Optimal Perturbations and Low-Frequency Oscillations in a Separated Boundary-Layer Flow

Abstract

A separated boundary-layer flow at the rear of a bump is considered. The flow undergoes two-dimensional low-frequency global oscillations known as 'flapping' typical for elongated separation bubbles. Computing equilibrium stationary states of the Navier-Stokes equations and performing a global instability analysis, the onset of the two-dimensional instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with large transient energy growth. At larger time a global oscillation is found accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The low-frequency of the resulting beating behavior is retrieved when applying the optimal initial perturbation to Navier-Stokes time integration. Exploring the possibility of model reduction using temporal two-dimensional instability modes, a low-dimensional feedback controller is designed, performing a Petrov/Galerkin projection based on the bi-orthogonality between the direct and adjoint modes. The low-dimensional controller is shown to be able to damp the full linear instability dynamics.