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The adjoint-based sensitivity analyses well explored in hydrodynamic stability studies are extended to calculate the sensitivity of forces acting on an aerofoil with respect to wall transpiration. The magnitude of the sensitivity quantifies the controllability of the force, and the distribution of the sensitivity represents a most effective control when the control magnitude is small enough. Since the sensitivity to streamwise control is one order smaller than that to the surface-normal one, the work is concentrated on the normal control. In direct numerical simulations of flow around a NACA0024 aerofoil, the unsteady controls are far less effective than the steady control owing to the lock-in effect. At a momentum coefficient of 0.0008 and a maximum control velocity of 3.6% of the free-stream velocity, the steady surface-normal control reduces drag by 20% or enhances lift by up to 140% at  Re =1000. A suction around the low-pressure region on the upper surface upstream of the separation point is found to reduce drag and enhance lift. At higher Reynolds numbers, the uncontrolled flow becomes three dimensional and the sensitivity diverges owing to the chaotic dynamics of the flow. Then the mechanism identified at lower Reynolds numbers is exploited to obtain the control, which is localized and can be generated by a limited number of actuators. The control to reduce drag or enhance lift is found to suppress unsteadiness, e.g. vortex shedding and three-dimensional developments. For example, at  Re =2000 and  α =10°, the control with a momentum coefficient of 0.0001 reduces drag by 20%, enhances lift by up to 200% and leads to a steady controlled flow.

Sensitivity of forces to wall transpiration in flow past an aerofoil