Boundary layer receptivity to pairs of unsteady oblique freestream vortical modes is studied in direct numerical simulation of flow over a flat plate with an elliptic leading edge. The freestream is perturbed by three types of oblique Fourier modes, differing in the magnitude of the three vorticity components. The vortical modes excite steady boundary layer streaks, and the associated receptivity mechanism is nonlinear in the forcing amplitude. Leading edges with two different aspect ratios are considered. It is found that the streak amplitudes obtained are largely unaffected by the leading edge bluntness. Whereas linear receptivity is the predominant mechanism at low forcing frequencies, the nonlinear mechanism becomes important when high-frequency vortices are present in the freestream. Nonlinear receptivity is therefore expected to contribute significantly to the excitation of boundary layer streaks by freestream turbulence. 

Adjoint-based iterative methods are employed in order to compute linear optimal disturbances in the case of a spatially growing boundary layer around an elliptic leading edge. The Lagrangian approach is used where an objective function is chosen and constraints are assigned. The optimization problem is solved using power iterations combined with a matrix-free formulation, where the state is marched forward in time with a standard DNS solver and backward with the adjoint solver until a chosen criterion is fulfilled. We consider the global and the upstream localized optimal initial condition leading to the largest possible energy amplification at time T . We find that the two-dimensional initial condition with the largest potential for growth is a Tollmien-Schichting-like wave packet that includes the Orr mechanism and is located inside the boundary layer, downstream of the leading edge. The localized optimal initial condition method allows a more precise systematic study of leading edge effects; we propose it a new method to study receptivity. We find the two-dimensional disturbances are inefficient at triggering an unstable eigenmode. The three-dimensional disturbances exploit the lift up mechanism; both the global and upstream localized disturbances give significant growth. These findings support the hypothesis of high receptivity to three-dimensional disturbances. 

Receptivity of the two-dimensional boundary layer on a flat plate with elliptic leading edge is studied by numerical simulation. Vortical perturbations in the oncoming free stream are considered, impinging on two leading edges with different aspect ratio to identify the effect of bluntness. The relevance of the three vorticity components of natural free-stream turbulence is illuminated by considering axial, vertical and spanwise vorticity separately at different angular frequencies. The boundary layer is most receptive to zero-frequency axial vorticity, triggering a streaky pattern of alternating positive and negative streamwise disturbance velocity. This is in line with earlier numerical studies on non-modal growth of elongated structures in the Blasius boundary layer. We find that the effect of leading-edge bluntness is insignificant for axial free-stream vortices alone. On the other hand, vertical free-stream vorticity is also able to excite non-modal instability in particular at zero and low frequencies. This mechanism relies on the generation of streamwise vorticity through stretching and tilting of the vertical vortex columns at the leading edge and is significantly stronger when the leading edge is blunt. It can thus be concluded that the non-modal boundary-layer response to a free-stream turbulence field with three-dimensional vorticity is enhanced in the presence of a blunt leading edge. At high frequencies of the disturbances the boundary layer becomes receptive to spanwise free-stream vorticity, triggering Tollmien-Schlichting (T-S) modes and receptivity increases with leading-edge bluntness. The receptivity coefficients to free-stream vortices are found to be about 15% of those to sound waves reported in the literature. For the boundary layers and free-stream perturbations considered, the amplitude of the T-S waves remains small compared with the low-frequency streak amplitudes.