docs(jfm): establish traceable manuscript planning baseline
Track the research dossiers, section freezes, supporting manuscript materials, and round-aware agent controls so future drafting decisions can be reviewed across both repository mirrors. Co-authored-by: Cursor <cursoragent@cursor.com>
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{}
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\chapter{Introduction} \label{Chap: introduction}
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This is a latex template for Ph.D. thesis in The Hong Kong Polytechnic University. It can be also used for other universities or degrees.
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\section{Commands}
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This is an example for renew commands, \eg, we can use \textbackslash eg to generate \eg. The same thing goes to \etc.
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We can use \textbackslash rev\{\} to mark your text: \rev{revised text}.
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\section{Citation}
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This is an example for citation \cite{pan2012investigation}.
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\section{Figures}
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This is a figure, as shown in \fref{fig:figure}.
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\begin{figure}[t]
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\centering
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\includegraphics[width=0.5\textwidth]{Figures/Illustration.pdf}
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\caption{Illustration of a figure.}
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\label{fig:figure}
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\end{figure}
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Figure \ref{fig:subfigure} shows an example of subfigures (\fref{fig:subfigure1} and \fref{fig:subfigure2}).
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\begin{figure}[t]
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\centering
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\subfloat[Subfigure 1]{
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\includegraphics[width=0.45\columnwidth]{Figures/fr_ear_left.pdf}
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\label{fig:subfigure1}
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}
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\subfloat[Subfigure 2]{
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\includegraphics[width=0.45\columnwidth]{Figures/fr_ear_right.pdf}
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\label{fig:subfigure2}
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}
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\caption{Frequency response with and without ears.}
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\label{fig:subfigure}
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\end{figure}
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This is an example for the minipage (\fref{fig:minipage1} and \fref{fig:minipage2}).
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\begin{figure}[t]
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\begin{minipage}{0.48\linewidth}
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\centering
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\includegraphics[width=0.58\columnwidth]{Figures/OE_cardioid.pdf}
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\caption{Minipage 1.}
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\label{fig:minipage1}
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\end{minipage}
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\hspace{5pt}
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\begin{minipage}{0.48\linewidth}
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\centering
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\includegraphics[width=0.58\columnwidth]{OE_sameDisDiffDeg.pdf}
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\caption{Minipage 2.}
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\label{fig:minipage2}
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\end{minipage}
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\end{figure}
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\section{Table}
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A table example is shown in \tref{tab:table}. It is a three-line table.
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\begin{table}[t]
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\renewcommand{\arraystretch}{1.1}
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\centering
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\caption{PolyU rankings.}
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\begin{tabular}{ccccc}
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\toprule
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Rankings &
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\begin{tabular}[c]{@{}c@{}}QS \\ World University\end{tabular} &
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\begin{tabular}[c]{@{}c@{}}QS \\ Asia University \end{tabular} &
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\begin{tabular}[c]{@{}c@{}}THE \\ World University \end{tabular} &
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\begin{tabular}[c]{@{}c@{}}THE \\ Asia University\end{tabular} \\ \midrule
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2022 &
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66 &
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25 &
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91 &
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15 \\
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2021 &
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75 &
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25 &
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129 &
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23 \\ \bottomrule
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\end{tabular}
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\vspace{10pt}
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\label{tab:table}
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\end{table}
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\section{Examples}
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\kant[1-3]
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\chapter{Introduction} \label{Chap: introduction}
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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.
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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).
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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.
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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.
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\begin{figure}[t]
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\centering
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\includegraphics[width=0.75\textwidth]{Figures/choi2014paraxial.png}
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\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}}
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\label{fig:choi2014paraxial}
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\end{figure}
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\section{Passive Cloaking in Optics and Electromagnetics}
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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.
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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.
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\begin{figure}[t]
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\centering
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\includegraphics[width=0.5\textwidth]{Figures/pendry2006controlling2.png}
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\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}}
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\label{fig:pendry2006controlling2}
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\end{figure}
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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.
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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.
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\begin{figure}[t]
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\centering
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\includegraphics[width=0.5\textwidth]{Figures/pendry2006controlling.png}
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\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}}
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\label{fig:pendry2006controlling}
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\end{figure}
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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).
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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.
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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.
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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.
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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.
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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.
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\section{Active Cloaking and Illusion in Wave Systems}
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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}.
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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.
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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.
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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.
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\begin{figure}[t]
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\centering
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\includegraphics[width=0.5\textwidth]{Figures/lin2021active.png}
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\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}}
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\label{fig:lin2021active}
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\end{figure}
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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.
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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.
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\section{Passive Cloaking in Fluids}
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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.
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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.
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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 $\Phi$:
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\begin{equation}\label{eq:euler}
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\Delta^2 \Phi=0
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\end{equation}
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subject to a linearized dynamic boundary condition on the mean free surface ($z=0$):
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\begin{equation}\label{eq:surface}
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\Phi_{tt}+g\Phi=0, \quad \text{on}\ z=0
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\end{equation}
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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.
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\begin{equation}\label{eq:surface2}
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\eta(x,y)=(\rm{i}\omega g)\phi,\quad \phi_z - K\phi = 0,\quad K=\omega^2/g, \text{on}\ z=0
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\end{equation}
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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
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\begin{equation}\label{eq:wave}
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\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)}
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\end{equation}
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and \ref{eq:surface2} provided that the unique positive real root k of
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\begin{equation}\label{eq:wave2}
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K\equiv \omega^2 / g = k \tanh kh
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\end{equation}
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corresponding to $h= h_0$ is assigned to $k_0$. The equation \ref{eq:wave2} is called the
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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 $c = \omega/k$.
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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.
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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.
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\begin{figure}[t]
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\centering
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\includegraphics[width=0.75\textwidth]{Figures/zou2019broadband.png}
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\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}}
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\label{fig:zou2019broadband}
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\end{figure}
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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.
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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.
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\begin{figure}[t]
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\centering
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\includegraphics[width=0.75\textwidth]{Figures/chen2024multilayered.png}
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\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}}
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\label{fig:chen2024multilayered}
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\end{figure}
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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.
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\section{Active Flow Control and Cloaking}
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|
||||
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.
|
||||
@@ -0,0 +1,116 @@
|
||||
\chapter{Problem Description and Methodology} \label{Chap: problem}
|
||||
|
||||
This chapter delineates the fundamental problem addressed in this dissertation: the active control of a complex, non-linear fluid flow to achieve hydrodynamic cloaking and illusion. We begin by outlining the overall research framework, which integrates a data-driven control agent with a high-fidelity fluid dynamics environment. Subsequently, we introduce the specific physical system under investigation—the "fluidic pinball"—and provide a detailed rationale for its selection as an ideal testbed. Finally, we connect this fundamental research to significant real-world challenges in marine engineering and bio-inspired stealth, establishing the broader context and potential impact of the work.
|
||||
|
||||
\section{Overall Research Framework}
|
||||
|
||||
The core of this investigation is a closed-loop control system designed to manipulate a complex fluid wake. This framework is conceptually divided into two principal components:
|
||||
|
||||
The Control Agent: An intelligent agent based on Deep Reinforcement Learning (DRL). This agent's role is to learn, through a process of trial-and-error interaction with the environment, an optimal control policy. It receives information about the state of the fluid flow from a set of virtual sensors and, based on its learned policy, determines the appropriate actuation commands to achieve a specific objective (cloaking or illusion).
|
||||
|
||||
The Fluid Environment: A high-fidelity numerical simulation of the fluidic pinball system. This environment, governed by the incompressible Navier-Stokes equations, serves as the physical world in which the DRL agent operates. It simulates the fluid dynamics resulting from the agent's actions and provides the sensory feedback necessary for the learning process. While current work is conducted in a numerical environment, the framework is designed to be directly translatable to a physical experiment.
|
||||
|
||||
The synergy between these two components allows for the discovery of complex, non-intuitive control strategies that would be intractable to design using traditional model-based methods, paving the way for advanced wake engineering.
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{Figures/framework.png}
|
||||
\caption{DRL learning framework. DRL continuously controls the rotation and learns from interactions.}
|
||||
\label{fig:framework}
|
||||
\end{figure}
|
||||
|
||||
\section{Fluidic Pinball: A Versatile Actuation and Sensing Platform}
|
||||
|
||||
The physical system at the heart of this study is the fluidic pinball, a configuration that has emerged as a canonical and richly informative benchmark within the active flow control researches.
|
||||
|
||||
The fluidic pinball consists of three identical circular cylinders of diameter $D$, arranged with their centers forming an equilateral triangle that points upstream. As detailed in the draft and illustrated below, the gap distance between adjacent cylinders is set to $0.5D$. This specific staggered arrangement creates a compact and highly interactive system.
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{Figures/domain.pdf}
|
||||
\caption{Geometry of the CFD environment.}
|
||||
\label{fig:domain}
|
||||
\end{figure}
|
||||
|
||||
The computational domain $\Omega$ is a rectangular channel defined in Cartesian coordinates $(x,y)$, with the origin at the inlet centerline:
|
||||
|
||||
\begin{equation}\label{eq:domain}
|
||||
\Omega = \left\{ [x,y]^{\top} \in \mathbb{R}^2 : x \in [0,64], \, y \in [-12.8,12.8] \right\} \setminus \bigcup_{i=1}^{3} \mathcal{C}_i
|
||||
\end{equation}
|
||||
|
||||
where $\mathcal{C}_i$ represents the cylindrical regions satisfying $(x - x_i)^2 + (y - y_i)^2 \geq (D/2)^2$. The cylinder centers are positioned at:
|
||||
|
||||
\begin{equation}\label{eq:pinball}
|
||||
\begin{array}{lll}
|
||||
\text{Front cylinder:} & x_1 = 30, & y_1 = 0 \\
|
||||
\text{Bottom cylinder:} & x_2 = 30 + 1.5\cos(30^\circ), & y_2 = -0.75 \\
|
||||
\text{Top cylinder:} & x_3 = 30 + 1.5\cos(30^\circ), & y_3 = 0.75
|
||||
\end{array}
|
||||
\end{equation}
|
||||
|
||||
This staggered arrangement creates a compact actuator array capable of redirecting fluid particles analogous to a pinball mechanism, hence the nomenclature "fluidic pinball".
|
||||
|
||||
Boundary conditions are specified as follows: Uniform velocity $U_0$ at the center of inlet ($x=0, y=0$), zero-pressure outflow at the outlet ($x=64$), and no-slip conditions at the top/bottom walls ($y=\pm12.8$). Crucially, the cylinder surfaces employ no-slip boundary conditions with prescribed circumferential velocities:
|
||||
|
||||
\begin{equation}\label{eq:cylinderboundary}
|
||||
\mathbf{u}\big|_{\partial\mathcal{C}_i} = a_i \left( -\sin\theta_i, \cos\theta_i \right)^\top
|
||||
\end{equation}
|
||||
|
||||
where $\theta_i$ denotes the angular position relative to each cylinder's center. The control inputs $a = [a_1, a_2, a_3]^\top$ correspond to the rotational speeds of the front ($U_F$), bottom ($U_B$), and top ($U_T$) cylinders, with positive values indicating counter-clockwise rotation. This triple-actuation system provides independent control authority over shear layer development and vortex shedding patterns.
|
||||
|
||||
Wake monitoring is achieved through three velocity sensors positioned downstream at:
|
||||
|
||||
\begin{equation}\label{eq:sensors}
|
||||
\begin{array}{lll}
|
||||
\text{Sensor 1:} & x_4 = 40, & y_4 = -2 \\
|
||||
\text{Sensor 2:} & x_5 = 40, & y_5 = 0 \\
|
||||
\text{Sensor 3:} & x_6 = 40, & y_6 = 2
|
||||
\end{array}
|
||||
\end{equation}
|
||||
|
||||
These probes measure instantaneous velocity components $(u,v)$ to quantify flow restoration (cloaking) or wake replication (illusion) by comparing against reference fields. To evaluate control robustness under perturbed inflow conditions, auxiliary disturbance generators may be positioned upstream. For instance, a cylindrical bluff body of variable diameter may be placed at $(x_d,y_d) = (10,y_d)$ to generate Kármán vortex streets or isolated vortices that interact with the pinball system. The sensor data and disturbance configurations collectively establish the observability framework for evaluating hydrodynamic invisibility and wake deception performance.
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\includegraphics[width=0.75\textwidth]{Figures/beem2015wake.png}
|
||||
\caption{A schematic of a seal chasing a fish by using its whiskers to “feel” for the flow features left in the fish wake. \cite{beem2015wake}}
|
||||
\label{fig:beem2015wake}
|
||||
\end{figure}
|
||||
|
||||
The velocity sensors are directly inspired by biological precedent. In aquatic ecosystems where vision is limited, hydrodynamic sensing is a primary tool for navigation and survival. Predators like harbor seals use their whiskers, and fish their lateral lines, to perceive the minute velocity and pressure fluctuations that constitute the wake of nearby organisms \cite{beem2015wake}. This natural reliance on flow-field information provides a strong justification for our use of velocity sensors as the basis for both feedback to the control agent and the evaluation of cloaking and illusion performance. By controlling the flow as perceived by these sensors, we are effectively managing the very hydrodynamic signature that a natural observer would detect.
|
||||
|
||||
The flow is governed by the incompressible Navier-Stokes equations, and for the purposes of this study, all simulations are conducted at a Reynolds number of $Re_D = 50$. This regime is chosen to ensure the flow remains two-dimensional while being sufficiently complex to exhibit the rich vortex dynamics essential for this research.
|
||||
|
||||
Control authority is achieved by independently rotating each of the three cylinders. The tangential velocity at the surface of each cylinder serves as a control input, $a = [a_1, a_2, a_3]^\top$, allowing for precise and independent manipulation of the shear layers separating from each body. This triple-actuation system is exceptionally versatile, enabling a wide range of control mechanisms.
|
||||
|
||||
To monitor the downstream wake and provide feedback to the control agent, an array of three virtual velocity sensors is positioned downstream of the pinball configuration. These sensors measure the instantaneous velocity components, providing the necessary data to quantify the degree of flow restoration for cloaking or the fidelity of wake replication for illusion.
|
||||
|
||||
The choice of the fluidic pinball is deliberate and strategic, motivated by its exceptionally rich and well-documented flow physics. As established in the comprehensive studies by Deng et al. \cite{deng2021galerkin} and others, the unforced fluidic pinball exhibits a surprisingly complex array of dynamic behaviors. As the Reynolds number is increased, the flow undergoes a series of bifurcations: it transitions from a globally stable steady state to a periodic, symmetric vortex shedding pattern via a supercritical Hopf bifurcation (at $Re_H \approx 18$), and subsequently to an asymmetric vortex shedding state after a pitchfork bifurcation (at $Re_{PF} \approx 68$). This progression results in a complex multi-attractor landscape with multiple stable and unstable solutions, making it a formidable and highly relevant challenge for any advanced control system.
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\includegraphics[width=0.75\textwidth]{Figures/deng2021galerkin.png}
|
||||
\caption{Lift coefficients at different Reynolds numbers (a) of the symmetric steady solutions us (black curve), the asymmetric steady (blue and red curve), the asymmetric steady solutions exemplified with the vorticity field at Re= 100 (b). \cite{deng2021galerkin}}
|
||||
\label{fig:deng2021galerkin}
|
||||
\end{figure}
|
||||
|
||||
Furthermore, the controllability of the fluidic pinball is well-established. The independent rotation of the cylinders allows for the implementation of numerous fundamental control mechanisms, including stagnation point manipulation, boat-tailing effects, base bleed simulation, and the exploitation of the Magnus effect. The system's response to both open-loop and closed-loop control has been investigated for performance-based objectives. Notably, Feng et al. \cite{feng2023control} successfully applied both brute-force search and DRL to discover control strategies for minimizing and tracking the hydrodynamic forces (lift and drag) on the pinball. This body of work provides a strong foundation, confirming the pinball as an ideal platform to explore the next frontier of flow manipulation.
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\includegraphics[width=0.75\textwidth]{Figures/feng2023control.png}
|
||||
\caption{Result of DRL-based flow control for forces tracking. The $C_{D0}$ in the time domain for four force tracking problems, vortical wake visualization of the last time step after convergence, and the corresponding actions generated by agents. (A) The changing of $C_{D0}$ that retain at 0, 1, 2, and 3 sharing the same legend. (B), (C), (D), (E) The vortical wake extracted in the last time step from four tests, respectively, sharing the same color bar in subfigure A and the corresponding actions. \cite{feng2023control}}
|
||||
\label{fig:feng2023control}
|
||||
\end{figure}
|
||||
|
||||
\section{Control Objective}
|
||||
|
||||
This research pivots from the traditional AFC paradigm of force optimization to the novel and more ambitious goal of comprehensive wake engineering. While prior work, such as that of Feng et al. \cite{feng2023control}, focused on manipulating integrated quantities like lift and drag, this research addresses the far more complex challenge of controlling the entire spatio-temporal structure of the downstream flow field. Our objectives are twofold:
|
||||
|
||||
Hydrodynamic Cloaking: To discover and implement an active control policy that commands the three rotating cylinders to effectively cancel the pinball's intrinsic wake. The goal is to restore the downstream flow characteristics to a state that is nearly indistinguishable from the undisturbed background flow, thereby rendering the object hydrodynamically "invisible" to downstream observers.
|
||||
|
||||
Hydrodynamic Illusion: To extend the concept of control beyond mere cancellation. Here, the objective is to actively sculpt the pinball's wake so that it convincingly mimics the hydrodynamic signature of a different object. This involves not only matching the overall flow field but also replicating key identifying features, such as the characteristic vortex shedding frequency (Strouhal number) of the target object, effectively creating a hydrodynamic disguise.
|
||||
|
||||
The primary application of this research lies in advancing the capabilities of underwater vehicles through hydrodynamic stealth and deception. The ability to actively cloak a vehicle's wake, or to generate an illusion that makes it appear as a different, non-threatening object, has direct and significant implications for the design of next-generation low-observability platforms for defense, scientific research, and environmental monitoring. This work addresses the broader challenge of managing disruptive hydrodynamic wakes, a problem of consequence in many marine applications. For instance, large offshore infrastructure is known to fundamentally alter local hydrodynamics, which can impact sensitive ecosystems \cite{bugnot2021current, degraer2020offshore}. While mitigating such large-scale environmental effects is a potential long-term goal, the immediate utility of the precision control demonstrated herein is in providing mobile underwater systems with an unprecedented ability to manage their hydrodynamic presence.
|
||||
|
||||
We will demonstrate the systematic achievement of these objectives across a spectrum of increasingly complex background flows, including a steady uniform flow, a periodic Kármán vortex street, and a transient, non-periodic isolated vortex.
|
||||
@@ -0,0 +1,236 @@
|
||||
% \chapter{Methodology} \label{Chap: methodology}
|
||||
|
||||
\section{Numerical Environment}
|
||||
|
||||
In this study, the flow dynamics is described by the incompressible NS equations with a force term
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
\nabla \cdot \boldsymbol{u}=0, \\
|
||||
\rho_f\left(\frac{\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u} \cdot \nabla \boldsymbol{u}\right)=-\nabla p+\mu \nabla^2 \boldsymbol{u}+\boldsymbol{f}_e,
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
Where $\rho_f$ is the fluid density, $mu$ is the dynamic viscosity, $p$ the pressure, $\bf{u}$ the velocity, and $\bf{f}_e$ the body force acting on the fluid by the structure. $\rho_f$ is the fluid density, $mu$ is the dynamic viscosity, $p$ the pressure, $\bf{u}$ the velocity, and $\bf{f}_e$ the body force acting on the fluid by the structure.
|
||||
|
||||
The hydrodynamic simulations are performed using the lattice Boltzmann method (LBM), a mesoscopic approach particularly suited for unsteady flow simulations involving complex boundaries and moving interfaces. The method solves the discrete Boltzmann equation with collision and streaming processes governed by the multi-relaxation-time (MRT) formulation, which enhances numerical stability compared to single-relaxation-time models. The He–Luo velocity model is adopted to ensure strict incompressibility conditions at low Mach numbers, with the discrete evolution equation expressed as:
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
f_i(\mathbf{x} + \mathbf{c}_i \Delta t, t + \Delta t) - f_i(\mathbf{x}, t) = -\mathbf{M}^{-1} \mathbf{S} \left[ \mathbf{m}(\mathbf{x},t) - \mathbf{m}^{(eq)}(\mathbf{x},t) \right]
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
where $f_i$ denotes the particle distribution function along the $i$-th discrete velocity direction $\mathbf{c}_i$, $\mathbf{M}$ is the transformation matrix mapping distribution functions to moment space, and $\mathbf{S}$ is the diagonal relaxation matrix. The equilibrium moments $\mathbf{m}^{(eq)}$ are constructed to recover the incompressible Navier-Stokes equations through Chapman-Enskog expansion. A D2Q9 lattice structure is employed for two-dimensional simulations, with the lattice speed $c_s = c/\sqrt{3}$ where $c = \Delta x / \Delta t$ defines the ratio of lattice spacing to time step.
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\includegraphics[width=0.50\textwidth]{Figures/d2q9.png}
|
||||
\caption{Lattice arrangement for D2Q9 scheme.}
|
||||
\label{fig:d2q9}
|
||||
\end{figure}
|
||||
|
||||
Figure~\ref{fig:domain} illustrates the computational domain and boundary condition implementation. The primary simulation domain spans $1280 \times 512$ lattice units. Inflow boundary conditions at $x=0$ implement a Dirichlet velocity specification through a modified bounce-back scheme with momentum correction, enforcing $u_x = U_0 = 0.01c$ and $u_y = 0$. Top and bottom walls at $y = \pm 12.8D$ employ analogous bounce-back treatments for no-slip conditions. At the outflow boundary ($x = 64D$), a convective condition is imposed:
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
\frac{\partial \phi}{\partial t} + U_0 \frac{\partial \phi}{\partial x} = 0
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
where $\phi$ represents velocity components, effectively minimizing vorticity reflections. The cylindrical surfaces utilize curvilinear boundary treatments based on the ghost-node interpolation method, which has demonstrated second-order spatial accuracy in previous studies. This approach reconstructs distribution functions at boundary-intersected links using Lagrangian interpolation, maintaining consistency with the no-slip condition for arbitrarily rotating cylinders. The rotational velocity at cylinder surfaces is imposed through the interpolated boundary velocity in the bounce-back scheme, with the circumferential velocity $\mathbf{u}_{\text{wall}} = a_i (-\sin\theta, \cos\theta)^\top$ for the $i$-th cylinder as defined in last chapter.
|
||||
|
||||
% \begin{table}
|
||||
% \begin{center}
|
||||
% \def~{\hphantom{0}}
|
||||
% \begin{tabular}{lccc}
|
||||
% $D/\delta x$ & $C_D$ & $C_L$ & $St$ \\[3pt]
|
||||
% 10 & $xxxx\pm~xx$ & $\pm~xxxx$ & xxxx\\
|
||||
% 20 & $xxxx\pm~xx$ & $\pm~xxxx$ & xxxx\\
|
||||
% 40 & $xxxx\pm~xx$ & $\pm~xxxx$ & xxxx\\
|
||||
% \end{tabular}
|
||||
% \caption{Mesh convergence results}
|
||||
% \label{tab:mesh}
|
||||
% \end{center}
|
||||
% \end{table}
|
||||
|
||||
% Mesh convergence studies were conducted to determine appropriate spatial resolution, with results summarized in Table~\ref{tab:mesh}. Three resolutions were evaluated at $Re_D = 50$: $D/\delta x = 10$, $20$, and $40$, where $\delta x$ denotes lattice spacing. Key hydrodynamic parameters including mean drag coefficient $\overline{C_D}$, root-mean-square lift coefficient $C_{L,\text{rms}}$, and Strouhal number $St$ were compared against benchmark data from previous investigations. The medium resolution ($D/\delta x = 20$) provided optimal balance between computational efficiency and accuracy, with relative errors below x\% for all monitored parameters compared to high-resolution simulations. This resolution was consequently adopted for the extensive reinforcement learning training phase. For final deterministic validation runs, the high-resolution mesh ($D/\delta x = 40$) was employed to ensure solution fidelity.
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{Figures/architecture.png}
|
||||
\caption{Research environment framework.}
|
||||
\label{fig:architecture}
|
||||
\end{figure}
|
||||
|
||||
To address the competing demands of computational accuracy and training efficiency in deep reinforcement learning applications, a specialized GPU-accelerated solver was developed. This implementation employs a hybrid architecture coupling CUDA C kernels for performance-critical computations with Python-based orchestration. During initialization, Python dynamically generates configuration-specific CUDA kernel code based on user-defined parameters, including collision operator selection (MRT in this study) and boundary condition types. The PyCUDA framework compiles these kernels to PTX code at runtime and manages all host-device memory transfers. During simulation execution, control variables are asynchronously transferred to the GPU via PyCUDA streams, allowing concurrent computation and data movement. This design achieves approximately 90\% GPU utilization efficiency by overlapping data transfers with kernel execution. The Python layer maintains full control over simulation state, enabling runtime modification of domain geometry, physical parameters, or even fundamental algorithms without process restart. This capability proves particularly valuable for reinforcement learning workflows, where environmental interactions require dynamic adaptation. On the specified hardware platform (AMD EPYC 7302 CPU with NVIDIA V100 GPU), this implementation completes one pinball flow-through period (approximately xxxx time steps) in under x minutes at training resolution, and approximately x minutes at high resolution for validation cases. While this framework offers significant advantages in flexibility and integration with machine learning ecosystems, certain limitations warrant acknowledgment. Dynamic kernel compilation introduces initial overhead during startup phases, though this becomes negligible in long-running simulations. Frequent host-device communication, while optimized through stream concurrency, may impose bottlenecks for extreme-scale problems exceeding available GPU memory. Furthermore, the dependency on PyCUDA may present compatibility challenges in high-performance computing environments requiring pure MPI implementations. Despite these constraints, the architecture delivers an effective compromise between computational performance and research flexibility, particularly suited for data-driven fluid dynamics investigations.
|
||||
|
||||
\section{Deep Reinforcement Learning}
|
||||
|
||||
The task of achieving hydrodynamic cloaking and illusion in the non-linear, multi-attractor regime of the fluidic pinball presents a formidable control problem. The system is characterized by a high-dimensional state space, strong non-linearities, and complex, time-delayed responses to actuation. Traditional model-based control methods are ill-suited for this challenge, as deriving an accurate and tractable analytical model of the wake dynamics is practically impossible. Consequently, a data-driven, model-free approach is not only advantageous but essential for discovering the sophisticated control policies required.
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{Figures/brunton2020machine2.png}
|
||||
\caption{Machine learning algorithms may be categorized into supervised,unsupervised,and semisupervised,depending on the extent and type of information available for the learning process. Abbreviations: PCA,principal component analysis; POD,proper orthogonal decomposition. \cite{brunton2020machine}}
|
||||
\label{fig:brunton2020machine2}
|
||||
\end{figure}
|
||||
|
||||
The application of machine learning to fluid dynamics is not a recent phenomenon, with early uses in flow-field analysis dating back to the 1990s. In the context of feedback control, several data-driven paradigms have been explored. Pioneering work by Lee et al. \cite{lee1997application} successfully employed Neural Networks (NNs) for turbulence control, demonstrating their theoretical power to approximate arbitrary non-linear control laws. However, this flexibility often comes at the cost of exorbitant computational or experimental resources needed to optimize a vast number of parameters. Other methods, such as Genetic Algorithms (GAs), have also been deployed but typically require the structure of the control law to be prespecified, limiting their ability to discover truly novel strategies in problems with complex, unknown dynamics.
|
||||
|
||||
In recent years, Reinforcement Learning (RL), and particularly its deep learning variant, Deep Reinforcement Learning (DRL), has emerged as a transformative approach for such complex control problems. Its success has rapidly expanded from games to challenging physical systems, including robotics, autonomous flight, and the simulation of biological locomotion. Unlike supervised learning, which requires labeled data, or the aforementioned methods that may require some prior system knowledge, RL learns an optimal policy through direct, trial-and-error interaction with its environment. This "end-to-end" learning capability is perfectly suited for discovering control strategies in systems where the underlying physics are too complex to model explicitly. As noted by Brunton et al. \cite{brunton2020machine}, the success of RL in fluid mechanics is critically dependent on a thoughtful formulation of the problem, where the states, actions, and rewards are chosen to reflect the governing physical mechanisms.
|
||||
|
||||
In light of these advantages, this dissertation adopts a DRL framework to tackle the challenges of hydrodynamic cloaking and illusion. This approach allows the control system to autonomously discover effective manipulation strategies for the fluidic pinball's rotational actuators, navigating the complex flow physics to achieve the desired global wake structure without any prior knowledge of the governing equations.
|
||||
|
||||
The control strategy for hydrodynamic cloaking and wake illusion is formulated through a DRL paradigm, distinguished from conventional model-based approaches by its independence from prior fluid dynamic knowledge. This model-free methodology employs proximal policy optimization (PPO), an actor-critic algorithm particularly effective for continuous action spaces. The learning process evolves through iterative interactions between the DRL agent and the fluidic environment, commencing with randomized exploratory actions and progressively refining control policies via reward-driven optimization. As depicted in Fig.~\ref{fig:framework}, the framework comprises three fundamental components: (i) the environment state $s_t$ captured by velocity sensors (\ref{eq:sensors}) and instantaneous hydrodynamic forces acting on the pinball; (ii) the control action $a_t = [U_F, U_B, U_T]^\top$ dictating rotational velocities of the three cylinders; and (iii) the reward function $r_t$ quantifying control performance.
|
||||
|
||||
\begin{figure}[t]
|
||||
\centering
|
||||
\includegraphics[width=0.75\textwidth]{Figures/brunton2020machine1.png}
|
||||
\caption{Deep reinforcement learning scheme \cite{brunton2020machine}}
|
||||
\label{fig:brunton2020machine1}
|
||||
\end{figure}
|
||||
|
||||
A critical innovation resides in the neural network architecture, where sinusoidal activation functions ($\sin$, $\cos$) replace conventional ReLU or tanh units. This design is motivated by the spectral characteristics of vortex-dominated flows, as demonstrated in Fig.~\ref{fig:act_func} through periodic signal reconstruction tests. Standard activations exhibit significant phase errors and amplitude decay when approximating vortical signatures, whereas trigonometric functions preserve spectral fidelity with mean squared errors reduced by x\% compared to next-best alternatives. The actor and critic networks thus adopt the form:
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
h_1 = \sin(\mathbf{W}_1 s_t + \mathbf{b}_1), \quad h_2 = \cos(\mathbf{W}_2 h_1 + \mathbf{b}_2), \quad a_t = \mathbf{W}_a h_2 + \mathbf{b}_a
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
where $\mathbf{W}$ and $\mathbf{b}$ denote trainable weights and biases, enabling explicit representation of phase-dependent flow features essential for vortex synchronization.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=0.75\textwidth]{Figures/act_func.pdf}
|
||||
\caption{Activation function affecting NN performance}
|
||||
\label{fig:act_func}
|
||||
\end{figure}
|
||||
|
||||
The reward function constitutes the second pivotal innovation, addressing the fundamental challenge of perceptual manipulation in fluid flows. Rather than relying solely on delayed sensor-based metrics, we integrate direct force measurements with flow similarity assessments to accelerate policy convergence. For hydrodynamic cloaking, the reward combines force minimization and sensor alignment:
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
r_{\text{cloak}} = \exp\left( -w_{C_D} \left| \sum_{i=1}^3 C_{D_i} \right| - w_{C_L} \left| \sum_{i=1}^3 C_{L_i} \right| - w_D \sum_{j=1}^3 \mathcal{D}(v_j, v_j^{\text{ref}}) \right)
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
where $C_{D_i}$, $C_{L_i}$ are drag and lift coefficients of individual cylinders, $v_j$ denotes velocity at sensor $j$, and $\mathcal{D}$ quantifies the dynamic time warping (DTW) distance between measured and reference signals. The exponential form creates a steep reward gradient near optimal states, simultaneously satisfying physical constraints (vanishing net force implies minimal flow disturbance) and perceptual objectives (sensor alignment ensures invisibility to downstream observers).
|
||||
|
||||
For wake illusion, the reward structure maintains analogous components but replaces force minimization with target matching:
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
r_{\text{illusion}} = \exp\left( -w_F \left| \| \mathbf{F}_{\text{pinball}} \| - \| \mathbf{F}_{\text{target}} \| \right| - w_D \sum_{j=1}^3 \mathcal{D}(v_j, v_j^{\text{target}}) \right)
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
where $\mathbf{F}_{\text{target}}$ corresponds to hydrodynamic forces on the emulated cylinder.
|
||||
|
||||
Training proceeds in discrete episodes, each spanning $400T_0$ ($T_0 = D/U_0$) to encompass multiple vortex shedding cycles. State observations and control updates occur at $0.8T_0$ intervals, balancing responsiveness with computational tractability. The PPO algorithm updates policy parameters $\theta$ by maximizing the clipped objective:
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
\mathcal{L}(\theta) = \mathbb{E}_t \left[ \min\left( \rho_t(\theta) \hat{A}_t, \text{clip}(\rho_t(\theta), 1-\epsilon, 1+\epsilon) \hat{A}_t \right) \right]
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
where $\rho_t$ is the probability ratio between new and old policies, $\hat{A}_t$ denotes advantage estimates from the critic network, and $\epsilon=0.2$ prevents destabilizing updates. Upon convergence (typically after $10^3$ episodes), deterministic control enforces the mean action $a_t = \mathbb{E}[ \pi_\theta(s_t) ]$ without exploratory noise, ensuring reproducible performance in validation studies. This framework effectively bridges short-term physical constraints with long-term perceptual objectives, enabling the discovery of non-intuitive control strategies for complex flow manipulation.
|
||||
|
||||
\begin{table}
|
||||
\begin{center}
|
||||
\def~{\hphantom{0}}
|
||||
\begin{tabular}{lc}
|
||||
Parameter & Value/Method \\[3pt]
|
||||
Network architecture (actor and critic) & $64 \times 64$ \\
|
||||
Activation function & Sin \\
|
||||
Actuations per $T_0$ & 1.25 \\
|
||||
Length of each episode & $600 T_0$ \\
|
||||
Optimizer & Adam \\
|
||||
Learning rate (actor) & $3 \times 10^{-4}$ \\
|
||||
Learning rate (critic) & $4 \times 10^{-4}$ \\
|
||||
\end{tabular}
|
||||
\caption{Hyper-parameters used in the DRL}
|
||||
\label{tab:drl}
|
||||
\end{center}
|
||||
\end{table}
|
||||
|
||||
\section{Dynamic Time Warping}
|
||||
|
||||
The analysis of time-ordered data is a cornerstone of modern science and engineering, with applications spanning a vast array of disciplines. A fundamental challenge in this domain is quantifying the similarity between two temporal sequences, especially when these sequences are subject to temporal distortions such as stretching, compression, or phase shifts. While simple point-wise distance metrics like the Euclidean distance are effective for sequences of identical length and perfect alignment, they often fail in real-world scenarios where processes may unfold at varying rates. For instance, the same word may be spoken at different speeds, a person's signature may be written faster or slower, or a biological process may progress at different rates in different individuals. To address this challenge, a more robust and flexible similarity measure is required.
|
||||
|
||||
One of the most powerful and widely adopted solutions to this problem is the Dynamic Time Warping (DTW) algorithm. First introduced in the context of speech recognition by Sakoe and Chiba \cite{sakoe1978dynamic}, DTW is a dynamic programming-based method that finds the optimal non-linear alignment between two time series. Instead of comparing the i-th point of one sequence to the i-th point of another, DTW calculates an optimal "warping path" that maps the time axis of one sequence to the other, minimizing the cumulative distance between their corresponding, aligned points. This ability to non-linearly warp time allows DTW to find the true similarity between two sequences, irrespective of temporal misalignments.
|
||||
|
||||
The versatility and effectiveness of DTW have led to its adoption far beyond its origins in speech processing. It has become an essential tool in fields requiring robust pattern matching in time-series data. Notable applications include the recognition of online handwriting and signatures, where individual writing speeds can vary significantly \cite{tappert1990state}, and in gesture recognition from video feeds, where movements may not be perfectly synchronized \cite{corradini2001dynamic}. In bioinformatics, DTW has proven superior to simple clustering for aligning gene expression time series, enabling researchers to map corresponding biological states even when they unfold at different rates \cite{aach2001aligning}. Furthermore, the algorithm is a key component in modern information retrieval systems, such as query-by-humming song recognition \cite{zhu2003warping}, time-series database searching \cite{zhu2003warping}, and time-series clustering \cite{niennattrakul2007clustering}, solidifying its status as a fundamental algorithm for analyzing dynamic, time-ordered data.
|
||||
|
||||
The objective of DTW is to compare two sequences, \(X=\left(x_{1}, x_{2}, \ldots, x_{N}\right)\) and \(N \in \mathbb{N}\). In our context, $X$ represents the time series of velocity vectors from the target flow (e.g., the undisturbed vortex street or the wake of a target cylinder), and $Y$ is the corresponding time series from our DRL-controlled flow. The elements $x_n$ and $y_m$ are the velocity vectors $(u_x, u_y$ recorded by the sensors at discrete time steps.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{Figures/dtw1.png}
|
||||
\caption{(a) Illustration of a warping path and (b) its interpretation for some sequence $X$ of length $N = 9$ and some sequence $Y$ of length $M = 7$. Each cell $(n, m)$ belonging to the warping path is indicated by a red dot and corresponds to an alignment between the elements $x_n$ and $y_m$ indicated by a red bidirectional arrow. \cite{muller2015fundamentals}}
|
||||
\label{fig:dtw1}
|
||||
\end{figure}
|
||||
|
||||
The first step is to define a local cost measure, $c(x_n,y_m)$, which quantifies the dissimilarity between any two points in the sequences. For our vector-valued velocity data, the Euclidean distance is a natural choice:
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
c(x_n,y_m)=\|x_n-y_m\|
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
Evaluating this cost for every pair of elements from $X$ and $Y$ yields and $N\times M$ cost matrix, $C$, where $C(n,m)=c(x_n, y_m)$. This matrix represents the landscape of local dissimilarities between the two signals. Regions of low cost (dark areas in visualizations) indicate moments where the two signals are locally similar. The goal of DTW is to find an optimal alignment, or warping path, through this cost matrix. A warping path $P=(p_1,\dots,p_L)$ is a sequence of index pairs $p_\ell=(n_\ell,m_\ell)$ that maps the indices of $X$ to the indices of $Y$. This path must satisfy three critical conditions:
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
&\text{Boundary condition: }p_{1}=(1,1) \text{ and } p_{L}=(N, M).\\
|
||||
&\text{Monotonicity condition: }n_{1} \leq n_{2} \leq \ldots \leq n_{L} \text{ and } m_{1} \leq m_{2} \leq \ldots \leq m_{L}.\\
|
||||
&\text{Step size condition: }p_{\ell+1}-p_{\ell} \in\{(1,0),(0,1),(1,1)\} \text{ for } \ell \in[1: L-1].\\
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
The total cost of a given warping path $P$ is the sum of the local costs of all the cells it passes through:
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
c_{P}(X, Y)=\sum_{\ell=1}^{L} \mathbf{C}\left(n_{\ell}, m_{\ell}\right) .
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
An optimal warping path, $P^\star$, is the path that minimizes this total cost. The DTW distance, denoted as $\operatorname{DTW}(X,Y)$, is formally defined as the total cost of this optimal path:
|
||||
|
||||
\begin{equation}
|
||||
\begin{gathered}
|
||||
\operatorname{DTW}(X, Y) =c_{P^{*}}(X, Y)=\min \left\{c_{P}(X, Y) \mid P \text { is an }(N, M) \text { warping path }\right\} .
|
||||
\end{gathered}
|
||||
\end{equation}
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{Figures/dtw2.png}
|
||||
\caption{DTW algorithm based on dynamic programming. \cite{muller2015fundamentals}}
|
||||
\label{fig:dtw2}
|
||||
\end{figure}
|
||||
|
||||
While the number of possible warping paths is exponential, the optimal path and its cost can be found efficiently in $O(NM)$ time using dynamic programming. This method works by building up an accumulated cost matrix, $\mathbf{D}$, where each element $\mathbf{D}(n,m)$ stores the minimum accumulated cost for aligning the prefixes $X(1:n)$ and $Y(1:m)$. The matrix $\mathbf{D}$ is computed recursively:
|
||||
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\mathbf{D}(n, m)=\mathbf{C}(n, m)+\min \left\{\begin{array}{l}
|
||||
\mathbf{D}(n-1, m-1) \\
|
||||
\mathbf{D}(n-1, m) \\
|
||||
\mathbf{D}(n, m-1)
|
||||
\end{array}\right.
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
The final DTW distance is the value in the top-right corner of this matrix, $\operatorname{DTW}(X,Y)=\mathbf{D}(N,M)$. Once the accumulated cost matrix is computed, the optimal warping path $P^\star$ can be found by backtracking from $(N,M)$ to $(1,1)$, at each step choosing the neighboring cell that led to the minimum accumulated cost.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{Figures/dtw3.pdf}
|
||||
\caption{The DTW algorithm exhibits broad sensitivity to various types of signal distortion.}
|
||||
\label{fig:dtw3}
|
||||
\end{figure}
|
||||
@@ -0,0 +1,348 @@
|
||||
\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.
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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.
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\begin{figure}[htbp!]
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\centering
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\includegraphics[width=0.80\textwidth]{Graphs/reduce_obs.pdf}
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\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.}
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\label{fig:reduce_obs}
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||||
\end{figure}
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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.
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@@ -0,0 +1,24 @@
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\chapter{Conclusion and Future Work} \label{Chap: Conclusion}
|
||||
|
||||
This report has ventured into the nascent and challenging intersection of active flow control and hydrodynamic perception management. Moving beyond the traditional objectives of performance optimization, such as drag reduction or vibration suppression, this work has established a comprehensive framework for achieving active hydrodynamic cloaking and illusion in a non-linear, vortical flow regime. By leveraging the power of model-free Deep Reinforcement Learning, we have demonstrated that it is possible to command a simple actuator system—the fluidic pinball—to perform sophisticated wake engineering tasks that were previously considered intractable.
|
||||
|
||||
The primary contribution of this thesis is the successful demonstration of both high-fidelity hydrodynamic cloaking and, for the first time to our knowledge, targeted hydrodynamic illusion in a complex, unsteady flow environment. We began by validating our DRL framework in a baseline steady flow, where the agent learned to completely suppress vortex shedding and nullify the net hydrodynamic forces on the pinball, effectively rendering it invisible to the mean flow. The true capability of the framework was then proven in more demanding scenarios. We demonstrated robust cloaking in the presence of complex background flows, including periodic Kármán vortex streets and aperiodic, transient vortices. In these cases, the DRL agent discovered non-trivial, multi-frequency control policies to allow these structures to pass through the control region with their fundamental characteristics preserved.
|
||||
|
||||
Building upon this success, we introduced and validated the novel concept of hydrodynamic illusion. The DRL agent successfully learned to manipulate the pinball's wake to convincingly mimic the distinct hydrodynamic signature of a different bluff body. This was achieved with remarkable precision, matching not only the overall wake structure but also the target's characteristic Strouhal number. Our investigation into mimicking targets of varying sizes revealed a fascinating control strategy, where the agent shifted to higher-frequency actuation to generate larger-scale flow structures, opening new questions about the underlying physics of control. Finally, we have taken the crucial first steps toward bridging the gap between simulation and reality by designing and performing preliminary tests on a physical experimental platform, identifying key engineering challenges and proposing viable solutions, such as reducing the observation space required for effective control.
|
||||
|
||||
While the results presented herein mark a significant step forward, they also illuminate a vast and exciting landscape for future research. The work completed thus far serves as a strong foundation upon which several promising research avenues can be built. These future directions aim to enhance the performance and applicability of the current framework, deepen our physical understanding of the learned control strategies, and accelerate the discovery process.
|
||||
|
||||
\begin{figure}[htbp!]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth, trim=0 0 0 3cm, clip]{Graphs/gantt_chart.pdf}
|
||||
\caption{Gantt chart visualizes the timeline for the PhD research.}
|
||||
\label{fig:gantt_chart}
|
||||
\end{figure}
|
||||
|
||||
One of the most immediate priorities is to enhance the control performance and robustness, particularly at higher Reynolds numbers. Our results indicated a clear degradation in both cloaking and illusion performance as the Reynolds number increased beyond 200. Future work could explore more advanced DRL algorithms, such as introduce transformer layer, which may be better suited to capturing the longer time dependencies and more chaotic dynamics of higher-Re flows. Furthermore, a curriculum learning approach, where the agent is progressively trained on tasks of increasing difficulty (e.g., gradually increasing the Reynolds number), could significantly improve the agent's ability to generalize. Investigating more actuation methods, such as change the position of the pinball, could also provide the additional control authority needed to manipulate more energetic flows.
|
||||
|
||||
A second critical direction is to improve the interpretability of the learned control policies and extract physical insights. The DRL agent, while effective, currently operates as a "black box." A significant future contribution would be to distill the complex, neural network-based policy into a simpler, physically interpretable "white-box" model. Techniques such as symbolic regression or sparse identification of nonlinear dynamics (SINDy) could be employed to find a concise mathematical expression that approximates the agent's control law. This would not only demystify the agent's decision-making process but could also lead to the discovery of new, fundamental principles of flow control. Understanding why the agent chooses a specific multi-frequency actuation or why it shifts to higher harmonics for certain illusion tasks would be of immense value to the broader fluid mechanics community.
|
||||
|
||||
Finally, the integration of Reduced-Order Models (ROMs) into the training pipeline presents a powerful opportunity to dramatically accelerate future research. The primary bottleneck in this work was the computational expense of high-fidelity CFD simulations. By developing an accurate and computationally inexpensive ROM of the fluidic pinball system, for example using a POD-Galerkin approach, we could create a surrogate environment for DRL training. An agent could undergo thousands or even millions of learning episodes on the ROM in a fraction of the time required by the full simulation. This "ROM-in-the-loop" training would enable more extensive exploration of the parameter space, faster development of control policies for new tasks, and more robust hyperparameter tuning. The policy learned on the ROM could then be efficiently fine-tuned on the high-fidelity model or transferred directly to the physical experiment, creating a highly efficient sim-to-real workflow.
|
||||
|
||||
In conclusion, this report has pushed the frontier of active flow control into the realm of perception management. By demonstrating the feasibility of hydrodynamic cloaking and illusion through deep reinforcement learning, we have opened the door to a new class of intelligent fluid systems. The future work outlined here promises not only to refine these capabilities but also to deepen our fundamental understanding of flow control, ultimately paving the way for transformative applications in marine engineering, underwater robotics, and environmental science.
|
||||
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