\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.