Optimizing control of stationary cross-flow vortices excited by surface roughness

Abstract

Stationary cross-flow vortices are excited within the swept Hiemenz boundary layer via surface roughness and actively controlled using an optimally configured control device. Control is modeled using localized wall motion, but in practice the optimization strategy could be applied to other laminar flow control technologies. A sensor-control iterative procedure, based on solutions of the forward and adjoint linearized Navier–Stokes equations, is applied to both feedforward and feedback loop systems. The former strategy only allows the control settings to be configured once, while the latter approach permits the repeated reoptimization of the control device. Surface roughness establishes a stationary cross-flow disturbance with a predefined set of flow conditions, but an unknown amplitude and phase. A sensor measures the local amplitude of the perturbation and relays the information to the control mechanism. Solutions of the adjoint linearized Navier–Stokes equations are coupled with the sensor measurements to configure and optimize the control mechanism, and establish an anti-phase wave that brings about destructive wave interference. The amplitude of the stationary cross-flow instability is reduced by an order 103 for the feedforward system, while amplitude reductions of the order 103 per iteration and 108 overall are realizable for the feedback modeling approach. Similar levels of flow control are realizable for a multiple controller configuration. However, stationary cross-flow disturbances could not be eliminated indefinitely. Inevitably, the cross-flow instability started to grow again, albeit at a considerably lower magnitude. The analysis is extended to include the effects of systematic error in the sensors measuring capability.