Stabilization of Tollmien—Schlichting Waves by Mode Interaction

Abstract

Decreasing skin friction in boundary layers attached to aircraft wings can have an impact in both fuel consumption and pollutant production, which are becoming crucial to reduce operation costs and meet environmental regulations, respectively. Skin friction in turbulent boundary layers is about ten times that of laminar boundary layers. Thus, an obvious method to reduce friction drag is to delay transition to turbulence, which is a fairly involved process in real aircraft wings [J98]. Transition sis promoted either by Tollmien—Schlichting (TS) and Klebanov (K) modes [K94], with the former playing an essential role. Various methods (e.g., suction [SG00,ZLB04], wave cancellation [WAA01,LG06]) have been proposed to reduce TS modes in laminar boundary layers. Mode interaction methods have been successfully used in fluid systems to control related instabilities, such as the Rayleigh—Taylor instability [LMV01]. Here, we present some recent results on using these methods to control TS modes in a compressible, 2D boundary layer over a flat plate at zero incidence. A given unstable TS mode can be stabilized by coupling its spatial evolution with that of a second selected stable TS mode, in such a way that the stable mode takes energy from the unstable one and gives a stable coupled evolution of both modes. The coupling device is a wavetrain in the boundary layer, with appropriate wavenumber and frequency, which can be created by an array of oscillators on the wall, and promotes both (i) parametric coupling between the stable and unstable TS modes and (ii) a mean flow that is also stabilizing. Three differences with wave cancelation methods are relevant. Namely, (a) nonlinear terms play an essential role in the process; (b) the unstable TS mode is stabilized (its growth rate is decreased), not just canceled; and (c) stabilization does not depend on the phase of the incoming wave, which implies that active control is not necessary.