% for the DRL-generated illusion.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/vort_illusion.pdf}
     \caption{Vorticity contours demonstrating illusion performance. Panels show the target flow field (top), the DRL-generated illusion (middle), and the uncontrolled pinball's natural wake (bottom). The controlled pinball successfully replicates the target vortex street with high fidelity.}
     \label{fig:vort_illusion}
\end{figure}

The relative error fields in \fref{fig:error_illusion} further underscore this success. The uncontrolled pinball (bottom) produces a wake that differs from the target across nearly the entire downstream domain, with errors consistently above 50\%. The DRL-generated illusion (top), however, confines significant errors to a very small region immediately surrounding the pinball itself, demonstrating precise control over the far-wake structure.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/error_illusion.pdf}
     \caption{Instantaneous error fields relative to the target flow for the illusion task. (Top) The DRL-controlled illusion maintains low error in the far-wake. (Bottom) The uncontrolled pinball's natural wake differs significantly from the target.}
     \label{fig:error_illusion}
\end{figure}

The phase portraits of the sensor signals in \fref{fig:sensor_illusion} provide a dynamic perspective. While the uncontrolled pinball produces a stable limit cycle (right), its shape and location in phase space differ substantially from the target (top). The DRL-generated illusion (left), however, produces limit cycles that are nearly identical to the target's, with only a minor, barely perceptible difference in the signal from the central sensor.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/sensor_illusion.pdf}
     \caption{Phase portraits of sensor velocity signals for the illusion task. The DRL-generated illusion (left) almost perfectly replicates the limit cycles of the target flow (top), a significant improvement over the uncontrolled case (right).}
     \label{fig:sensor_illusion}
\end{figure}

The frequency spectra in \fref{fig:freq_illusion} confirm the precise frequency matching. While the uncontrolled pinball's spectrum has a dominant peak, it is offset from the target frequency. The DRL controller not only shifts the primary peak to match the target's Strouhal number precisely but also adjusts the amplitudes of the harmonics to better replicate the overall signal shape, resulting in a spectrum that is a near-perfect match to the target.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/freq_illusion.pdf}
     \caption{Frequency analysis of the y-velocity component for the illusion task. The DRL controller precisely matches both the fundamental frequency and the harmonic structure of the target signal.}
     \label{fig:freq_illusion}
\end{figure}

The control strategy for illusion, shown in \fref{fig:action_illusion}, reveals interesting characteristics. The mean rotational speeds are significantly lower than in the cloaking scenarios, and the oscillation amplitudes are also smaller. This is likely because the target object itself has a drag-inducing effect, so the controller does not need to work as hard to generate a wake. The front cylinder appears to be the primary actuator, with the rear two providing finer adjustments. The FFT of the control signals (right) shows a complex, multi-frequency policy, but with a dominant frequency that once again aligns perfectly with the target Strouhal number.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/action_illusion.pdf}
     \caption{Analysis of the control policy for the illusion task. (Left) Time series and probability distributions show lower mean speeds and amplitudes compared to cloaking. (Right) The control signal spectrum is complex but dominated by the target frequency.}
     \label{fig:action_illusion}
\end{figure}

The robustness of the illusion was tested at different Reynolds numbers (\fref{fig:re_illusion}). At $Re=50$, the similarity is an excellent 95.6\%. However, performance degrades at higher Reynolds numbers, dropping to 36.5\% at $Re=200$, where the generated wake shows significant deformation and decay. This, consistent with the cloaking results, indicates that controlling higher-Re flows remains a significant challenge.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/re_illusion.pdf}
     \caption{Effect of Reynolds number on illusion performance. High-fidelity illusion is achieved at lower Re, but performance degrades significantly as Re increases.}
     \label{fig:re_illusion}
\end{figure}

The control policies at different Reynolds numbers also differ, as seen in \fref{fig:action_re_illusion}. At $Re=200$, all three cylinders exhibit significant rotational oscillations of similar amplitude. At $Re=50$, however, the front cylinder dominates the actuation, while the rear cylinders have much smaller amplitudes. This suggests that achieving the control objective in lower-energy, less inertial flows requires less control effort.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/action_re_illusion.pdf}
     \caption{Control signal comparison at $Re=50$ and $Re=200$.}
     \label{fig:action_re_illusion}
\end{figure}

We further explored the pinball's ability to mimic cylinders of different sizes (\fref{fig:d_illusion}). It successfully mimicked a target 1.5 times its characteristic diameter with 97.1\% similarity. However, when attempting to mimic a target 3 times its diameter, the similarity dropped to 65.8\%. While the shape of the wake was still well-replicated, the vortex strength was visibly weaker, indicating a physical limitation on the amount of momentum the pinball can inject into the flow.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/d_illusion.pdf}
     \caption{Illusion performance when mimicking cylinders of different sizes. The plots show the generated wakes for targets of 1.5D and 3.0D.}
     \label{fig:d_illusion}
\end{figure}

A fascinating phenomenon was observed in the control signals for these different-sized illusions (\fref{fig:action_d_illusion}). When mimicking a target significantly larger than itself, the dominant frequency of the control signal shifted to a higher harmonic of the target Strouhal number. This suggests the agent discovered a more energy-efficient, high-frequency actuation strategy to generate the larger-scale, lower-frequency flow structures. The underlying physical mechanism for this frequency-doubling control strategy is a compelling area for future investigation.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/action_d_illusion.pdf}
     \caption{Control signals for mimicking different-sized targets. When mimicking the larger (3D) target, the control frequency shifts to a higher harmonic, indicating a change in control strategy.}
     \label{fig:action_d_illusion}
\end{figure}

\section{Illusion in Unsteady Flow}

As an early exploration of what may be considered the most challenging of this research, we briefly investigated the task of creating an illusion in an unsteady flow. The objective, shown in the conceptual framework of \fref{fig:schematic_illusion_unsteady}, is immensely difficult: to take an incoming periodic vortex street and actively cancel it to produce a steady, uniform outflow. This requires the controller to counteract a large-scale, energetic inflow using only the limited authority of cylinder rotation, representing a far greater challenge than any of the previous tasks.

\begin{figure}[h!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/schematic_illusion_unsteady.pdf}
     \caption{Conceptual framework for the most challenging task: creating a steady-flow illusion from an unsteady vortex-street inflow.}
     \label{fig:schematic_illusion_unsteady}
\end{figure}

Our initial results, presented in the error fields of \fref{fig:error_illusion_unsteady}, are promising but underscore the profound difficulty of this objective. The DRL-controlled case (top) demonstrates a significant improvement over the uncontrolled scenario (bottom), achieving a similarity of 64.6\% with the target steady flow, a substantial increase from the 16.5\% similarity of the uncontrolled case. A notable artifact in the controlled case is the presence of some error upstream of the pinball. This is attributed to the imperfect control policy, which not only manipulates the downstream flow but also generates pressure waves that propagate upstream, subtly altering the vortex shedding frequency of the disturbance-generating cylinder itself. This highlights that the control is not yet fully optimized. To isolate the control problem and eliminate this upstream feedback loop in future investigations, we plan to implement a fixed inflow boundary condition using a pre-recorded velocity field. Nevertheless, the primary goal of downstream wake cancellation shows clear progress. 

\begin{figure}[htbp!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/error_illusion_unsteady.pdf}
     \caption{Preliminary error fields for the unsteady-to-steady illusion task. The controller significantly reduces downstream error compared to the uncontrolled case, but achieving a perfectly steady state remains a challenge.}
     \label{fig:error_illusion_unsteady}
\end{figure}

The phase portraits in \fref{fig:sensor_illusion_unsteady} illustrate the progress and the remaining challenge. The uncontrolled signals (light color) are chaotic and occupy a large region of the phase space. The DRL-controlled signals (dark color) are successfully corralled into a much smaller region around the target fixed points (black circles), which represent a perfectly steady flow. While the controller has not yet learned to completely damp the oscillations, it has made significant progress in stabilizing the flow. This indicates that with further training and perhaps architectural refinements, achieving this difficult illusion may be possible.

\begin{figure}[htbp!]
     \centering
     \includegraphics[width=0.40\textwidth]{Graphs/sensor_illusion_unsteady.pdf}
     \caption{Phase portraits for the unsteady-to-steady illusion task. The DRL controller (dark lines) successfully confines the chaotic signals (light lines) to the vicinity of the target steady-state points (black circles).}
     \label{fig:sensor_illusion_unsteady}
\end{figure}

\section{Experiment Setup}

To enhance the credibility of our numerical findings and bridge the gap to real-world application, physical validation is essential. We have designed and initiated the construction of an experimental platform to verify our results for cloaking in a steady flow within a water tunnel environment.

To ensure a stable, low-Reynolds-number inflow, we opted for a towing-platform design, as depicted in the top-view and side-view schematics in \fref{fig:topview} and \ref{fig:sideview}. The entire assembly—including the fluidic pinball, control motors, sensors, and the DRL inference module—is mounted on a cart that travels along a linear rail system. The cart is driven by a timing belt connected to an external motor, allowing for precise control of the towing velocity. Flow field measurements are planned using a planar laser sheet and a camera for Particle Image Velocimetry (PIV) imaging through a side window. Figure 4.32 shows a 3D rendering of the complete experimental setup.

\begin{figure}[htbp!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/topview.pdf}
     \caption{Top-view schematic of the towing-platform experimental setup for water tunnel validation.}
     \label{fig:topview}
\end{figure}

\begin{figure}[htbp!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/sideview.pdf}
     \caption{Side-view schematic of the experimental setup, showing the PIV measurement configuration.}
     \label{fig:sideview}
\end{figure}

\begin{figure}[htbp!]
     \centering
     \includegraphics[width=0.70\textwidth]{Graphs/renderview.png}
     \caption{3D rendered model of the integrated experimental platform.}
     \label{fig:renderview}
\end{figure}

Significant engineering challenges arise from the physical constraints of the water tunnel (25cm width). To operate in the desired low-Re regime, the pinball cylinders must be small (preliminary design: 1cm diameter). This results in extremely small hydrodynamic forces, requiring force sensors with micro-Newton resolution. Furthermore, the need for three independently rotating cylinders in close proximity demands a compact sensor design that incorporates a bearing and ensures high concentricity to avoid spurious forces from eccentric rotation.

Given these unique and demanding requirements, custom instrumentation was necessary. \fref{fig:force_sensor} shows a custom-designed and fabricated device that integrates an air bearing with a 2D force sensor. The design is stackable to allow for simultaneous force measurement on all three cylinders. It is constructed from copper to minimize thermal expansion effects and uses semiconductor strain gauges in a half-bridge configuration to measure minute deformations while rejecting thermal drift. To process these signals, a custom low-noise, high-precision data acquisition system was also developed, as shown in \fref{fig:adc}. This system is designed to integrate seamlessly with our Python-based DRL training framework.

\begin{figure}[htbp!]
     \centering
     \includegraphics[width=0.70\textwidth]{Graphs/force_sensor.jpg}
     \caption{Photograph of the custom-designed, stackable device integrating an air bearing and a 2D force sensor for measuring micro-Newton forces.}
     \label{fig:force_sensor}
\end{figure}

\begin{figure}[htbp!]
     \centering
     \includegraphics[width=0.70\textwidth]{Graphs/adc.jpg}
     \caption{Photograph of the custom multi-channel, low-noise signal acquisition hardware.}
     \label{fig:adc}
\end{figure}

Unfortunately, initial assembly and testing have revealed several critical issues. The force sensor design suffers from internal stress concentration due to over-constraint and welding processes, leading to significant signal noise. Additionally, the manufacturing precision of the linear rail system is insufficient, causing velocity instability and vibrations during towing, which further contaminates the sensitive force measurements.

To overcome these hurdles, we are pursuing two parallel paths. The first is the iterative redesign and optimization of the experimental hardware. The second is a numerical exploration into reducing the number of observations required for the DRL agent, which could significantly lower the difficulty of experimental measurement. As shown in \fref{fig:reduce_obs}, we tested the cloaking performance when the number of state observations was reduced from the original 12 down to 9, 5, and even 2. The learning curves show that while performance degrades slightly, the agent can still achieve a high reward even with a drastically reduced set of inputs. This promising result suggests that by optimizing the sensor placement and control architecture, we may be able to achieve effective control in the physical experiment with a much simpler and less noise-prone measurement system.

\begin{figure}[htbp!]
     \centering
     \includegraphics[width=0.80\textwidth]{Graphs/reduce_obs.pdf}
     \caption{Learning curves showing the effect of reducing the number of state observations. The agent maintains strong learning performance even when the number of inputs is reduced from 12 (red reference) to 9, 5, and 2 (blue), suggesting a path towards simplified experimental validation.}
     \label{fig:reduce_obs}
\end{figure}

With the presentation of these numerical and preliminary experimental results, we have demonstrated the core contributions of this thesis. The final chapter will summarize these findings, draw conclusions, and outline promising avenues for future work.