DynamisLab/docs/My_Confirmation/Chapters/1.Introduction.tex

\chapter{Introduction} \label{Chap: introduction}

The invisible cloak, a concept long embedded in literary narratives, represents humanity's enduring fascination with the possibility of becoming undetectable. This captivating prospect has transitioned from the realm of fiction to a tangible and vibrant field of scientific inquiry, where researchers seek to manipulate physical fields to render objects imperceptible to external observers. Over the past few decades, this pursuit has yielded remarkable success, with various theories demonstrating feasible methods for achieving cloaking and the more advanced capability of illusion—making an object appear as something else—across diverse physical domains. These breakthroughs have been most prominent in fields governed by linear wave phenomena, such as optics, electromagnetics, and acoustics.

This introductory chapter aims to bridge the gap between these established concepts and our innovative exploration of cloaking and illusion within the highly non-linear environment of fluid dynamics. The primary objective is to provide the reader with a comprehensive understanding of the foundational principles of cloaking while systematically building the argument for the novelty and significance of this report. To achieve this, the chapter is structured along two fundamental axes: the nature of the governing physics (linear vs. non-linear) and the type of control strategy employed (passive vs. active).

Our review will commence by exploring the origins of Passive Cloaking in Optics and Electromagnetics, where theories like transformation optics and scattering cancellation laid the theoretical groundwork. We will then examine the progression to Active Cloaking and Illusion in Wave-based Systems, which introduced adaptive control to overcome the inherent limitations of passive designs. Following this, the discussion will pivot to the fluid domain, first by reviewing efforts in Passive Cloaking in Water Waves, a field that highlights both the potential and the challenges of applying cloaking concepts to fluids, often through linear approximations. Finally, we will survey the mature field of Active Flow Control, noting its traditional focus on performance optimization (e.g., drag reduction) rather than the perceptual management goals of cloaking.

Through this structured four-part review, we will progressively reveal a critical and compelling research frontier: the application of active control to achieve robust hydrodynamic cloaking and illusion in a flow regime dominated by non-linearity. This synthesis will underscore that extending these concepts from linear, wave-based physics to the complex dynamics of vortex shedding around a bluff body is not an incremental step, but a formidable challenge that constitutes the core contribution of this dissertation.

\begin{figure}[t]
     \centering
     \includegraphics[width=0.75\textwidth]{Figures/choi2014paraxial.png}
     \caption{Example of a practical paraxial cloak. (a)-(c) A hand is cloaked for varying directions, while the background image is transmitted properly. (d) On-axis view of the ray optics cloaking device. (e) Setup using practical, easy to obtain optics, for demonstrating paraxial cloaking principles. \cite{choi2014paraxial}}
     \label{fig:choi2014paraxial}
\end{figure}

\section{Passive Cloaking in Optics and Electromagnetics}

The academic journey towards realizing invisibility began in physical domains where the governing principles are described by linear wave equations, most notably optics and electromagnetics. In these fields, the concept of "passive cloaking" emerged as the first and most theoretically developed approach. Passive techniques are characterized by their reliance on static, engineered materials that require no external energy input or active feedback to function. These methods achieve invisibility by surrounding an object with a specially designed shell, or "cloak," which manipulates incident waves (such as light or microwaves) to guide them around the concealed region. To an external observer, the waves emerge on the other side as if they had propagated through empty space, leaving the object and the cloak itself completely undetected. The development of this field has been primarily driven by two elegant and powerful theoretical frameworks: Transformation Mapping Theory and Scattering Cancellation Theory.

The most profound and influential paradigm for passive cloaking is Transformation Optics (TO), a concept formally introduced in 2006 by John Pendry \cite{pendry2006controlling} and Ulf Leonhardt \cite{leonhardt2006optical} in separate seminal works. The theory's power lies in a remarkable insight: the form of Maxwell's equations, which govern electromagnetism, remains invariant under coordinate transformations. This means that a distortion in the coordinate system—akin to bending the fabric of space itself—can be mathematically replicated by a physical medium with precisely tailored material properties.

\begin{figure}[t]
     \centering
     \includegraphics[width=0.5\textwidth]{Figures/pendry2006controlling2.png}
     \caption{The relation between the reference space and real space in TO. (A) The reference space in a Cartesian coordinate system. There is a free space which there is no electromagnetic media filled in this space and so optical ray propagates in a straight line. (B) The real space: after the space transformation, the coordinate grid is curved. \cite{pendry2006controlling}}
     \label{fig:pendry2006controlling2}
\end{figure}

The quintessential cloaking design using this method involves a "push-forward" mapping. In a virtual, mathematical space, one defines a region that is compressed into an infinitesimally small point or line. Since waves cannot penetrate a point, this region is inherently invisible. The transformation then "stretches" or maps this point back into a finite volume (e.g., a sphere or cylinder) in the real, physical world. To make the physics of the real space behave like the physics of the virtual space, one must construct a metamaterial shell around this volume. The required material properties of this shell—specifically its permittivity and permeability—are dictated by the mathematics of the coordinate transformation and are typically highly anisotropic (direction-dependent), inhomogeneous (spatially varying), and may even possess singular values at the inner boundary. When fabricated, this shell compels electromagnetic waves to flow smoothly around the concealed central region, just as the coordinate lines were distorted, rendering any object placed inside invisible.

While the 2006 papers ignited widespread interest, the foundational ideas trace back further. Dolin \cite{dolin1961possibility} had shown the form-invariance of Maxwell's equations as early as 1961, and Ward and Pendry \cite{ward1996refraction} later used coordinate transformations to solve electromagnetic problems in cylindrical geometries in 1996. The generality of the transformation mapping approach has allowed it to be extended far beyond invisibility, enabling the design of other fascinating devices like field concentrators, rotators, and various illusion apparatuses. Furthermore, its principles have been successfully adapted to other linear wave systems, including acoustics and thermodynamics.

\begin{figure}[t]
     \centering
     \includegraphics[width=0.5\textwidth]{Figures/pendry2006controlling.png}
     \caption{A ray-tracing program has been used to calculate ray trajectories in the cloak. (A) A two-dimensional (2D) cross section of rays striking the system, diverted within the annulus of cloaking material contained within $R_1 < r < R_2$ to emerge on the far side undeviated from their original course. (B) A 3D view of the same process. \cite{pendry2006controlling}}
     \label{fig:pendry2006controlling}
\end{figure}

Despite its theoretical elegance, transformation-based cloaking faces a monumental practical challenge: the required materials, with their extreme anisotropic and inhomogeneous properties, are exceptionally difficult, if not impossible, to fabricate perfectly. This fabrication bottleneck spurred the development of a more pragmatic approach known as Scattering Cancellation Theory (SCT).

Instead of bending space, SCT operates on the principle of destructive interference. The goal is to design a cloaking shell that generates a scattered field that is precisely equal in magnitude and opposite in phase to the scattered field produced by the object it encloses. When these two scattered fields superimpose, they perfectly cancel each other out, resulting in a total scattered field of zero. To an observer, the absence of a scattered wave makes the object appear to have vanished.

The primary advantage of SCT is its practicality. It circumvents the need for complex metamaterials by typically employing simple, layered structures composed of homogeneous and isotropic bulk materials with conventional properties. The design process involves analytically solving the governing equations to determine the material parameters and geometry of the layers needed to achieve the desired cancellation for a specific object and incident wave. This method has been successfully demonstrated not only in electromagnetism but also in other fields like heat transfer, proving to be a robust and accessible alternative for creating passive cloaks.

Building upon the same principles of wave manipulation, researchers quickly realized that the goal could be shifted from simply eliminating an object's signature to actively misrepresenting it. This led to the concept of "illusion optics." Rather than engineering a scattered field of zero (cloaking), one could engineer a specific, non-zero scattered field that mimics the signature of a completely different object.

Early demonstrations of this concept showcased remarkable capabilities. For instance, Yang et al. \cite{yang2008superscatterer} designed a "superscatterer" using a shell of negative refractive material that made a small object appear electromagnetically much larger than its actual size. Conversely, Jiang et al. \cite{jiang2011shrinking} used metamaterials to create a "shrinking device," which made an object appear smaller. More advanced designs have since achieved the ability to create illusions of shifted positions or to function interchangeably as a cloak or an illusion device depending on specific parameters. These passive illusion devices highlight the sophisticated level of control achievable over linear wave phenomena through meticulous material design.

In summary, the foundational work on passive cloaking and illusion in linear physical systems has established a rich theoretical and practical landscape. Both the elegant, space-bending approach of Transformation Optics and the pragmatic, interference-based method of Scattering Cancellation Theory have proven highly effective. However, their success is fundamentally tied to the principle of linear superposition, and their static, passive nature inherently limits their performance to specific frequencies and conditions, a critical constraint that motivates the exploration of active control strategies.

\section{Active Cloaking and Illusion in Wave Systems}

While passive cloaking techniques represent a monumental theoretical achievement, they are constrained by fundamental physical principles that limit their practical utility. Passive cloaks, particularly those based on Scattering Cancellation Theory (SCT), are inherently resonant and thus operate effectively only within a narrow frequency band. More critically, as demonstrated by Monticone and Alù \cite{monticone2013cloaked}, the law of energy conservation dictates a trade-off: any linear, passive cloak that reduces scattering at one frequency must necessarily increase the total scattering when integrated across the entire electromagnetic spectrum. This "conservation of scattering" principle implies that a passive cloak, when interrogated by a broadband pulse, could paradoxically scatter more energy than the uncloaked object itself, making it highly detectable in transient regimes. The significant challenge of fabricating the exotic metamaterials required for transformation optics further compounds these bandwidth limitations \cite{chen2013broadening}.

To surmount these inherent constraints, the research paradigm shifted towards active cloaking. This approach represents a fundamental departure from static material design, instead employing a dynamic system of sensors, controllers, and active sources that can adapt in real-time. The core principle of active cloaking is conceptually akin to active noise cancellation: the system is designed to intelligently generate a secondary field that destructively interferes with and cancels the unwanted scattered field from an object. This strategy liberates the design from the narrow bandwidth limitations of passive resonant structures, offering the potential for true broadband invisibility and adaptability to varying incident fields.

One of the earliest and most influential theoretical frameworks for active cloaking was proposed by Miller \cite{miller2006perfect}, who sought to define the requirements for a "perfect" cloak. His design envisioned a shell of sensors and active sources enveloping the object to be hidden. The sensors measure the local properties of an incoming wave, and a controller calculates the precise output for the active sources to radiate a secondary field. This radiated field is engineered to achieve two simultaneous goals: first, to cancel the field scattered by the object, and second, to precisely replicate the original, unperturbed incident field in the exterior region, effectively erasing any trace of the object's presence. Miller's causal simulations, which involved an array of 3,264 source-sensor pairs to cloak a spherical volume from a pulse, highlighted the complexity of the approach but confirmed its theoretical viability. A key insight was that perfect, causal cloaking requires a non-local response—where the action of each source depends on information from other locations—a condition that active systems, unlike passive materials, can fulfill.

Subsequent research has refined this concept into more practical implementations. Guevara-Vasquez et al. \cite{vasquez2009active} developed an active shielding methodology for the 2D Helmholtz equation using a finite number of external active devices. By representing the devices mathematically as multipole source expansions derived from Green's theorem, they demonstrated that as few as three active sources could effectively create a "quiet zone" around an object while simultaneously canceling the far-field scattering. More recently, Lin \cite{lin2021active} has demonstrated a sophisticated active acoustic cloaking system for complex 3D objects by integrating the boundary element method with a modern control strategy. In this work, arrays of monopole control sources and error sensors are distributed around the rigid body. The optimal strength of the control sources is determined by a feedforward linear-quadratic regulator (LQR) algorithm, which minimizes the error signals to cancel the scattered acoustic field.

\begin{figure}[t]
     \centering
     \includegraphics[width=0.5\textwidth]{Figures/lin2021active.png}
     \caption{(a) Primary (uncontrolled) acoustic pressure fields scattering by a sphere due to a monopole source (left column) and a plane wave (right column), (b) cloaked pressure fields, (c) actively modified free fields to resemble the acoustic fields in (a) associated with the presence of a sphere. \cite{lin2021active}}
     \label{fig:lin2021active}
\end{figure}

Crucially, this same active control architecture can be repurposed to create active illusions. The versatility of the active system allows the objective to be shifted from merely canceling the scattered field to actively sculpting it into a new, desired form. As demonstrated by Lin \cite{lin2021active}, by modifying the cost function within the LQR controller, the system can be programmed to make an object generate the acoustic signature of a completely different object. For example, the system could make a complex object like a cube or a bird appear acoustically as a simple sphere, or misrepresent its size and orientation. This capability to generate arbitrary, deceptive wave signatures represents a significant leap beyond simple invisibility.

In conclusion, active cloaking and illusion in linear wave systems provide a powerful and flexible solution to the fundamental broadband limitations of passive methods. By replacing static metamaterials with adaptive sensor-actuator networks governed by sophisticated control algorithms, these systems can achieve robust, tunable control over scattered fields. However, this performance comes at the cost of increased system complexity and computational demand, and introduces new challenges such as maintaining control stability, particularly near system resonance frequencies. Having established these principles in linear, wave-based domains, we now turn our attention to the far greater challenge of applying them to the inherently non-linear world of fluid dynamics.

\section{Passive Cloaking in Fluids}

The successful transfer of cloaking principles from their origins in optics and electromagnetics to other wave-based fields like acoustics stands in stark contrast to the progress within the fluid dynamics community. Establishing a direct connection has proven to be a significant challenge, largely because the physics of fluid motion is fundamentally different. Unlike the linear, often homogeneous environments of wave propagation, fluid flows are governed by the inherently non-linear Navier-Stokes equations. They are characterized by complex phenomena such as viscosity, vorticity, and turbulence, which do not have direct analogues in linear wave theory.

Consequently, the initial forays into hydrodynamic cloaking have strategically avoided the full complexity of fluid dynamics. Instead, researchers have focused on specific physical regimes where the governing equations can be simplified to a linear or quasi-linear form, making them mathematically analogous to the wave problems already solved. This approach, while clever and insightful, has confined passive hydrodynamic cloaking to highly constrained scenarios, most notably linearized water waves and low-Reynolds-number flows.

One of the most visually appealing and well-studied areas for fluid cloaking is the manipulation of surface water waves. The appeal lies in the fact that under a specific set of idealizing assumptions, the problem becomes tractable. The classical theory of water waves typically assumes the fluid is inviscid (no internal friction) and the flow is irrotational (no vorticity). This allows the fluid velocity to be described by a scalar velocity potential, $\Phi$. Furthermore, by assuming the wave steepness is small, the boundary conditions at the free surface can be linearized. Under these conditions, the complex fluid dynamics problem simplifies to solving the Laplace equation for the potential $\Phi$:

\begin{equation}\label{eq:euler}
\Delta^2 \Phi=0
\end{equation}

subject to a linearized dynamic boundary condition on the mean free surface ($z=0$):

\begin{equation}\label{eq:surface}
\Phi_{tt}+g\Phi=0, \quad \text{on}\ z=0
\end{equation}

By considering time-harmonic solutions of the form $\Phi(x,y,z,t) = \Re\{\phi(x,y,z)\exp^{-\rm{i}\omega t}\}$ and $\zeta(x,y,t) = \Re\{\eta(x,y)\exp^{-\rm{i}\omega t}\}$ where $\omega$ is the assumed radian frequency and now $\phi$ and $\eta$ are frequency-dependent complex-valued functions incorporating information about the amplitude and the phase of the fluid motion.

\begin{equation}\label{eq:surface2}
\eta(x,y)=(\rm{i}\omega g)\phi,\quad \phi_z - K\phi = 0,\quad K=\omega^2/g, \text{on}\ z=0
\end{equation}

representing the time-independent free-surface elevation. Waves propagate on the surface of the water (they are akin to guided, interface or surface waves in other physical disciplines) with exponential decay in the direction of increasing depth away from the surface. Waves of amplitude $A$ over a flat bed of depth $h_0$ from infinity in a direction $\theta_0$ w.r.t. the positive x-axis are given by the potential

\begin{equation}\label{eq:wave}
\phi_{i n c}=\frac{-\mathrm{i} g A}{\omega} \frac{\cosh k_0\left(z+h_0\right)}{\cosh k_0 h_0} \mathrm{e}^{\mathrm{i} k_0\left(x \cos \theta_0+y \sin \theta_0\right)}
\end{equation}

and \ref{eq:surface2} provided that the unique positive real root k of

\begin{equation}\label{eq:wave2}
K\equiv \omega^2 / g = k \tanh kh
\end{equation}

corresponding to $h= h_0$ is assigned to $k_0$. The equation \ref{eq:wave2} is called the
water wave dispersion relation and encodes information on how waves of different frequencies travel at different speeds and how both relate to the fluid depth. As usual, the phase speed is given by $c = \omega/k$.

We shall refer to the shallow water regime as being when $kh\ll 1$ or $\lambda\gg h$ where $\lambda= 2\pi/k$ is the wavelength. In this case \ref{eq:wave2} shows that $k \approx \omega/\sqrt{gh}$ and $c\approx \sqrt{gh}$. Water waves are therefore non-dispersive in the shallow water approximation. The resulting framework is mathematically similar to that of acoustics or electromagnetics, allowing for the direct application of cloaking theories.

Within this linearized framework, passive cloaking has been successfully demonstrated. Inspired by transformation optics, researchers have designed cloaks for shallow water waves, where the wave behavior is non-dispersive. More practical approaches have focused on passive structural design. For example, Zou et al. \cite{zou2019broadband} designed and experimentally validated a broadband water wave cloak that functions by converting incident plane waves into guided waves that travel along a channel with a carefully engineered depth profile, before converting them back to plane waves. This method, achieved by simply modifying the bathymetry of the seabed, effectively renders a protected region calm without requiring exotic metamaterials. Such successes, however, are intrinsically linked to the validity of the linear water wave theory and are designed to manipulate surface wave patterns, not the underlying bulk flow.

\begin{figure}[t]
     \centering
     \includegraphics[width=0.75\textwidth]{Figures/zou2019broadband.png}
     \caption{The concept of a broadband cloak for water waves: (a) The schematic plot of the top view of the cloak device and three-dimensional sketch of gradient index metamaterials (GIMs). (b) The side view of the waveguide cloak with GIMs. (c) The experimental photographs of the real cloak device with two GIMs in the water tank. \cite{zou2019broadband}}
     \label{fig:zou2019broadband}
\end{figure}

Another avenue for passive hydrodynamic cloaking has been explored in the regime of very low Reynolds numbers ($Re \ll 1$), where viscous forces dominate over inertial forces. In these scenarios, such as Hele-Shaw flow or flow through porous media, the non-linear terms of the Navier-Stokes equations become negligible, and the flow is governed by linear equations like the Stokes or Darcy equations, which again bear a strong resemblance to the Laplace equation.

This mathematical convenience has enabled the design of passive cloaks for creeping flows. For instance, Chen et al. \cite{chen2024multilayered} recently developed a multilayered, homogeneous hydrodynamic cloak designed to operate in free fluid flow at low Reynolds numbers. Drawing inspiration from Hele-Shaw flow, they constructed a passive shell from concentric layers of alternating heights. Through structural optimization, this device was shown to achieve near-perfect cloaking, guiding the slow-moving fluid smoothly around a central object. While this represents a significant achievement in microfluidic flow control, its success is fundamentally predicated on operating within a linear flow regime, where vortex shedding and inertial effects are absent.

\begin{figure}[t]
     \centering
     \includegraphics[width=0.75\textwidth]{Figures/chen2024multilayered.png}
     \caption{Experimental demonstration of the homogeneous hydrodynamic cloak. The simulated pressure fields around the obstacle without cloak (a) and with cloak (d). The corresponding simulated velocity field with streamlines around the obstacle without cloak (b) and with cloak (e). The experimental results of the flow fields with streamlines of the obstacle without cloak (c) and with cloak (f). \cite{chen2024multilayered}}
     \label{fig:chen2024multilayered}
\end{figure}

In summary, the field of passive hydrodynamic cloaking has made tangible progress by identifying and exploiting specific physical niches where fluid dynamics can be treated as a linear system. These efforts, whether by linearizing the physics of water waves or restricting the context to creeping flows, have successfully translated the core ideas of transformation optics and scattering cancellation into the fluid domain. However, they also clearly delineate the current frontier. These methods are, by design, inapplicable to the vast majority of engineering and natural flows that are characterized by moderate to high Reynolds numbers, where non-linear effects, flow separation, and vortex shedding are not just present, but are the dominant features of the flow. This critical gap underscores the need for a different paradigm, one that can confront non-linearity directly: active flow control.

\section{Active Flow Control and Cloaking}

To confront the challenges posed by the non-linear, vortical flows that render passive methods ineffective, we turn to the mature and powerful discipline of Active Flow Control (AFC). Within fluid mechanics, AFC has a rich history, driven by significant industrial and environmental needs since Prandtl's \cite{krazer1905verhandlungen} foundational work on boundary layer manipulation. AFC technologies utilize external energy input through actuators (e.g., jets, suction/blowing, moving surfaces) and often employ sensors and feedback loops to dynamically alter a flow field for a desired outcome\cite{zhao2023review}.

\begin{figure}[t]
     \centering
     \includegraphics[width=0.75\textwidth]{Figures/zhao2023review1.png}
     \caption{Flow past a cylinder with two jets symmetrically located on the two side of the cylinder. \cite{zhao2023review}}
     \label{fig:zhao2023review1}
\end{figure}

Historically, the objectives of AFC have been almost exclusively performance-oriented and localized. The primary goals have been to optimize the interaction between a fluid and a body, such as reducing drag on vehicles to improve fuel efficiency, enhancing lift on airfoils, suppressing detrimental flow-induced vibrations (FIV) and noise, or promoting mixing in chemical reactors. A vast array of AFC strategies have been developed to achieve these ends. These include the use of secondary flows, such as steady or synthetic jets, to energize the boundary layer and delay flow separation. For example, Feng and Wang \cite{feng2010circular} demonstrated that a synthetic jet at the rear of a cylinder could symmetrize the wake and suppress vortex shedding. More complex systems like windward-suction-leeward-blowing (WSLB), investigated by Wang et al. \cite{wang2016active}, have shown remarkable effectiveness, achieving near-complete suppression of vibrations on bluff bodies. Other prominent methods involve mechanical actuation, such as imposing rotary oscillations on a cylinder or using smaller, rotating control rods in its vicinity to disrupt the formation of large-scale vortices, as reviewed extensively by Zhao et al. \cite{taheri2023enhancement}.

\begin{figure}[t]
     \centering
     \includegraphics[width=0.5\textwidth]{Figures/zhao2023review2.png}
     \caption{VIV suppression by (a) a single rotating rod and (b) two rotating rods. \cite{zhao2023review}}
     \label{fig:zhao2023review2}
\end{figure}

Despite the immense success and sophistication of AFC, a critical conceptual gap has persisted between its traditional application and the goals of cloaking. Conventional AFC research, while highly effective at tasks like drag reduction, has seldom ventured into the territory of perception management. The ambition to not merely optimize a local force or suppress a vibration, but to actively manipulate the entire downstream flow field to perfectly restore the undisturbed, free-stream conditions, represents a radically different and more demanding challenge. This goal, which defines hydrodynamic cloaking, requires the global cancellation of the object's entire hydrodynamic signature, including its velocity deficit, turbulence, and vortical structures.

The immense complexity of this task, which involves controlling a high-dimensional, non-linear, and time-varying system, has made traditional model-based control approaches intractable. This challenge has naturally directed attention towards advanced, data-driven control paradigms, particularly Deep Reinforcement Learning (DRL). DRL offers a model-free framework capable of discovering sophisticated, non-intuitive control policies by directly interacting with a complex environment. A landmark study by Rabault et al. \cite{rabault2019artificial} first demonstrated the power of this approach, successfully training a DRL agent to manipulate jets on a cylinder to stabilize the Kármán vortex street and achieve an 8\% drag reduction at a Reynolds number of 100.

Building on this methodological advance, the first explicit attempt to use AFC for hydrodynamic stealth was presented by Ren et al. \cite{ren2021bluff}. They employed a DRL agent to control a series of WSLB actuators on a cylinder with the express goal of hiding its hydrodynamic traces. Their results were remarkable: the DRL-trained policy learned to manipulate the flow in such a way that the velocity deficit in the near wake was reduced by 99.5\%, effectively erasing the most prominent signature of the cylinder's presence. While born from the lineage of drag and vibration control, their work represents the first tangible step towards true active hydrodynamic cloaking in a non-linear regime.

\begin{figure}[t]
     \centering
     \includegraphics[width=0.75\textwidth]{Figures/ren2021bluff.png}
     \caption{Left: Learning process represented by the variation of cost function values against episode number. The four insets show WSLB actuations generated by the DRL agent at different stages of the learning. Right: Instantaneous wake patterns and measured velocity profiles at the selected four stages. The progressive disappearance of velocity deficit well demonstrates the effectiveness of DRL trained wake control.\cite{ren2021bluff}}
     \label{fig:ren2021bluff}
\end{figure}

However, even with this breakthrough, the field remains in its infancy. While the work of Ren et al. \cite{ren2021bluff} represents a significant step towards active cloaking, the logical and far more complex extension—hydrodynamic illusion—remains virtually unexplored. The ability to not just erase a wake, but to actively sculpt it to convincingly mimic the hydrodynamic signature of a completely different object, requires an even greater level of precision and control.

This dissertation aims to fill this void. We will advance the state-of-the-art by developing and implementing an AFC system capable of achieving not only robust hydrodynamic cloaking but also, for the first time, targeted hydrodynamic illusion in a non-linear, vortical flow environment. To achieve this ambitious goal, we will leverage a Deep Reinforcement Learning framework to discover control strategies for the complex, unsteady wake of cylinder. The specific design of this experimental and computational framework, and the precise formulation of the control problem, will be the subject of the following chapter.