Adjoint-based iterative methods are employed to compute linear optimal disturbances in 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 optimisation problem is solved using power iterations combined with a matrix-free formulation, where the state is marched forward in time with a standard direct numerical simulation solver and backward with the adjoint solver until a chosen convergence criterion is fulfilled. We consider the global and, more relevant to receptivity studies, the upstream localised 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-Schlichting-like wave packet that includes the Orr mechanism and is located inside the boundary layer downstream of the leading edge. Three-dimensional optimal disturbances induce streaks by the lift-up mechanism. Requiring the optimal initial condition to be localised upstream of the plate enables us to better study the effects of the leading edge on the boundary layer receptivity mechanisms. Two-dimensional upstream disturbances are inefficient at triggering unstable eigenmodes, whereas three-dimensional disturbances induce streamwise streaks with significant growth.

Subcritical transition to turbulence requires finite-amplitude perturbations. Using a nonlinear optimisation technique in a periodic computational domain, we identify the perturbations of plane Couette flow transitioning with least initial kinetic energy for Re <= 3000. We suggest a new scaling law E-c = O(Re-2.7) for the energy threshold vs. the Reynolds number, in quantitative agreement with experimental estimates for pipe flow. The route to turbulence associated with such spatially localised perturbations is analysed in detail for Re = 1500. Several known mechanisms are found to occur one after the other: Orr mechanism, oblique wave interaction, lift-up, streak bending, streak breakdown, and spanwise spreading. The phenomenon of streak breakdown is analysed in terms of leading finite-time Lyapunov exponents of the associated edge trajectory.

We perform numerical simulations of control of a separating laminar boundary layer by means of blowing and suction at the wall in the form of traveling waves. Separation is imposed by prescribing accelerating and decelerating free-stream velocity for the flow over a flat plate. We find that downstream traveling waves already at very low amplitudes are able to eliminate the separation and induce a turbulent but attached boundary layer flow. Upstream traveling waves of relatively higher amplitudes only slightly reduce separation while keeping the flow laminar. The amplitude of the blowing/suction needed to achieve such significant effects are considerably smaller than those previously considered for drag reduction and transition delay in plane geometries.

We determine the initial condition on the laminar-turbulent boundary closest to the laminar state using nonlinear optimization for plane Couette flow. Resorting to the general evolution criterion of nonequilibrium systems we optimize the route to the statistically steady turbulent state, i.e., the state characterized by the largest entropy production. This is the first time information from the fully turbulent state is included in the optimization procedure. We demonstrate that the optimal initial condition is localized in space for realistic flow domains.

Adjoint-based iterative methods are employed in order to compute linear optimal disturbances in 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 optimisation 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 convergence criterion is fulfilled. We consider the global and the upstream localised optimal initial condition leading to the largest possible energy amplification at time T. We found that the twodimensional initial condition with the largest potential for growth is a Tolmien-Schlichting-like wave packet that includes the Orr mechanism and is located inside the boundary layer, downstream of the leading edge. Three-dimensional disturbances induce streaks by the lift-up mechanism. Localised optimal initial condition enables us to better study the effects of the leading edge; with this approach we propose a new method to study receptivity. Two-dimensional upstream disturbances, are inefficient at triggering an unstable eigenmode. The three-dimensional disturbances instead induce elongated streamwise streaks; both the global and upstream localised disturbances give significant growth. This advocates for high receptivity to three-dimensional disturbances.

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. 

The global linear stability of the flat-plate boundary-layer flow to three-dimensional disturbances is studied by means of an optimization technique. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. Both optimization problems are solved using a Lagrange multiplier technique, where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearized Navier-Stokes equations. The approach proposed here is particularly suited to examine convectively unstable flows, where single global eigenmodes of the system do not capture the downstream growth of the disturbances. In addition, the use of matrix-free methods enables us to extend the present framework to any geometrical configuration. The optimal initial condition for spanwise wavelengths of the order of the boundary-layer thickness are finite-length streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths, it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. This mechanism is dominant for the long computational domain and thus for the relatively high Reynolds number considered here. Three-dimensional localized optimal initial conditions are also computed and the corresponding wave packets examined. For short optimization times, the optimal disturbances consist of streaky structures propagating and elongating in the downstream direction without significant spreading in the lateral direction. For long optimization times, we find the optimal disturbances with the largest energy amplification. These are wave packets of Tollmien-Schlichting waves with low streamwise propagation speed and faster spreading in the spanwise direction. The pseudo-spectrum of the system for real frequencies is also computed with matrix-free methods. The spatial structure of the optimal forcing is similar to that of the optimal initial condition, and the largest response to forcing is also associated with the Orr/oblique wave mechanism, however less so than in the case of the optimal initial condition. The lift-up mechanism is most efficient at zero frequency and degrades slowly for increasing frequencies. The response to localized upstream forcing is also discussed.