Modelling and control of fluid flows and marine structures
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The contributions of this thesis fall naturally into two main categories: Part I: Feedback control of fluid flows, and; Part II: Modelling and control of marine structures. Main Con tributions of Part I Part I presents new results on stabilization (for the purpose of drag reduction or vortex shedding suppression) and destabilization (for the purpose of mixing) of channel, pipe and cylinder flows. In order to provide a stand-alone reference on this topic, the thesis also contains a comprehensive review of the research carried out in this field over the last decade or so, along with introductory chapters on fluid mechanics and control theory. The review also serves the purpose of placing the contributions by the author into the wider context of the field. The contributions by the author are the following: Section 4.3.1: A new boundary feedback control law for stabilization of the parabolic equilibrium flow in 2D channel flow is derived using Lyapunov stability theory. The controller uses pressure measurements taken on the channel wall, and applies actuation in the form of wall transpiration, that is, suction and blowing of fluid across the wall. Although the analysis is valid for small Reynolds numbers, only, simulations indicate that the control is very effcient in stabilizing the flow at Reynolds numbers several orders of magnitude higher. The pressure-based control law performed much better than other Lyapunov-based control laws studied. Section 4.3.3: The simple pressure-based control law derived in Section 4.3.1 is generalized to the 3D pipe flow. As for the 2D channel flow, the analysis is valid for small Reynolds numbers, only. Section 4.3.4: The pressure-based feedback control law derived in Section 4.3.1 for the 2D channel flow results in flow transients with instantaneous drag far lower than that of the corresponding laminar flow. In fact, for the first time, instantaneous total drag in a constant-mass- flow 2D channel flow is driven to negative levels. The physical mechanisms by which this phenomenon occur is explained, and the possibility of achieving sustained drag reductions to below the laminar level by initiating such low-drag transients on a periodic basis is explored. The results add to the evidence that the laminar ow represents a fundamental limit to the drag reduction achievable by wall transpiration. Section 4.4: A state feedback controller that achieves global asymptotic stabilization of a nonlinear Ginzburg-Landau model of vortex shedding from bluff bodies is designed using backstepping. Stabilization is obtained in two steps. First, the upstream and downstream parts of the system are shown to exhibit the inputto- state stability property with respect to certain boundary input terms governed by the core flow in the vicinity of the bluff body. Second, a finite difference approximation of arbitrary order of the core flow is stabilized using the backstepping method. Consequently, all the states in the core flow are driven to zero, including the boundary input terms of the upstream and downstream subsystems. The control design is valid for any Reynolds number, and simulations demonstrate its performance. Section 5.2: For thefirst time, active feedback control is used to enhance mixing by exploiting the natural tendency in the flow to mix. By applying the pressurebased feedback control law derived for stabilizing the 2D channel flow in Section 4.3.1, with the sign of the input reversed, enhanced instability of the parabolic equilibrium flow is obtained, which leads rapidly to highly complex flow patterns. The mixing enhancement is quantified using various diagnostic tools. Section 5.3: A Lyapunov based boundary feedback controller for achieving mixing in a 3D pipe flow governed by the Navier-Stokes equation is designed. It is shown that the control law maximizes a measure of mixing that incorporates stretching and folding of material elemen ts, while at the same time minim izing the control effort and the sensing effort. The penalty on sensing results in a static output- feedback control law (rather than full-state feedback). A lower bound on the gain from the control effort to the mixing measure is also deriv ed. For the openloop system, input/output-to-state stability properties are established, which show a form of detectability of mixing in the interior of the pipe from the chosen outputs on the wall. The effectiveness of the optimal control in achieving mixing enhancement is demonstrated in numerical sim ulations. Simulation results also show that the spatial changes in the control velocity are smooth and small, promising that a low number of actuators will suffice in practice. Section 5.4: Motivated by the mixing results for channels and pipes in Sections 5.2 and 5.3, a simulation study that investigates the feasibility of enhancing particle dispersion in the wake of a circular cylinder is carried out. For a subcritical case, vortex shedding is successfully provoked using feedback. Main Contributions of Part II Part II deals with modelling and control of slender marine structures and marine vessels. Chapter 8: A new finite element model for a cable suspended in water is developed. Global existence and uniqueness of solutions of the truncated system is shown for a slightly simplified equation describing the motion of a cable with negligible added mass and supported by fixed end-points. Based on this, along with well known results on local existence and uniqueness of solutions for symmetrizable hyperbolic systems, a global result for the initial-boundary value problem is conjectured. The FEM model for the cable is assembled to give a model of a multi-cable mooring system, whic h, in turn, is coupled to a rigid body model of the floating vessel. The result is a coupled dynamical model of a moored v essel, which can be applied to applications such as turret-based moored ships, or tension leg platforms. As a simple application of the sim ulator, controlling the line tensions dynamically as an additional means of station keeping is explored. Chapter 9: Output feedback tracking control laws for a class of Euler-Lagrange systems subject to nonlinear dissipative loads are designed. By imposing a monotone damping condition on the nonlinearities of the unmeasured states, the common restriction that the nonlinearities be globally Lipschitz is removed. The proposed observer-controller scheme renders the origin of the error dynamics uniformly globally asymptotically stable, in the general case. Under certain additional assumptions, the result continue to hold for a simplified control law that is less sensitive to noise and unmodeled phenomena.