This paper describes a scenario of transition from laminar to turbulent flow in a spatially developing boundary layer over a flat plate. The base flow is the Blasius non-parallel flow solution; it is perturbed by optimal disturbances yielding the largest energy growth over a short time interval. Such perturbations are computed by a nonlinear global optimization approach based on a Lagrange multiplier technique. The results show that nonlinear optimal perturbations are characterized by a localized basic building block, called the minimal seed, defined as the smallest flow structure which maximizes the energy growth over short times. It is formed by vortices inclined in the streamwise direction surrounding a region of intense streamwise disturbance velocity. Such a basic structure appears to be a robust feature of the base flow since it is practically invariant with respect to the initial energy of the perturbation, the target time, the Reynolds number and the dimensions of the computational domain. The minimal seed grows very rapidly in time while spreading, and it triggers nonlinear effects which bring the flow to turbulence in a very efficient manner, through the formation of a turbulence spot. This evolution of the initial optimal disturbance has been studied in detail by direct numerical simulations. Using a perturbative formulation of the Navier–Stokes equations, each linear and nonlinear convective term of the equations has been analysed. The results show the fundamental role of the streamwise inclination of the vortices in the process. The nonlinear coupling of the finite amplitude disturbances is crucial to sustain such streamwise inclination, as well as to generate dislocations within the flow structures, and local inflectional velocity distributions. The analysis provides a picture of the transition process characterized by a sequence of structures appearing successively in the flow, namely, $ \mrm{\Lambda} $ vortices, hairpin vortices and streamwise streaks. Finally, a disturbance regeneration cycle is conceived, initiated by the fast nonlinear amplification of the minimal seed, providing a possible scenario for the continuous regeneration of the same fundamental flow structures at smaller space and time scales.

Recent studies have suggested that in some cases transition can be triggered by some purely nonlinear mechanisms. Here we aim at verifying such an hypothesis, looking for a localized perturbation able to lead a boundary-layer flow to a chaotic state, following a nonlinear route. Nonlinear optimal localized perturbations have been computed by means of an energy optimization which includes the nonlinear terms of the Navier-Stokes equations. Such perturbations lie on the turbulent side of the laminar-turbulent boundary, whereas, for the same value of the initial energy, their linear counterparts do not. The evolution of these perturbations toward a turbulent flow involves the presence of streamwise-inclined vortices at short times and of hairpin structures prior to breakdown.

This paper provides the analysis of bursting and transition to turbulence in a Couette flow, based on the growth of nonlinear optimal disturbances. We use a global variational procedure to identify such optimal disturbances, defined as those initial perturbations yielding the largest energy growth at a given target time, for given Reynolds number and initial energy. The nonlinear optimal disturbances are found to be characterized by a basic structure, composed of inclined streamwise vortices along localized regions of low and high momentum. This basic structure closely recalls that found in boundary-layer flow (Cherubini et al., J. Fluid Mech., vol. 689, 2011, pp. 221–253), indicating that this structure may be considered the most ‘energetic’ one at short target times. However, small differences in the shape of these optimal perturbations, due to different levels of the initial energy or target time assigned in the optimization process, may produce remarkable differences in their evolution towards turbulence. In particular, direct numerical simulations have shown that optimal disturbances obtained for large initial energies and target times induce bursting events, whereas for lower values of these parameters the flow is directly attracted towards the turbulent state. For this reason, the optimal disturbances have been classified into two classes, the highly dissipative and the short-path perturbations. Both classes lead the flow to turbulence, skipping the phases of streak formation and secondary instability which are typical of the classical transition scenario for shear flows. The dynamics of this transition scenario exploits three main features of the nonlinear optimal disturbances: (i) the large initial value of the streamwise velocity component; (ii) the streamwise dependence of the disturbance; (iii) the presence of initial inclined streamwise vortices. The short-path perturbations are found to spend a considerable amount of time in the vicinity of the edge state (Schneider et al., Phys. Rev. E, vol. 78, 2008, 037301), whereas the highly dissipative optimal disturbances pass closer to the edge, but they are rapidly repelled away from it, leading the flow to high values of the dissipation rate. After this dissipation peak, the trajectories do not lead towards the turbulent attractor, but they spend some time in the vicinity of an unstable periodic orbit (UPO). This behaviour led us to conjecture that bursting events can be obtained not only as homoclinic orbits approaching the UPO, as recently found by van Veen & Kawahara (Phys. Rev. Lett., vol. 107, 2011, p. 114501), but also as heteroclinic orbits between the equilibrium solution on the edge and the UPO.