The use of global modes to understand transition and perform flow control

Abstract

The stability of nonparallel flows is considered using superposition of global modes. When perturbed by the worst case initial condition, these flows often exhibit a large transient growth associated with the development of wave packets. The global modes of the systems also provide a good starting point for the design of reduced order models used to control the growing disturbances. Three recent investigations are reviewed. The first example is the growth of a wave packet on a falling liquid sheet. The optimal perturbation analysis shows that the worst case initial condition is a localized disturbance that creates a propagating wave packet that hits the downstream end, regenerating a wave packet upstream through a global pressure pulse. Second, we consider two-dimensional disturbances in the Blasius boundary layer. It is found that a wave packet is optimally excited by an initial condition consisting of localized backward leaning Orr structures. Finally, the control of a globally unstable boundary-layer flow along a shallow cavity is considered. The disturbance propagation is associated with the development of a wave packet along the cavity shear layer, unstable to the Kelvin–Helmholtz mechanism, followed by a global cycle related to the two unstable global modes. Direct numerical simulations of this flow are coupled to a measurement feedback controller, which senses the wall shear stress at the downstream lip of the cavity and provides the actuation at the upstream lip. A reduced order model for the control is obtained by a projection on the least stable global eigenmodes. The linear-quadratic-Gaussian controller is run in parallel to the Navier–Stokes time integration and it is shown to damp out the global oscillations.