Optimal free-stream perturbation to an airfoil flow

Free-stream perturbations induce large coherent structures, crucial to the laminar–turbulent transition on flight vehicles and turbine blades, significantly affecting drag, heat transfer, noise, etc. This paper computes the optimal free-stream perturbations that undergo the strongest amplification using an adjoint method, specifically for the incompressible flow around a NACA0012 airfoil, in contrast to sharp (or no) leading-edge configurations. These perturbations are examined at different angles of attack (0 Superscript ring ∘ ∘ – 6 Superscript ring 6 ∘ 6∘ ) and Reynolds numbers ( italic Re equals 1000 Re = 1000 Re = 1000 – 8000 8000 $8000$ ). At low angles of attack, two extreme points of energy growth emerge in the frequency–spanwise wavenumber plane, both on the axes. One has a spanwise wavenumber of zero, corresponding to two-dimensional unsteady modes in the wake, and the other has a frequency of zero, representing three-dimensional steady boundary-layer modes. The former is driven by the inflection point of the base velocity profile in the wake and is stronger, whereas the latter is related to the lift-up non-modal stability mechanism and directly influences the flow on the airfoil surfaces. The distribution of the three-dimensional steady optimal perturbations, similarly to the thickness of the base flow boundary layer, follows a italic Re Superscript negative 0.5 Re − 0.5 Re-0.5 scaling law. This scaling enables the projection of optimal profiles obtained at low italic Re Re Re to high ones for engineering applications, where the direct calculation is prohibitively expensive. Subsequently, the linear optimal perturbation at italic Re equals 100 000 Re </mml:mrow