Optimal rotary control of the cylinder wake in the laminar regime

Abstract

In this paper we develop the Optimal Control Approach to the rotary control of the cylinder wake. We minimize the functional which represents the sum of the work needed to resist the drag force and the work needed to control the flow, where the rotation rate φ̇(t) is the control variable. Sensitivity of the functional to control is determined using the adjoint equations. We solve them in the “vorticity” form, which is a novel approach and leads to computational advantages. Simulations performed at Re=75 and Re=150 reveal systematic decrease of the total power and drag achieved using a very small amount of control effort. We investigate the effect of the optimization horizon on the performance of the algorithm and the correlation of the optimal controls with the changes of the flow pattern. The algorithm was also applied to the control of the subcritical flow at Re=40, however, no drag reduction was achieved in this case. Based on this, limits of the performance of the algorithm are discussed.

Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models

“…To supplement this main result, we will also give an estimate of the computational savings that can be obtained by a POD ROM based optimal control approach compared with the more "classical" approach where the Navier-Stokes equations are used for constraints [38][39][40]. Consequently in this study our main concern is not to determine the control law with the maximum energetic efficiency as it can be characterized for example by the Power Saving Ratio (PSR) [40, for a definition or hereafter in § 5.2.3].…”

“…Recently, due to the maturity of control theory, optimization methods and computational fluid dynamics, optimal and suboptimal approaches attracted increased attention in flow control setting [35][36][37]. For example, in [38][39][40] the optimal control theory was used with the twodimensional Navier-Stokes equations as the state equation to control by rotary oscillation the unsteady wake of the cylinder (see table 1 for the characteristics of these approaches). An attractive element of the optimal control approach is that the control design is explicitly based on the cost functional.…”

Adjoint-based optimization of PDE systems with alternative gradients

“…Since Riesz representation (5) does not apply in nonHilbert spaces, we employ a more general procedure to extract the steepest descent directions in Banach spaces which follows the proposal first made by Neuberger in [15]. This procedure will involve a nonlinear transformation of the adjoint field equivalent to a nonlinear change of variables in iterative procedure (4). Further-more, by extracting the steepest descent directions in a continuous family of nested spaces we will allow this change of the metric to vary in the course of iterations (4).…”

“…• shape optimization in aerodynamics (see, e.g., [1,2]), • flow control for drag reduction, (see, e.g., [3,4]), • variational data assimilation in dynamic meteorology known as 4DVAR (see, e.g., [5]), • mixing enhancement (see, e.g., [6]). …”

Nonlinear Preconditioning in Problems of Optimal Control for Fluid Systems

“…• shape optimization with application to aircraft design, e.g., Mohammadi and Pironneau (2001); Martins et al (2004), • flow control for drag reduction, e.g., Bewley et al (2001); Protas and Styczek (2002), • variational data assimilation in dynamic meteorology, e.g., Kalnay (2003),…”