Flow control

In the case of the Blasius boundary layer it is by now accepted that more than one route to transition is possible. When free-stream disturbances are very small, transition is conceivably preceded by the exponential amplification of Tollmien-Schlichting waves. On the other hand, when somewhat larger perturbations are present, the near-wall region becomes populated by elongated streaks of low and high streamwise velocity.

Early theory was able to explain the presence of the streaks on the basis of a physical argument, named the lift-up effect. Today, most attention has shifted to the fact that the linearized operator governing the amplification of infinitesimal disturbances is highly non-normal and, as a consequence, significant transient amplification of non-modal disturbances can occur even in subcritical flow configurations.

It is reasonable to argue that transition can be spatially delayed if the amplification of the streaks is reduced. A simple way to do that is to employ suction/blowing at the wall to control the growth of the streaks. The term optimal control is here used to denote the best way of controlling a certain initial perturbation. In the context of boundary-layer instabilities, the perturbation kinetic energy is usually taken as an indicator of the level of disturbances, so that the control can be optimized by requiring, for instance, the energy at the end of the flat plate (or the integral of the energy over the streamwise length) to be the lowest.

The simplest approach to identifying the optimal control of a given flow is to impose the worst initial condition and then optimize the control. This procedure is indeed generally known by the name of "optimal control".

Another interesting topic related to flow control is the determination of the linear response of a turbulent system to small-enough perturbations. In view of the flow-control possibilities offered by modern MEMS technology, in fact, the linear-response function can help considerably in controller design, by answering such a basic question as which effects are to be felt here and now if a wall actuator has been moved there and a given time ago. Even though a turbulent flow is a nonlinear phenomenon, a linear response either in the frequency or the time domain can be defined if perturbations are small enough. Using the Direct Numerical Simulation (DNS) tools nowadays available it is possible to describe the complete mean response of a turbulent channel flow to small external disturbances. Space-time impulsive perturbations can be applied at one channel wall, and the linear response describes their mean effect on the flow field as a function of spatial and temporal separations. The turbulent response is shown to differ from the response of a laminar flow with the turbulent mean velocity profile as the base flow.

For more details, see the following related publications.

Luchini, P., Quadrio, M. & Zuccher, S. 2006 The phase-locked mean impulse response of a turbulent channel flow. Physics of Fluids, 18, 121702.

Zuccher, S., Luchini, P. & Bottaro A. 2004 Algebraic Growth of Blasius Boundary Layer: Optimal and Robust Control in the Nonlinear Regime. Journal of Fluid Mechanics 513, 135-160.

Zuccher, S. 2002 Receptivity and control of flow instabilities in a boundary layer, Ph.D thesis, Politecnico di Milano, Milano.