Physics and control of wall turbulence
Seminar Room 1, Newton Institute
It has been generally accepted that nonlinearity is an essential characteristic of turbulent flows. Consequently, except for special situations in which a linear mechanism is expected to play a dominant role (e.g., rapidly straining turbulent flows to which the rapid distortion theory can be applied), the role of linear mechanisms in turbulent flows has not received much attention. Even for transitional flows, a common notion is that the most a linear theory can provide is insight into the early stages of transition to turbulence. But several investigators have recently shown that linear mechanisms play an important role even in fully turbulent, and hence fully nonlinear, flows. Examples of such studies include: optimal disturbances in turbulent boundary layers (Butler \& Farrell 1993); transient growth due to non-normality of the Navier-Stokes system (Reddy \& Henningson 1993); applications of a linear control theory to transitional and turbulent channel flows (Joshi \etal 1997); and a numerical experiment (Kim \& Lim 2000) demonstrating that near-wall turbulence could not be maintained in turbulent channel flow when a linear mechanism was artificially suppressed.
\medskip Turbulent channel flow is analyzed from a linear system point of view. After recasting the linearized Navier-Stokes equations into a state-space representation, the singular value decomposition (SVD) analysis is applied to the linear system, with and without control input, in order to gain new insight into the mechanism by which various controllers are able to accomplish the viscous drag reduction in turbulent boundary layers. We examine linear-quadratic-regulator (LQR) controllers that we have used, as well as the opposition control of Choi \etal (1994), which has been a benchmark for comparison of various control strategies. The performance of control is examined in terms of the largest singular values, which represent the maximum disturbance energy growth ratio attainable in the linear system under control. The SVD analysis shows a similarity between the trend observed in the SVD analysis (linear) and that observed in direct numerical simulations (nonlinear), thus reaffirming the importance of linear mechanisms in the near-wall dynamics of turbulent boundary layers. It is shown that the SVD analysis of the linearized system can indeed provide useful insight into the performance of linear controllers. Other issues, such as the effect of using the evolving mean flow as control applied to a nonlinear flow system (a.k.a. gain scheduling) and high Reynolds-number limitation, can be also investigated through the SVD analysis. Finally, time permitting, a linear Floquet analysis of a channel flow with periodic control, which had been shown to sustain skin-friction drag below that of a laminar channel, will be discussed to elucidate the drag reducing mechanism.