Active flow control often exploits disturbance amplification mechanisms to achieve desired flow properties. Recently, theoretical predictions of optimal control based on stability analysis have gained traction. However, these methods are limited in their ability to predict nonlinear control strategies, such as burst-mode actuation for separated flows, which involve intermittent and high-amplitude forcing. To address this limitation, we developed a nonlinear optimal forcing analysis based on optimal perturbation theory. This method is specifically designed to capture non-harmonic forcing patterns and the nonlinear temporal evolution of the disturbance field. We applied this method to the two-dimensional high-subsonic, low-Reynolds number flow around a NACA0012 airfoil to reattach the separated flow and investigate the onset mechanism of low-frequency oscillation. The analysis identified an optimal temporal forcing pattern characterized by damped oscillation. This forcing exploits flow amplification mechanisms over the separation bubbles, promoting the formation of spanwise vortices in the shear layer. When implemented as a periodic forcing concentrated at the separated point, these vortices were stably generated, resulting in a significant lift increase via momentum exchange. A key finding is that the application of this optimal forcing induced long-term changes in the flow field, driven by the transient emergence of low-frequency oscillations. Furthermore, we explored the intermittent application of this forcing and found that an appropriate duty cycle can enhance the lift coefficient while reducing energy consumption.
Taniguchi et al. (Tue,) studied this question.