Linear iterative method for closed-loop control of quasiperiodic flows

Abstract

This work proposes a feedback-loop strategy to suppress intrinsic oscillations of resonating flows in the fully nonlinear regime. The frequency response of the flow is obtained from the resolvent operator about the mean flow, extending the framework initially introduced by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to study receptivity mechanisms in turbulent flows. Using this linear time-invariant model of the nonlinear flow, modern control methods such as structured ${\mathcal{H}}_{\infty }$-synthesis can be used to design a controller. The approach is successful in damping self-sustained oscillations associated with specific eigenmodes of the mean-flow spectrum. Despite excellent performance, the linear controller is however unable to completely suppress flow oscillations, and the controlled flow is effectively attracted towards a new dynamical equilibrium. This new attractor is characterized by a different mean flow, which can in turn be used to design a second controller. The method can then be iterated on subsequent mean flows, until the coupled system eventually converges to the base flow. An intuitive parallel can be drawn with Newton’s iteration: at each step, a linearized model of the flow response to a perturbation of the input is sought, and a new linear controller is designed, aiming at further reducing the fluctuations. The method is illustrated on the well-known case of two-dimensional incompressible open-cavity flow at Reynolds number $Re=7500$, where the fully developed flow is initially quasiperiodic (2-torus state). The base flow is reached after five iterations. The present work demonstrates that nonlinear control problems may be solved without resorting to nonlinear reduced-order models. It also shows that physically relevant linear models can be systematically derived for nonlinear flows, without resorting to black-box identification from input–output data; the key ingredient being frequency-domain models based on the linearized Navier–Stokes equations about the mean flow. Applicability to amplifier flows and turbulent dynamics has, however, yet to be investigated.

Predicting the response of turbulent channel flow to varying-phase opposition control: Resolvent analysis as a tool for flow control design

“…It should be mentioned that the largest drag increase among all DNS runs was observed for ∠Â d = 3π/4 and y + d = [20,25]. The Re τ of these runs increased by more than a factor of 2 and the simulations developed numerical instabilities due to insufficient grid resolution before a statistically steady state was reached.…”

“…Yeh and Taira [24] used the gains of the uncontrolled resolvent operator as proxy to identify adequate control parameters for separation control over an airfoil and showed that parameter combinations with high uncontrolled gain correspond to favorable controller settings. Leclerq et al [25] considered and controlled a two-dimensional incompressible open-cavity flow using structured H ∞ synthesis on a sequence of linear flow models, where each flow model of the sequence is based on the resolvent operator about the mean of the previous controlled flow. Their study shows that the controller is able to successfully suppress the oscillations in a sequence of five controllers.…”

“…The varying-phase opposition control scheme (9) introduced in the preceding section is used as a test to evaluate the capabilities of the resolvent model for controller design. To this end, a total of 50 varying-phase opposition control runs covering a parameter range of five sensor locations y + d = [5,10,15,20,25] and ten phase shifts ∠Â d = [−3π/4, −π/2, −3π/8, −π/4, −π/8, 0, π/8, π/4, π/2, 3π/4] were performed in the subsampled resolvent model and the DNS. The raw DR data were subsequently interpolated using bilinear splines and normalized with the respective maximum DR to generate the two contour maps of ξ shown in Fig.…”

Feedback Control of Coaxial Atomization Based on the Spray Liquid Distribution

“…The response of coaxial two-fluid atomizers to a wide range of swirling and non-swirling gas co-flows has been investigated extensively (Aliseda et al, 2008;Eggers and Villermaux, 2008;Lasheras et al, 1998;Marmottant and Villermaux, 2004), providing the theoretical and experimental basis for this feedback control work. Additionally, over the last decade data reduction methods have been extensively applied for the reduced order description and control of fluid flows (Blanchard and Sapsis, 2019;Grenga et al, 2018;Hervé et al, 2012;Huang et al, 2017;Krolick and Owkes, 2018;Leclercq et al, 2019;Mohan and Gaitonde, 2017;Rabault et al, 2019;Rowley and Dawson, 2017;Schmid et al, 2011;Tallet et al, 2016). Here, we present a method to perform feedback closed-loop control of a spray atomizer where the control input consists of measurements of light attenuation across the depth of the spray at 10 gas diameters downstream of the nozzle and where the control actuation is on the swirl and no-swirl gas flow rates (defined over ranges that span the desired momentum ratios and swirl ratios of interest).…”

A Ginzburg–Landau model for linear global modes in open shear flows

“…Ψ in (3.2)) (Bagheri et al 2009)? (ii) Can the GLE work on non-modal (Chen & Rowley 2011), quasi-periodic (Leclercq et al 2019) or multi-mode systems (Carini, Auteri & Giannetti 2015)? (iii) Can nonlinear effects be ignored (Leclercq et al 2019)?…”