Adaptive and model-based control theory applied to convectively unstable flows

International audienceResearch on active control for the delay of laminar-turbulent transition in boundary layers has made a significant progress in the last two decades, but the employed strategies have been many and dispersed. Using one framework, we review model-based techniques, such as linear-quadratic regulators, and model-free adaptive methods, such as least-mean square filters. The former are supported by a elegant and powerful theoretical basis, whereas the latter may provide a more prac-tical approach in the presence of complex disturbance envi-ronments, that are difficult to model. We compare the meth-ods with a particular focus on efficiency, practicability and robustness to uncertainties. Each step is exemplified on the one-dimensional linearized Kuramoto-Sivashinsky equation, that shows many similarities with the initial linear stages of the transition process of the flow over a flat plate. Also, the source code for the examples are provided. 1 Introduction The key motivation in research on drag reduction is to develop new technology that will result in the design of ve-hicles with a significantly lower fuel consumption. The field is broad, ranging from passive methods, such as coating surfaces with materials that are super-hydrophobic or non-smooth [1], to active methods, such as applying wall suction or using measurement-based closed-loop control [2]. This work positions itself in the field of active control methods for skin-friction drag. In general, the mean skin friction of a turbulent boundary layer on a flat plate is an order of magni-tude larger compared to a laminar boundary layer. One strat-egy to reduce skin-friction drag is thus to push the laminar-turbulent transition on a flat plate downstream [3]. Differ-ent transition scenarios may occur in a boundary layer flows, depending on the intensity of the external disturbances act-ing on the flow, [4]. Under low levels of free-stream turbu-lence and sufficiently far downstream, the transition process is initiated by the linear growth of small perturbations called Tollmien-Schlichting (TS) waves [3]. Eventually, these per-turbations reach finite amplitudes and breakdown to smaller scales via nonlinear mechanisms [5]. However, in presence of stronger free-stream disturbances, the exponential growth of TS waves are bypassed and transition may be directly triggered by the algebraic growth of stream-wise elongated structures, called streaks [4]. One may delay transition by damping the growth of TS waves and/or streaks, and thus postpone their nonlinear breakdown. This strategy enables the use of linear theory for control design. Fluid dynamists noticed in the early 90's, that many of the emerging concepts in hydrodynamic stability theory al-ready existed in linear systems theory [6, 7]. For example, the analysis of a system forced by harmonic excitations is referred to as signalling problem by fluid dynamicists, while control theorists analyze the problem by constructing a Bode diagram, [8]; similarly, a large transient growth of a fluid system corresponds to large norm of a transfer function and matrix with stable eigenvalues can be called either globally stable or Hurtwitz, [5, 9]. However, the systems theoretical approach had taken one step further, by "closing the loop", i.e providing rigorous conditions and tools to modify the linear system at hand. It was realized by fluid dynamists that the extension of hydro-dynamic stability theory to include tools and concepts from linear control theory was natural [10, 11, 12]. A long series of numerical investigations addressing the various aspects of closed-loop control of transitional [13, 14, 15] and turbulent flows [16, 17, 18] followed in the wake of these initial contri-butions. At the same time, research on active control fo