Periodic Control of Turbulent Boundary Layers

Abstract

Abstract : Over the last two and a half years, control strategies were developed to reduce viscous drag in turbulent boundary layers, in particular, by expanding on the recent discovery that simple two dimensional, streamwise sinusoidal waves traveling upstream sustain a channel flow with sublaminar drag. This simple open-loop control of wall-bounded blowing and suction modifies the Reynolds shear stress distribution to directly reduce drag. This reduction is predicted by linear theory and has been confirmed with nonlinear direct numerical simulations (DNS). However, the traveling waves also induce instabilities in the channel flow. Channel flow dynamical equations linearized about a periodic flow induced by the traveling wave shows unstable modes. For small amplitude traveling waves, the linearized dynamics are fairly accurate in predicting the instabilities. Significant progress has been made in understanding the instability of the flow field induced by the upstream traveling wave using Floquet analysis, developing an appropriate linearized dynamical system for which the control design is based, and the implementation of these controllers to suppress these secondary instabilities for small amplitude traveling waves. Although the linearized dynamics are periodic in a fixed or laboratory coordinate frame, the system dynamics are time-invariant in a coordinate frame moving with the traveling wave. This allowed an enormous simplification of the controller syntheses and based on the moving coordinate frame, linear quadratic regulators (LQR) are shown to suppress these secondary instabilities. Currently, the linear models and the feedback controls designed on them are limited to somewhat small amplitude traveling waves. Since even for small amplitude traveling waves, the coupling induced by the traveling wave between wave number pairs creates a very large dynamical system. An approach to model reduction, which might have enormous benefit in constructing reduced models for large am