S Cherubini

P De Palma

J.-Ch Robinet

Non linear optimals in the asymptotic suction boundary layer: transition thresholds and symmetry breaking

Keywords:

 for other transition scenarios. For low to moderate Reynolds numbers, these thresholds are found to scale as Re -2 , suggesting a new scaling law for transition in the ASBL.

I. INTRODUCTION

Drag reduction in external and internal flow is a fundamental topic in fluid mechanics since it is a key issue for improving the performance of engineering systems, increasing energy saving, and reducing environmental impact. Flow suction through the wall was among the first techniques applied to control the structure of the boundary layer in order to reduce drag. The idea, already employed by Prandtl and described in his first paper in 1904 [START_REF] Schlichting | Boundary layer theory[END_REF], was born together with the concept of boundary layer itself, described for the first time in 1904 by Prandtl during the Third International Mathematics Congress at Heidelberg [START_REF] Anderson | Ludwig prandtl's boundary layer[END_REF]. About thirty years later, such a technique was employed to delay transition over aircraft wings by reducing the boundary layer thickness and inducing a fuller velocity profile close to the wall [START_REF] Schlichting | Boundary layer theory[END_REF]. The influence of suction on the stability of the boundary layer was studied by analytical methods considering uniform suction at wall. In particular, a very simple exponential solution for the velocity was derived by Meredith and Griffith (1938) [START_REF] Schlichting | Boundary layer theory[END_REF] which would be valid at a sufficiently high distance from the leading edge of a flat plate. This solution of the Navier-Stokes equation is known as the asymptotic suction boundary layer (ASBL) [START_REF] Schlichting | Boundary layer theory[END_REF] and is considered a suitable model to study boundary layers subject to active control by suction and to investigate the transition mechanism. [START_REF] Hocking | Non-linear instability of the asymptotic suction velocity profile[END_REF] [START_REF] Hocking | Non-linear instability of the asymptotic suction velocity profile[END_REF] demonstrated that the critical Reynolds number is about two orders of magnitude higher than that of the Blasius boundary layer (BBL). In fact, the normal velocity term in the Orr-Sommerfeld and Squires equations stabilizes the Tollmien-Schlichting (TS) waves, producing an effective way of damping their asymptotic growth. This effect has been widely employed in different forms in the design of aircraft wing (see, for example, [START_REF] Joslin | Aircraft Laminar Flow Control[END_REF] [START_REF] Joslin | Aircraft Laminar Flow Control[END_REF]).

More recently, the development of the optimal transient growth analysis has renewed the interest in the study of the ASBL. In fact, it is well known that for a sufficiently high level of free-stream turbulence (FST), a bypass route to transition may occur in the boundary layer which corresponds to the growth of linear optimal perturbations (LOP) [START_REF] Fransson | Transition induced by free stream turbulence[END_REF][START_REF] Matsubara | Disturbance growth in boundary layers subjected to free-stream turbulence[END_REF]. This mechanism is based on the development of streamwise-aligned structures composed by alternating low and high velocity streaks observed for the first time by Klebanoff [START_REF] Klebanoff | Effects of freestream turbulence on the laminar boundary layer[END_REF]. The algebraic growth of the streaks due to the lift-up effect [START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF] leads eventually to secondary instability and break-up to turbulence [START_REF] Andersson | On the breakdown of boundary layer streaks[END_REF][START_REF] Brandt | Transition in a boundary layers subject to free-stream turbulence[END_REF][START_REF] Hoepffner | Transient growth on boundary layer streaks[END_REF]. From a numerical point of view, LOP were computed for several shear flows [START_REF] Andersson | Optimal disturbances and byoass transition in boundary layers[END_REF][START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF][START_REF] Corbett | Optimal perturbations for boundary layers subject to stream-wise pressure gradient[END_REF][START_REF] Luchini | Reynolds number indipendent instability of the Blasius boundary layer over a flat surface: optimal perturbations[END_REF][START_REF] Reddy | Energy growth in viscous channel flows[END_REF]. In all of these works, optimal perturbations are defined as those initial flow states yielding the largest amplification of the disturbance energy over a time/space interval and can be computed using a variational optimization approach [START_REF] Farrell | Optimal excitation of perturbations in viscous shear flow[END_REF][START_REF] Luchini | Adjoint Equations in Stability Analysis[END_REF]. For the case of the boundary layer at low Reynolds number, such optimal structures consists of pairs of streamwise aligned counter-rotating vortices producing streamwise streaks by the lift-up effect, in perfect agreement with the above experimental findings. The same mechanisms have been studied in the ASBL. [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layers[END_REF] [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layers[END_REF] performed an experimental analysis about the development of forced TS waves and about the algebraic growth of disturbances induced by free-stream turbulence.

They confirmed the damping of TS waves due to suction and were able to suppress transition in both cases. Using a local approach, [START_REF] Fransson | Optimal linear growth in the asymptotic suction boundary layer[END_REF] [START_REF] Fransson | Optimal linear growth in the asymptotic suction boundary layer[END_REF] computed LOP for the ASBL and compared their results with experiments. They observed a significant transient growth, although smaller than in the case of the BBL. This indicates that the strong effect of the damping of the energy growth of TS waves obtained by suction is not achieved in the case of the algebraic growth. [START_REF] Bystrom | Optimal disturbances in suction boundary layers[END_REF] [START_REF] Bystrom | Optimal disturbances in suction boundary layers[END_REF] computed LOP for the semi-suction boundary layer in order to take into account the presence of a small region free of suction close to the leading edge of the flat plate. With this approach, they could improve the agreement of the results with experimental data, demonstrating that the optimal energy growth is indeed obtained in the upstream region without suction. Finally, [START_REF] Levin | Transition thresholds in the asymptotic suction boundary layer[END_REF] [START_REF] Levin | Transition thresholds in the asymptotic suction boundary layer[END_REF] studied the energy thresholds for transition to turbulence in the ASBL, for Re = 500, 800, 1200, with perturbations having the form of oblique waves, streamwise vortices, or random noise; whereas, [START_REF] Levin | Turbulent spots in the asymptotic suction boundary layer[END_REF] [START_REF] Levin | Turbulent spots in the asymptotic suction boundary layer[END_REF] analyzed the energy threshold for the same Reynolds numbers, in the case of localized disturbances, and investigated the formation and evolution of turbulent spots.

The ASBL has been also considered for testing and validating the recent dynamical system theory of turbulence, which analyze the fundamental role of non linearities in transition to and sustainment of turbulence for shear flows. Such a theory relies on: 1) the observation of the existence of exact coherent states, which can be unstable fixed points, periodic orbits or chaotic solutions of the Navier-Stokes equations, having a few unstable directions (see Ref. [START_REF] Eckhardt | Turbulence transition of pipe flow[END_REF][START_REF] Hof | Experimental Observation of Nonlinear Traveling Waves in Turbulent Pipe Flow[END_REF][START_REF] Nagata | Three-dimensional finite amplitude solutions in plane Couette flow[END_REF][START_REF] Viswanath | Stable manifolds and the transition to turbulence in pipe flow[END_REF][START_REF] Waleffe | Three-dimensional states in plane shear flow[END_REF][START_REF] Wedin | Exact coherent structures in pipe flow: travelling wave solutions[END_REF]); 2) the idea that such states constitute the skeleton of transition and regeneration processes of turbulence and can be used to understand its nature. In particular, for analyzing the transition process, it is interesting to study those flow perturbations confined on the boundary between the laminar and the turbulent states, called the edge of chaos [START_REF] Cherubini | Edge states in a boundary layer[END_REF][START_REF] Duguet | Minimal transition thresholds in plane Couette flow[END_REF][START_REF] Schneider | Turbulence Transition and the Edge of Chaos in Pipe Flow[END_REF][START_REF] Skufca | Edge of chaos in a parallel shear flow[END_REF]. Those perturbations can be very dangerous, being the closest ones to the laminar state capable of triggering transition. On the edge of chaos, one or more relative attractors may live, called edge states [START_REF] Skufca | Edge of chaos in a parallel shear flow[END_REF], which can be fixed points [START_REF] Schneider | Laminar-turbulent boundary in plane Couette flow[END_REF], periodic orbits [START_REF] Kreilos | Edge states for the turbulence transition in the asymptotic suction boundary layer[END_REF] or chaotic states [START_REF] Eckhardt | Turbulence transition of pipe flow[END_REF]. Concerning the ASBL, [START_REF] Kreilos | Edge states for the turbulence transition in the asymptotic suction boundary layer[END_REF] [START_REF] Kreilos | Edge states for the turbulence transition in the asymptotic suction boundary layer[END_REF] investigated the structure of the edge-states for turbulence transition, identifying a periodic orbit embedded in the laminar-turbulent boundary. Such a solution shows the same basic mechanisms of transient growth interactions between streamwise-aligned vortices and streaks which characterize many shear flows. Furthermore, it captures also the bursting phenomenon typical of the BBL. Khapko et al. [START_REF] Khapko | Complexity of localised coherent structures in a boundary-layer flow[END_REF][START_REF] Khapko | Localized edge states in the asymptotic suction boundary layer[END_REF] have investigated the dynamics restricted to the laminarturbulent boundary, describing the complex spatio-temporal dynamics of different localized edge states for several streamwise wavelength. They found that all these states have the same structure, consisting of a localised pair of low-and high-speed streaks flanked with streamwise vortices. Investigating the structure of the relative attractors embedded in the edge of chaos allows to identify the typical shape and dynamics of the coherent structures consituting the skeleton of turbulence. However, for unraveling the main features of the most effective (in terms of both time and energy) path to transition, the minimal-energy states on the laminar-turbulent boundary should be analyzed [START_REF] Cherubini | Minimal perturbations approaching the edge of chaos in a Couette flow[END_REF][START_REF] Rabin | Triggering turbulence efficiently in plane Couette flow[END_REF].

Very recently, the problem of finding the minimal energy perturbation on the edge of turbulence has been investigated by solving the non linear optimal growth problem for finiteamplitude initial perturbations (see [START_REF] Kerswell | An optimisation approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar[END_REF] for a review). Those perturbations which optimize at a given (target) time the growth of a functional linked to transition (the kinetic energy or the dissipation for instance), called non linear optimal perturbations (NLOPs), have been found for a pipe flow [START_REF] Pringle | Using nonlinear transient growth to construct the minimal seed for shear flow turbulence[END_REF][START_REF] Pringle | Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos[END_REF]; a boundary layer flow [START_REF] Cherubini | Rapid path to transition via nonlinear localized optimal perturbations[END_REF][START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF]; and a Couette flow [START_REF] Cherubini | Nonlinear optimal perturbations in a Couette flow: bursting and transition[END_REF][START_REF] Duguet | Minimal transition thresholds in plane Couette flow[END_REF][START_REF] Monokrousos | Non-equilibrium thermodynamics and the optimal path to turbulence in shear flows[END_REF][START_REF] Rabin | Triggering turbulence efficiently in plane Couette flow[END_REF].

By optimizing the energy at large target times and bisecting the initial energy to bring the perturbation close to the laminar-turbulent boundary, the perturbation of minimal energy capable of bringing the flow to the edge state and then to transition, called the minimal seed of turbulent transition can be found [START_REF] Rabin | Triggering turbulence efficiently in plane Couette flow[END_REF]. When small target times are considered, a different procedure should be used for finding minimal-energy perturbations on the edge of chaos, directly targeting the neighbourhood of the edge state in a finite time [START_REF] Cherubini | Minimal perturbations approaching the edge of chaos in a Couette flow[END_REF].

In all cases, the NLOPs are characterized by a very different structure with respect to the linear optimal ones and largely outgrow them in energy due to non linear mechanisms [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF][START_REF] Pringle | Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos[END_REF].

For the boundary-layer and the Couette flow, the NLOPs are characterized by a similar fundamental structure, composed of a localized array of vortices and low-momentum regions of typical length scale, capable of maximizing the energy growth most rapidly. Cherubini et al. (2011) [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF][START_REF] Cherubini | A purely nonlinear route to transition approaching the edge of chaos in a boundary layer[END_REF] have discussed the contribution of non linear effects in such a strong energy growth, showing that non linearity is crucial to sustain the growth of such optimal perturbations. The knowledge of these non-linear mechanisms may allow one to design effective control strategies to delay transition by using wall suction [START_REF] Cherubini | Nonlinear control of unsteady finiteamplitude perturbations in the Blasius boundary-layer flow[END_REF]. The aim of the present paper is to extend the analysis of the NLOP to the case of the ASBL, following the approach that the authors have employed for the BBL, discussing similarities and differences between these two cases, and highlighting the role of the suction velocity.

The paper is organized as follows. In the second section we define the problem and describe the non linear optimization method. In the third section, divided into three parts, a thorough discussion of the results of the non linear optimization analysis is provided. In particular, in the first part, the focus is on the characterization of the NLOP with respect to the LOP, also by comparing with the results already obtained for the BBL and Couette flows.

The second part provides an analysis varying the Reynolds number, achieving supercritical values; whereas, the third part deals with the optimal route to turbulence, computed by the DNS. Finally, concluding remarks are provided.

II. PROBLEM FORMULATION

A. Governing equations and numerical method

The behaviour of an incompressible flow is governed by the Navier-Stokes (NS) equations:

∂u ∂t + (u • ∇)u = -∇p + 1 Re ∇ 2 u, ∇ • u = 0, (1)
where u = (u, v, w) is the velocity vector and p indicates the pressure term. Dimensionless variables are defined with respect to the inflow boundary-layer displacement thickness δ * and the freestream velocity, U ∞ , so that the Reynolds number is Re = U ∞ δ * /ν, ν being the kinematic viscosity. A Cartesian coordinate system is considered, x, y and z being the streamwise, wall-normal and spanwise directions, respectively. The asymptotic suction boundary-layer flow is defined as the flow over a flat plate with a uniform wall-normal suction velocity V S = 1/Re applied along the wall. Therefore, imposing no-slip conditions for the streamwise and spanwise components of the velocity at the wall, one obtains from equations

(1) the following solution [START_REF] Griffith | The possible improvement in aircraft performance due to boundary layer suction[END_REF]:

U = ((1 -e -y ), -V S , 0) T . (2)
The numerical simulations are performed by integrating the NS equations with the following boundary conditions: at the bottom boundary, no-slip conditions for the x and z components of the velocity and homogeneous suction for the y component, v = -V S ; at the upper boundary, the z component of the velocity and of the vorticity are set to zero and homogeneous injection is imposed for the y component of the velocity; in the streamwise and spanwise directions, periodicity is imposed for the three velocity components.

The analysis has been performed using five values of the Reynolds number (see table I), obtained varying the suction velocity V S . The NS equations are discretized by a finitedifference fractional-step method using a staggered grid [START_REF] Verzicco | A finite-difference scheme for the three-dimensional incompressible flows in cylindrical coordinates[END_REF]. A second-order-accurate centered space discretization is used. Performing a grid-convergence analysis, a mesh made up by 451 × 100 × 61 points has been selected for the reference domain at Re = 610 with dimensions L x = 100, L y = 20 and L z = 10.5. The spanwise dimension has been chosen very close to the one used in [START_REF] Levin | Transition thresholds in the asymptotic suction boundary layer[END_REF] for determining transition thresholds, whereas the streamwise length is much longer to avoid interaction of the flow structures with its own tail for long target times. Since the NLOP has been found to localize more and more with increasing Reynolds number, the domain length has been reduced accordingly, in order to reduce the computational cost. Table I summarizes the domain lengths and the corresponding grid points selected for different Reynolds numbers after validations with respect to larger domains.

B. Non linear optimization

The non linear behavior of a perturbation q = (u ′ , v ′ , w ′ , p ′ ) T evolving in the laminar asymptotic suction boundary-layer flow is analyzed by solving the NS equations written in perturbative formulation with respect to the steady state solution, Q = (U, P ) T , with U

given by equation [START_REF] Andersson | Optimal disturbances and byoass transition in boundary layers[END_REF]. A zero perturbation boundary condition is imposed for the three velocity components at the y-constant boundaries, whereas periodicity of the perturbation is forced in the spanwise and streamwise directions. The goal is to find the perturbation at t = 0 providing the largest disturbance growth at a given target time, T . At this purpose, a Lagrange multiplier technique is used [START_REF] Cherubini | Rapid path to transition via nonlinear localized optimal perturbations[END_REF][START_REF] Luchini | Adjoint Equations in Stability Analysis[END_REF][START_REF] Pringle | Using nonlinear transient growth to construct the minimal seed for shear flow turbulence[END_REF][START_REF] Zuccher | Algebraic growth in a blasius boundary layer: optimal and robust control by mean suction in the nonlinear regime[END_REF] to perform a constrained optimization of the perturbation energy. The disturbance energy density is defined as

E(t) = 1 2V V u ′ 2 (t) + v ′ 2 (t) + w ′ 2 (t) dV = 1 2V u ′ (t) • u ′ (t) , (3)
where V is the volume of the computational domain. Given an initial energy E(0) = E 0 , we aim at finding the shape and amplitude of an initial perturbation q 0 which induces at target time T the largest energy gain E(T )/E 0 ; threfore, the objective function of the optimization procedure is ℑ = E(T )/E(0). The Lagrange multiplier technique consists in searching for extrema of an augmented functional, L, with respect to every independent variable, the three-dimensional incompressible NS equations and the value of the initial energy being imposed as constraints. The augmented functional reads:

L = E(T ) E(0) - T 0 u † • ∂u ′ ∂t -u ′ • ∇U + U • ∇u ′ + u • ∇u ′ -∇p ′ - ∇ 2 u ′ Re dt - T 0 p † • ∇u ′ dt -λ E 0 E(0) -1 . (4)
where (u † , p † , λ) are the Lagrange multipliers, e.g. the adjoint variables. Integrating by parts and setting to zero the first variation of L with respect to (u ′ , p ′ ) leads to the adjoint equations plus the compatibility condition (which are provided in Ref. [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF]). The adjoint equations are linked to the direct ones by the presence of the direct variables in the advection terms, so that the whole flow field needs to be stored at each time step, requiring a remarkable storage capacity. The gradient of the augmented functional with respect to the initial perturbation q 0 is forced to vanish by means of a conjugate gradient algorithm as detailed in Ref. [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF]. A coupled iterative approach similar to that used in [START_REF] Zuccher | Algebraic growth in a blasius boundary layer: optimal and robust control by mean suction in the nonlinear regime[END_REF] and [START_REF] Pringle | Using nonlinear transient growth to construct the minimal seed for shear flow turbulence[END_REF] is used to solve the problem, relying on the forward and backward solution of the direct and adjoint NS equations, respectively, and on the update of the initial perturbation in the conjugate gradient direction at each iteration, until convergence is reached. A detailed description of the optimization technique and of its convergence properties is provided in Ref. [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF] for the case of the BBL flow and in Ref. [START_REF] Cherubini | Nonlinear optimal perturbations in a Couette flow: bursting and transition[END_REF] for the Couette flow.

III. RESULTS

A. Non linear optimal perturbations at Re = 610

The non linear optimization has been performed at Reynolds number Re = 610. This rather low Reynolds number (compared to the critical one for the ASBL) has been chosen for comparison purpose with the BBL case of ref. [START_REF] Cherubini | Rapid path to transition via nonlinear localized optimal perturbations[END_REF]. Figure 2 shows the value of the optimal energy gain versus the target time for an initial energy E 0 = 3.0 × 10 -7 . The dashed line refers to the results of a linear optimization, whereas the solid line represent the non linear optimization. As also observed for the BBL flow [START_REF] Cherubini | Rapid path to transition via nonlinear localized optimal perturbations[END_REF], the non linear optimal energy gain is remarkably larger than the corresponding linear one for T > 50. The influence of the parameter E 0 on the value of the optimal energy is shown in Figure 3, for three values of the target time. It appears that a non linearity threshold value of the initial energy exists from which strong differences are observed in the non linear optimal energy with respect to the linear one (compare the solid lines with the dashed ones). Such a threshold decreases when the target time increases, as one can observe by comparing the solid lines in Figure 3, converging towards a value, E 0 = 1.2 × 10 -7 , which might be close to the energy of the minimal seed for this Reynolds number (i.e., the perturbation of minimal energy on the laminar-turbulent boundary). Table II provides a comparison between the energy gains obtained for the BBL and the ASBL at T = 75 (the behavior is similar for different target times) for three optimizations: a linear optimization and two non linear optimizations with E 0 = 1.2 × 10 -7 and E 0 = 3.0 × 10 -7 , respectively. The results indicate that a significant reduction of the optimal energy growth is obtained in the linear case; however, wall suction (red), and T = 200 (green) versus the initial energy E(0), using the non linear optimization (solid lines with symbols) and the linear optimization (dashed lines). is much more effective in damping the growth of non linear optimal perturbations.

Test case Linear

E 0 = 1.2 × 10 -7 E 0 = 3.0 × 10 -
Crossing the non linearity threshold also yields large modifications in the shape of the optimal perturbations. This can be observed in Figure 4, which provides the optimal initial perturbations obtained for the ASBL at Re = 610 and T = 75, for two values of the initial energy, E 0 . For the lowest one, E 0 = 1.2 × 10 -7 (top frame), the perturbation is similar to that obtained by the linear optimization in a BBL flow [START_REF] Cherubini | Optimal wave packets in a boundary layer and initial phases of a turbulent spot[END_REF], being characterized by alternated vortices elongated in the streamwise direction (black and white surfaces), localized in two different positions along the flat plate. Due to weak non linear effects, which are non-negligible for such values of the initial energy, some spanwise modulations are present on the streamwise perturbation (green surfaces). Concerning the amplitudes, the largest perturbation velocity component is the spanwise one (|w max | = 0.0027), followed by the wall-normal (|v max | = 0.0025) and the streamwise one (|u max | = 0.0003). One can notice that the streamwise perturbation is one order of magnitude lower than the others, meaning that for this value of the initial energy the mechanism of growth is still very close to the linear optimal one, based on the lift-up of the streamwise base flow velocity by the vortices given by the wall-normal and spanwise perturbation. However, as one can observe in figure 4, the shape of the optimal perturbation changes remarkably between E 0 = 1.2 × 10 -7 and

E 0 = 3.0 × 10 -7 .
The most striking difference is the strong localization of the disturbance in both the streamwise and spanwise direction. In fact, for initial energies larger than the non linearity threshold, a strong localization of the initial perturbation leads to larger amplitudes (for the same initial energy), triggering non linear effects that induce a remarkable increase of the energy gain at target time. In fact, for an increase of the initial energy of a factor 2.5, we observe at t = 0 an increase of the velocity magnitudes of a factor of about 12 for v and w, whereas a factor 80 is obtained for u. These values of the perturbation velocity components, together with the particular shape of the disturbance, are able to trigger non linear effects which allow a much larger energy growth than in the linear case.

This strong localization appears to be a typical feature of NLOP in shear flows, since it has been also observed for the pipe [START_REF] Pringle | Using nonlinear transient growth to construct the minimal seed for shear flow turbulence[END_REF][START_REF] Pringle | Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos[END_REF], the BBL [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF], and the Couette flow [START_REF] Cherubini | Nonlinear optimal perturbations in a Couette flow: bursting and transition[END_REF][START_REF] Monokrousos | Non-equilibrium thermodynamics and the optimal path to turbulence in shear flows[END_REF][START_REF] Rabin | Triggering turbulence efficiently in plane Couette flow[END_REF].

Furthermore, not only the extension, but also the structure of the perturbation changes remarkably. For E 0 = 3.0×10 -7 , the optimal perturbation is composed by three streamwisealternated vortices showing a finite inclination with respect to the streamwise direction (black and white surfaces), whereas in the quasi-linear case at E 0 = 1.2 × 10 -7 the vortices are streamwise-aligned. On both flanks of such inclined vortices, localized patches of finiteamplitude streamwise disturbance are observed (green surfaces). Concerning the relative magnitude of the velocity perturbations, the largest perturbation velocity component is the spanwise one (|w max | = 0.033), followed by the streamwise (|u max | = 0.03) and the wallnormal one (|v max | = 0.024). These values are similar to those found for the Couette flow [START_REF] Cherubini | Nonlinear optimal perturbations in a Couette flow: bursting and transition[END_REF], whereas, for the BBL flow at the same Re, the largest component has been found to be the streamwise one, whose value is about half of the maximum value found here for the ASBL, for an initial energy just above the non linearity threshold, see [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF]. It is worth noticing that, for all of these flows, in the linear case the streamwise velocity component at initial time is from one to two orders of magnitude lower than the spanwise and the streamwise ones, whereas in the non linear case all of the components are of the same order, meaning that different mechanisms are responsible for the growth of the perturbation energy. The structure of the NLOP found here shows some similarities with that found for the Couette flow (compare with Figure 5 of Ref. [START_REF] Cherubini | Nonlinear optimal perturbations in a Couette flow: bursting and transition[END_REF]) and with that obtained for the BBL flow (see Figure 4, bottom). For all of these shear flows, the NLOP is characterized by streamwise-inclined vortical structures and finite-amplitude patches of streamwise disturbance. However, while for the Couette and the ASBL flow (at least at low Reynolds number) the optimal disturbance does not show any particular symmetry, for the BBL it is symmetrical with respect to a z = const axis. In fact, one can see in Figure 4 (bottom frame), that the NLOP for the Blasius flow at E 0 = 1.2 × 10 -7 is composed by a basic structure similar to that of the ASBL, but the disturbance is symmetric with respect to a z-aligned axis.

The flow structures can be better analyzed by taking x-constant slices of the NLOP, as As proposed in Ref. [START_REF] Pringle | Using nonlinear transient growth to construct the minimal seed for shear flow turbulence[END_REF], the disturbance of minimum amplitude capable of triggering turbulence is defined as the minimal seed for a given Reynolds number. Bisecting the value of the initial energy at T = 200, and checking whether the obtained NLOP is able to induce transition, we have found the energy level of the minimal seed to be about 1.277 × 10 -7

for Re = 610. The corresponding maximum amplitudes of the velocity components are |u| max = 0.029, |v| max = 0.031, |w| max = 0.031, very close to the values found at lower target time (even if the wall-normal component is now slightly larger than the streamwise one). The minimal seed is sandwiched between the NLOPs shown in figure 7 (a) and (b), for E 0 = 1.2 × 10 -7 and E 0 = 1.35 × 10 -7 , both showing the basic structure provided in figure 4 (middle frame). It is worth to notice that the NLOP keeps the same structure of the minimal seed also for values of the initial energy slightly lower than the minimal seed energy. For larger values of the initial energy, local maxima can be found, as in the case of the Couette flow [START_REF] Cherubini | Nonlinear optimal perturbations in a Couette flow: bursting and transition[END_REF]. Furthermore, for energies larger than the minimal seed one, convergence on the optimum is not assured (see [START_REF] Rabin | Triggering turbulence efficiently in plane Couette flow[END_REF]), since transition might be observed at target time.

However, since for T = 200 transition is still not observed for initial energies not much higher than the minimal seed one, we have been able to perform two optimizations with two different initial conditions, in order to check the existence of such local maxima. Thus, we have chosen E 0 = 2.7 × 10 -7 (almost double than the energy of the minimal seed), with two different initializations, namely, i) the minimal seed structure, and ii) a symmetrical initial disturbance constructed by mirroring in the spanwise direction the minimal seed structure with respect to the streamwise axis. The energy gain for the first optimization is E(T )/E 0 = 1641, and the optimal shape is shown in figure 7 (c flow, but this does not lead to a symmetrisation of the optimal (at least for Re = 610).

The results discussed above show that

B. Reynolds number analysis

The laminar profile of the ASBL given by equation ( 2) is linearly stable for Re < 54382 [START_REF] Hocking | Non-linear instability of the asymptotic suction velocity profile[END_REF], which is about 100 times the critical Reynolds for a Blasius boundary layer [START_REF] Schlichting | Boundary layer theory[END_REF]. Thus, the value of Re = 610 employed in the previous analysis is largely subcritical for the ASBL.

Therefore, to generalize our results, we have extended the analysis to larger Reynolds numbers, namely, Re = 1200, 2500, 5000, 10000. For each value of the Reynolds number but the latter, the value of the initial energy has been bisected in order to approximate the minimal seed with a two-digit accuracy. The solid line in Figure 8 shows the energy of the minimal seed, E min , versus the Reynolds number, for Re = 610, 1200, 2500, 5000. Whereas, the dashed lines in the figure reproduce the results of Ref. [START_REF] Levin | Transition thresholds in the asymptotic suction boundary layer[END_REF][START_REF] Levin | Turbulent spots in the asymptotic suction boundary layer[END_REF], for four different initial perturbation structures: i) random three-dimensional noise (NOISE); ii) streamwise vortices (SV), obtained by a local spatial optimization; iii) spatially extended oblique waves (OW), obtained by a local spatial optimization; iv) localized disturbances (LD) consisting of two alternated counter-rotating pairs of streamwise vortices. One can notice that the transition threshold provided by the minimal seed energy is amost two orders of magnitude lower than the energy thresholds found for spatially extended disturbances such as the streamwise vortices and the oblique waves. Moreover, E min is one order of magnitude lower than the minimal energy found for the localized perturbations selected in Ref. [START_REF] Levin | Turbulent spots in the asymptotic suction boundary layer[END_REF].

We can also observe in figure 8 that E min varies with Re roughly following a power law Re -2 , whereas Levin et al. [START_REF] Levin | Transition thresholds in the asymptotic suction boundary layer[END_REF] found a -2.1 exponent for the SV and NOISE perturbations and a -2.8 exponent for the OW one. It is worth to notice that in Ref. [START_REF] Duguet | Minimal transition thresholds in plane Couette flow[END_REF], computing minimal seeds, a power law Re -2.7 has been found for a Couette flow in a small domain (whereas the OW scenario was characterized by a -2 exponent for the same configuration).

Concerning the velocity amplitudes, for Re = 1200 the minimal seed is characterized by |u| max = 0.014, |v| max = 0.015, |w| max = 0.017. Whereas the minimal LD triggering transition in Ref. [START_REF] Levin | Turbulent spots in the asymptotic suction boundary layer[END_REF] was characterized by |v| max = 0.0124, very close to the minimal amplitudes found here, but |u| max = 0.0. Thus, the large difference in the transition thresholds can be linked on the complete absence of streamwise velocity disturbances in Ref. [START_REF] Levin | Turbulent spots in the asymptotic suction boundary layer[END_REF], which appears to be a crucial feature for inducing a rapid transition to turbulence using low-energy perturbations. Two other crucial elements which might explain the difference between the energy thresholds for the LD and the minimal seed are: i) the larger spatial extension of the LDs, which makes them more energetic than the minimal seed for similar associated amplitudes; ii) the fact that the vortices are perfectly aligned with the streamwise axis, and spanwise-symmetric with respect to this axis, whereas the perturbations inducing the largest growth by non linear mechanisms appear to be characterized by a finite inclination with respect to the streamwise axis, and do not show any spanwise symmetry. Thus, it appears that a non linear optimization is crucial to determine the order of magnitude of the minimal thresholds for transition to turbulence, and for accurately determining the shape and typical length scales of the minimal perturbation capable of inducing transition to turbulence.

As shown in Figure 9, the shapes of the minimal seed for Re = 1200 and Re = 2500 are very similar to the ones described in the previous section for Re = 610. They are composed III. In particular, the streamwise and spanwise lengths appear to be almost halved for an increase of Re of a factor two.

For a fluid with a given kinematic viscosity, the dependence on Re of the typical length scales of the wall-structures, such as the streaks, is more complex for the ASBL than for the BBL. For the ASBL the Reynolds number can be varied either by changing the freestream velocity U ∞ (keeping the suction velocity fixed), or by changing the suction velocity, thus modifying the displacement thickness δ * (but keeping U ∞ fixed). Yoshioka et al. [START_REF] Yoshioka | Free stream turbulence induced disturbances in boundary layers with wall suction[END_REF] have experimentally measured the typical length scales of wall structures induced by free streamturbulence in the ASBL, concluding that: i) if V S and δ * are kept constant, the spanwise spacing of the streaks varies with U -1 ∞ ; ii) if U ∞ is kept constant and δ * is changed, the dimensional spacing of the streaks remains constant. In the present work, we change the Reynolds number by keeping U ∞ constant, modifying the suction velocity V S ; therefore, we are increasing the reference length, δ * , when the Reynolds number increases. Thus, a decrease of a factor 2 of the minimal seed size corresponding to a twofold increase of the reference length δ * is consistent with the results in Ref. [START_REF] Yoshioka | Free stream turbulence induced disturbances in boundary layers with wall suction[END_REF], since the dimensional typical length scales of the streaks will not change with Re when U ∞ is kept constant. We can thus conclude that, at least for Reynolds numbers in the range [610, 10000], the typical length scales of the NLOPs change accordingly to the streaks spacing measured in Ref. [START_REF] Yoshioka | Free stream turbulence induced disturbances in boundary layers with wall suction[END_REF]. This explains why we have chosen to use smaller domain lengths for larger Re.

Another important feature of the minimal seed shown in figure 9 for Re = 1200 and 2500 is that, as for Re = 610, it does not present any symmetry in the spanwise direction. How-ever, when the Reynolds number is increased to Re = 5000, two types of non linear optimal solutions have been found. Figure 10 shows the result of the non linear optimization for two initial energies, E 0 = 2 × 10 -9 and E 0 = 4 × 10 -9 , for T = 100. One can observe that the latter is almost symmetrical, roughly corresponding to a mirroring of the former with respect to the streamwise axis. The minimal seed for this Reynolds number is sandwhiched between these two solutions with different symmetries. Further bisections between these two values indicate that the minimal seed is not symmetrical. However, the presence of a symmetrical non linear optimal is an indication of the change in the optimal dynamics that is observed at larger Reynolds numbers. In fact, for Re = 10000, only symmetrical optimal disturbances have been obtained. Figure 11 shows two of them for two energy levels and a short target time T = 50. This result has been verified by performing optimizations at a larger, supercritical Reynolds number, Re = 65000, for which a symmetrical minimal seed has been found as well, preserving the same structure. This indicates that a value of the Reynolds exists between 5000 and 10000 for which the NLOP changes from a nonsymmetric shape to a symmetric one. Therefore, for sufficiently high Reynolds numbers, the structure of the minimal seed becomes similar to the one of the BBL. This behavior can be explained considering that we are changing the Reynolds number Re = U ∞ /V S by modifying the suction velocity V s ; therefore, for high values of Re, the influence of V S tends to become smaller and the velocity profile tends to be closer to that of the BBL flow. At some "critical" Reynolds number, the modification of the ASBL profile towards a profile closer to the BBL one induces a change in the symmetry of the NLOP.

In the next section we will compare the route to turbulence at low Reynolds number for the ASBL and the BBL, in order to understand which differences in the base flow profile can induce different symmetries on the non linear optimal solutions, and link these results to the change of symmetry observed at high Reynolds numbers in the ASBL C. The route of the non linear optimal perturbations to turbulence

In this section, we analyze by DNS the evolution towards turbulence of the NLOP obtained for the ASBL with Re = 610 and T = 75, providing a comparison with the NLOP of the BBL for the same conditions [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF]. In order to achieve transition, the two perturbations have different energy, namely, E 0 = 3.0 × 10 -7 for the ASBL and E 0 = 1.2 × 10 -7 for the BBL. In the case of the BBL, being the flow non-parallel, we do not impose periodicity in the streamwise direction. Thus, we have used a domain two times longer in x than that used for the ASBL, in order to follow the evolution of the perturbation for a sufficiently long time before the disturbance leaves the domain. The number of grid points in the streamwise direction has been doubled as well, in order to maintain the same local grid resolution. A qualitative picture of the transition process initiated by the NLOP for the ASBL is given in figure 12, showing the streamwise vorticity (black and white surfaces) and velocity (blue and yellow) perturbations. At t = 40 (first frame), the initial vortices increase their strength and streamwise inclination. This first phase is similar to that found for the BBL, and appears to depend on the action of non linear coupling terms linked to the components of the streamwise vorticity, such as ww z , as explained in detail in Ref. [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF].

At the same time, the streamwise velocity perturbation increases its amplitude, due to a modified lift-up effect [START_REF] Cherubini | The minimal seed of turbulence transition in a boundary layer[END_REF]. In fact, since the initial vortices are inclined, the generated streaks are modulated in the streamwise direction, as shown in the second frame for t = 80. In particular, a main high-speed bent streak (yellow) is created, flanked by two weaker low-speed ones. On such streaks, localized patches of vorticity are observed (see the third frame for t = 140), which are originated from the splitting of the initial inclined vortices. The bent streaks continue to be fed by the streamwise vortices, elongating in the streamwise direction, as shown in the fourth frame for t = 200. However, in the regions of larger vorticity, stronger modulations of the streaks are induced, leading the wave packet to break-up starting from a localized region (x = 250 in the fifth frame for t = 260).

Such a scenario recalls the mechanism of secondary instability of streamwise streaks which triggers bypass transition in boundary-layer flows [START_REF] Brandt | Transition in a boundary layers subject to free-stream turbulence[END_REF][START_REF] Schoppa | Coherent structure generation in near-wall turbulence[END_REF]. In particular, since the initial disturbance is not symmetric, the streaks are characterized by sinuous oscillations, which represent the primary instabilities of streamwise streaks [START_REF] Andersson | On the breakdown of boundary layer streaks[END_REF][START_REF] Brandt | Transition in a boundary layers subject to free-stream turbulence[END_REF]. However, in the non linear optimal case, this mechanism is much more rapid than the one relying on the linear growth of streamwise-aligned streaks and successive saturation and secondary instability. In fact, the initial inclined vortices can create bent streaks in a short time, leading to break-up without experiencing secondary instability, due to their spanwise modulations [START_REF] Waleffe | On a self-sustaining process in shear flows[END_REF]. The whole transition process recalls the first phases of the disturbance evolution on the periodic orbit recently found by bisection in a small domain (see Ref. [START_REF] Kreilos | Edge states for the turbulence transition in the asymptotic suction boundary layer[END_REF]). However, since the NLOP considered here does not lay on the edge of chaos as the mentioned periodic orbit, transition to turbulence is reached after the bursting phase.

Despite the similarity of the initial optimal disturbances, the non linear route to transition described here shows important differences with respect to that found in the non-parallel case. In fact, for the BBL case, the perturbation maintains the initial symmetry of the NLOP up to large times (obviously, before turbulence is initiated). As shown in figure 13, the initial symmetric inclined vortices transport the flow momentum causing an amplification of the streamwise component of velocity along them and inducing the creation of low-and high-momentum zones showing a Λ and an X shape, respectively (see the blue and yellow surfaces in the first frame for t = 40). This Λ structure of the perturbation is maintained at larger time (see the second frame for t = 80), and the symmetric inclined vortices connect their fronts to create a Λ-vortex, which eventually turns into a hairpin vortex which leads the flow to break-up (third frame for t = 140). The formation of the hairpin for the BBL can be clearly seen in figure 14 at times t = 80 (a), t = 100, (b), and t = 160, where the green surfaces show the Q-criterion, highlighting the regions of high vorticity. At t = 80, one can observe that the initial vortices, tilted by the mean flow, are inclined at a small angle with respect to the streamwise axis (≈ 7 • ). However, at larger times, the vortices in the heart of the wave packet increase their inclination, reaching angles of about 35 • . This phase coincides with the formation of the hairpin vortex (two of them are visible in the second frame at t = 100) which grows in size and splits into smaller hairpin vortices, leading very quickly to a turbulent spot (see the third frame at t = 160). On the other hand, for the ASBL, although the initial vortices show a similar wall-normal inclination with respect to the streamwise direction, as shown in the first and second frame of figure 15, the head of the hairpin vortex cannot be created due to the lack of symmetry of the perturbation. Thus, the vorticity does not spread in space as in the BBL case, but remains localized in a narrow region in the streamwise direction (see the third frame at t = 160).

The differences between the transition paths in the ASBL and BBL case can be analyzed by extracting the rms values of the three components of the velocity perturbation, as shown in figure 16, the thick lines referring to the ASBL, the thin ones to the BBL. In the BBL flow, the three components of velocity grow more rapidly and achieve larger rms values than in the ASBL case (see figure 16 provided in figure 17 (a) and (b), respectively, one can observe that these two components of the vorticity perturbations are much more extended in space, and larger in magnitude in the non-parallel case than in the parallel one (both components are about 30%). In particular, the vorticity surfaces are localized at the head and legs of the hairpin vortices, explaining the larger growth of such terms with respect to the streamwise vorticity.

In order to better understand the differences between the two optimal transition paths, it is worth to analyze the evolution of the symmetric optimal perturbation superposed to the ASBL base flow at low Reynolds number. Thus, we inject the selected NLOP obtained for the BBL in the ASBL flow, and analyze its evolution in time. Figure 18 shows the streamwise vorticity (black and white) and velocity (yellow and blue) perturbations at t = 80 and t = 140. Comparing with figures 13 (b) and (c), one can observe that the vortices as well as the low-and high-momentum regions are straighten up and lose part of their inclination.

As a result, at t = 140 one can observe three alternated streaks with weak vortices on their flanks. In particular, even if the initial perturbation is symmetric with respect to a z = const axis, the hairpin vortex is not formed at t = 100. This is clearly shown in figure 19, providing the Q-criterion isosurfaces. One can see that at t = 80 (a) and at t = 100 (b) the inclination of the vortices is weak and begins to grow only at t = 160, turning eventually in a hairpin structure at t ≈ 200. This can be also inferred by analyzing in figure 20 the time evolution of the rms value of the three velocity and vorticity components, and comparing them with the BBL case. As provided by the thick lines in figure 20 (a), the initial growth of the velocity components of the NLOP injected into the ASBL is initially similar to the one characterizing the BBL. However, at t ≈ 40, the spanwise velocity component begins to decrease, and the wall-normal and the streamwise ones strongly decrease their growth rate. Concerning the vorticity components, at approximately the same time they all begin to decrease, the ω ′ z and the ω ′ y with a larger rate than the ω ′ x . This decrease is due to the absence of the hairpin vortex at t ≈ 100; in fact, it begins to grow only at t ≈ 200, when the growth of all of the vorticity components is observed. Thus, we can say that the mechanism of formation of the hairpin, which is responsible for the strong growth of all of the vorticity components in the BBL case, is delayed for the ASBL, making a "sinuous" instability mechanism more effective in inducing rapidly transition, like in the case of the Couette flow [START_REF] Cherubini | Nonlinear optimal perturbations in a Couette flow: bursting and transition[END_REF]. Thus, in the case of the ASBL a symmetrical optimal perturbation is less effective in inducing transition than a non-symmetrical one. As observed in figure 15, such a delay appears to be due to the lower wall-normal inclination of the vortices with respect to the streamwise axis. This can be linked to a simple mechanism of transport of the perturbation by the mean flow. In fact, injecting the same initial vortical structure in the ASBL and the BBL flows, in the latter case the highest part of the vortex will experience a lower streamwise base velocity as it is advected downstream, so it will decelerate with respect to its lowest part. This will induce an increase of the inclination of the initial vortex in the wall-normal direction, leading to a connection of the fronts of the two vortices, finally creating an hairpin structure. This is clearly shown in figure 21, providing the base flow vectors at Re = 610 for the ASBL (left frame) and the BBL (right frame) and the effect they have on the same vortical structure evolving in time up to t = 100. For the ASBL flow, the profile is fuller than the BBL one, so the mechanism of inclination by the mean flow is weaker, delaying the formation of the hairpin vortex from an initial symmetric perturbation. Therefore, the zones of strong velocity deficit (in blue in the figure) remain closer to the wall, delaying the formation of inflection points in the mean flow profile, and the consequent transition to turbulence. Moreover, we have verified that this effect is not due to the parallelism of the ASBL flow. In fact, a non-linear optimization performed for a parallel flow having the same Blasius profile with Re = 610 at all abscissae, gives a symmetric NLOP. This conjecture can be verified by visualizing the activation of the transport terms of the spanwise vorticity (characterizing the head of the hairpin) by the mean flow. Figure 22 shows one of these two transport terms, Uv ′ x , for the BBL (top frames) and the ASBL (bottom frames) at t = 100 (left frames) and t = 150 (right frames). It appears that, for the BBL, the terms is activated at the head of the hairpin vortices, spreading in space at larger time. Whereas, for the ASBL, they are characterized by a lower amplitude (lower than half the one for the BBL), and they are rapidly damped in time. This confirms that the transport of the spanwise vorticity by the mean flow is indeed the mechanism which can explain the large differences in the symmetry of the non linear optimal perturbations. Such a result shows that it is crucial to take into account the non linearity as well as the non-parallelism of a flow at the same time, for determining with accuracy the most effective route to transition.

For larger Reynolds numbers, since the suction velocity is much weaker, the velocity profile tends to be closer to that of the BBL flow. Thus, the transition scenario tends to become similar to the BBL one, generating an hairpin vortex at small times, for symmetrical initial perturbations. Figure 23 shows three snapshots of the evolution in time of the NLOP obtained for Re = 5000 and E 0 = 4.25 × 10 -9 , showing a behaviour very similar to the one recovered for the BBL in figure 13. In particular, one can observe the formation of Λ structures for the negative streamwise velocity component and the streamwise vorticity.

Figure 24 shows the generation of a train of hairpin vortices at t = 75 and = 100, as well as the base flow profiles, which appear much less full than in the previous case. This confirms that wall suction is a crucial parameter for the stabilisation of a boundary layer flow, since, when it is strong enough, it can modify the base flow profile and delay the formation of strongly growing vortical structures such as the hairpin vortices.

IV. SUMMARY

A variational procedure has been employed to find non linear optimal disturbances in the asymptotic suction boundary layer (ASBL) flow. These perturbations are defined as the ones yielding the largest energy growth at a given target time T , for a given Reynolds number Re.

The results have been compared with those obtained using the same approach in the case of the Blasius boundary layer (BBL) flow [START_REF] Cherubini | Rapid path to transition via nonlinear localized optimal perturbations[END_REF]. The influence of the different structure of the ASBL velocity profile with respect to the BBL on the non-linear optimal growth mechanism has been studied. It has been found that suction remarkably reduces the optimal energy gain in the non linear case. Moreover, the optimal perturbation obtained in the present case shares the same basic structure found for different shear flows such as the BBL and the Couette flows. However, unlike the BBL case, the optimal perturbation for the ASBL flow for low to moderate Reynolds number is not spanwise-symmetric. In particular, it has been found that a value of the Reynolds number exists between 5000 and 10000 for which the non linear optimal perturbation changes from a non symmetric shape to a symmetric one. Therefore, for sufficiently high Reynolds numbers (low suction velocity), the structure of the non linear optimal disturbance becomes similar to the one of the BBL. By bisecting the initial energy of the non linear optimal perturbations, minimal energy thresholds for subcritical transition to turbulence have been obtained. These energy thresholds are found to be 1 to 4 order of magnitude lower than the ones found in other transition scenarios such as secondary instability of elongated streamwise vortices, random noise, oblique waves and localized streamwise-aligned disturbances [START_REF] Levin | Transition thresholds in the asymptotic suction boundary layer[END_REF]. For 610 < Re < 5000, these thresholds are found to scale as Re -2 , suggesting a new scaling law for transition in the ASBL.

Finally, direct numerical simulations show that the different structure of the base flow with respect to the BBL leads to a different evolution of the initial perturbation. In fact, unlike the case of the BBL flow, for low to moderate values of the Reynolds number, the formation of hairpin vortices is not observed in the transition process before break-up to turbulence, and a sinuous transition scenario is observed. This appears to be due to the lower tilting of the vortices induced by the fuller velocity profile in the ASBL case, which delay the formation of hairpin vortices. However, when wall suction is not strong enough (i.e., at large Re, for a given freestream velocity) the vorticity transport and tilting mechanism giving birth to the hairpin vortex is observed again. Future work will aim at investigate the existence of a similar change of symmetry in the non-linear optimal perturbations for different shear flows such as the plane Poiseuille flow, as well as the influence of the independent parameters of the optimisation, such as the initial energy and the target time, on the optima perturbation structure.

Figure 1 :

 1 Figure 1: Sketch of the ASBL flow with a superposed localized disturbance.

Figure 2 :

 2 Figure 2: (Color online) Optimal energy gain versus target time T for Re = 610, E 0 = 3.0 × 10 -7 . The dashed line with triangles indicates the results of the linear optimization; the solid line with squares (red online) indicates the results of the non linear optimization.

Figure 3 :

 3 Figure 3: (Color online) Optimal energy for Re = 610 at target time T = 50 (black), T = 100

Figure 4 :

 4 Figure 4: (Color online) Initial perturbations obtained by the non linear optimization for the asymptotic suction boundary-layer at Re = 610 and target time T = 75: iso-surfaces of the optimal perturbations (grey, green online, for the negative streamwise component; dark and light gray for negative and positive streamwise vorticity, respectively) with initial energy E 0 = 1.2 × 10 -7 (top frame, surfaces for u ′ = -0.00017, ω ′ x = ±0.01) and E 0 = 3.0 × 10 -7 (middle frame, u ′ = -0.015, ω ′ x = ±0.1). Initial perturbations obtained by the non linear optimization for the Blasius boundary layer flow at Re = 610, target time T = 75, with initial energy E 0 = 1.2 × 10 -7 (bottom frame, u ′ = -0.01, ω ′ x = ±0.06). Axes are not in the same scale.

Figure 5 :

 5 Figure 5: (Color online) Contours and vectors of the velocity components of the NLOP at initial time obtained with Re = 610, T = 75 for the asymptotic suction boundary layer with E 0 = 3.0×10 -7 , on the planes x = 211 (a), and x = 215 (b); for the Blasius boundary layer with E 0 = 1.2×10 -7 , on the planes x = 228 (c), and x = 232 (d). Shaded contours indicate the streamwise disturbance velocity (dark, red online, for positive values; light, green online, for negative ones); vectors represent the wall-normal and the spanwise disturbance velocity components.

Figure 6 :

 6 Figure 6: (Color online) Contours and vectors of the velocity components of the NLOP at target time obtained with Re = 610, T = 75 for the asymptotic suction boundary layer with E 0 = 3.0×10 -7 on the planes x = 249 (a), and x = 253 (b); for the Blasius boundary layer with E 0 = 1.2× 10 -7 on the planes x = 267 (c), and x = 273 (d). Shaded contours indicate the streamwise disturbance velocity (dark, red online, for positive values; light, green online, for negative ones); vectors represent the wall-normal and the spanwise disturbance velocity components.

Figure 7 :

 7 Figure 7: (Color online) Isosurfaces of the initial perturbations obtained by the non linear optimization for the ASBL at Re = 610 and target time T = 200, with initial energies E 0 = 1.2 × 10 -7 (a), E 0 = 1.35 × 10 -7 (b), E 0 = 2.7 × 10 -7 (c) and a suboptimal for E 0 = 2.7 × 10 -7 initialized with a symmetric initial perturbation (d). Green and red, for the negative and positive streamwise velocity component; dark and light, for negative and positive streamwise vorticity, respectively, with values u ′ = -0.005, ω ′ x = ±0.05 (a-b), u ′ = -0.01, ω ′ x = ±0.08 (c), u ′ = -0.005, ω ′ x = ±0.05 (d).

Figure 8 :

 8 Figure 8: Minimal energy for turbulent transition for the asymptotic suction boundary layer at different Reynolds numbers (solid line). The dashed lines show the minimal energy for different transition scenarios, namely noise (NOISE), streamwise vortices (SV), oblique waves (OW), and localized disturbances (LD), extrapolated from data in [34].

Figure 9 :

 9 Figure 9: (Color online) Minimal seeds for the asymptotic suction boundary layer for Re = 1200 (left, with E 0 = 2.08 × 10 -8 ) and Re = 2500 (right, with E 0 = 4.16 × 10 -9 ) with target time T = 150: iso-surfaces of the negative streamwise component, u ′ = -0.0025 (green) and of the negative and positive streamwise vorticity, ω ′ x = -0.045 (black and white, respectively)

Figure 10 :

 10 Figure 10: (Color online) Non linear optimal perturbations for the asymptotic suction boundary layer at Re = 5000 for E 0 = 2.12 × 10 -9 (left) and E 0 = 4.24 × 10 -9 (right) with target time T = 100: iso-surfaces of the negative streamwise component, u ′ = -0.001 (green) and of the negative and positive streamwise vorticity, ω ′ x = ±0.045 (black and white, respectively)

Figure 11 :

 11 Figure 11: (Color online) Non linear optimal perturbations for the asymptotic suction boundary layer at Re = 10000 with E 0 = 2.5 × 10 -9 (top), and with E 0 = 1.25 × 10 -8 (bottom) with target time T = 50: iso-surfaces of the negative streamwise component, u ′ = -0.002 (green) and of the negative and positive streamwise vorticity, ω ′ x = ±0.045 (black and white, respectively)

Figure 12 :

 12 Figure 12: (Color online) Snapshots of the evolution in time of the selected NLOP for the ASBL: iso-surfaces of the streamwise velocity and vorticity perturbations (yellow and blue, for u ′ = ±0.15, respectively; black and white, ω ′ x = ±0.2, respectively) at t = 40, 80, 140, 200, 260 (from top to bottom).

Figure 13 :

 13 Figure 13: (Color online) Snapshots of the evolution in time of the selected NLOP for the BBL: iso-surfaces of the streamwise velocity and vorticity perturbations (yellow and blue, for u ′ = ±0.1, respectively; black and white, ω ′ x = ±0.2, respectively) at t = 40, 80, 140 (from top to bottom).

Figure 14 :

 14 Figure 14: (Color online) Snapshots of the evolution in time of the selected NLOP for the BBL: isosurfaces of the Q-criterion for t = 80, 100, 160 (from left to right) .

Figure 15 :

 15 Figure 15: (Color online) Snapshots of the evolution in time of the selected NLOP for the ASBL: isosurfaces of the Q-criterion for t = 80, 100, 160 (from left to right) .

  figure16 (b), in the BBL case all of the three components grow more rapidly; the largest differences between the two flows are recovered for the wall-normal and spanwise vorticity, which attains values almost one order of magnitude larger than in the ASBL case. This can be explained by observing that the vorticity components ω ′ z and ω ′ y have large values at the head and legs of the hairpin which characterize the BBL route to transition. In fact, plotting the ω ′ z and ω ′ y surfaces for the parallel and non-parallel flow cases at t = 100, as

Figure 16 :

 16 Figure 16: (Color online) Evolution in time of the rms values of the three components of velocity (solid lines for u ′ , dashed for v ′ , dashed-dotted for w ′ ) (a) and vorticity (solid lines for ω ′ z , dashed for ω ′ y , dashed-dotted for ω ′ x ) (b) for a DNS initialized by the selected NLOP for the ASBL (thick lines) and the BBL (thin lines).

Figure 17 :

 17 Figure 17: (Color online) Snapshots of the evolution in time of the selected NLOP for the ASBL (left frame) and the BBL (right frame): spanwise and wall-normal vorticity (blue surfaces for ω ′ z = 0.8, red ones for ω ′ y = 0.65.

Figure 18 :

 18 Figure 18: (Color online) Snapshots of the evolution in time of the selected NLOP obtained for the BBL, injected in the ASBL flow: iso-surfaces of the streamwise velocity and vorticity perturbations (yellow and blue, for u ′ = ±0.1, respectively; black and white, ω ′ x = ±0.2, respectively) at t = 80, 140 (from top to bottom).

Figure 19 :

 19 Figure 19: (Color online) Snapshots of the evolution in time of the selected NLOP obtained for the BBL, injected in the ASBL flow: isosurfaces of the Q-criterion for t = 80, 100, 160 (from left to right).

Figure 20 :

 20 Figure 20: (Color online) Evolution in time of the rms values of the three components of velocity (solid lines for u ′ , dashed for v ′ , dashed-dotted for w ′ ) (a) and vorticity (solid lines for ω ′ z , dashed for ω ′ y , dashed-dotted for ω ′ x ) (b) for a DNS initialized by the selected NLOP obtained for the BBL, injected into the ASBL (thick lines) and the BBL (thin lines) flows. .

Figure 21 :

 21 Figure 21: (Color online) Snapshots of the evolution in time of the selected NLOP obtained for the BBL at Re = 610, injected in the ASBL (left frame) and the BBL flow (right frame): isosurfaces of the Q-criterion for t = 100 and vectors of the base flow.

Figure 22 :

 22 Figure 22: (Color online) Snapshots of the evolution in time of the advection term U v ′ x obtained for the BBL (top frames) and the ASBL (bottom frames) at t = 100 (left frames) and t = 150 (right frames): isosurfaces U v ′ x = ±0.13 for the BBL, U v ′ x = ±0.06 for the ASBL.

Figure 23 :

 23 Figure 23: (Color online) Snapshots of the evolution in time of the symmetrical NLOP at Re = 5000, with initial energy E 0 = 4.25 × 10 -9 , extracted at t = 50, 100, 150, from top to bottom. The isosurfaces represent the streamwise velocity and vorticity perturbations (yellow and blue, for u ′ = ±0.015, 0.03, 0.1, from top to bottom; black and white, ω ′ x = ±0.15, 0.1, 0.25, from top to bottom).

Figure 24 :

 24 Figure 24: (Color online) Snapshots of the evolution in time of the selected NLOP obtained for the ASBL at Re = 5000, extracted at t = 75 (left frame) and t = 100 (right frame): isosurfaces of the Q-criterion (Q = 20) and vectors of the base flow.

Table I :

 I Domain lengths, grid points and target times chosen for the optimizations at different

	Reynolds numbers

Table II

 II

				7
	BBL	275.10	801.98	1104.1
	ASBL	125.16	125.15	158.37

: Comparison between energy gains at T = 75 for the BBL and the ASBL.

Table III :

 III Streamwise and spanwise dimensions, l x , l z of the minimal seed for the considered Reynolds numbers, measured as the largest spanwise and streamwise lengths of the flow regions where |u ′ | > 0.001 by streamwise alternated positive and negative vortices with a finite inclination with respect to the streamwise and wall-normal axis. It is worth to notice that, even if the basic structure is the same, the minimal seed is much more localized for higher values of the Reynolds number, as shown in table

	Re = 610 1200 2500 5000 10000
	l x = 18.2 8.6 4.1 2.9 1.7
	l z = 8.6 4.4 2.7 1.94 1.3

Acknowledgments

Some computations have been performed on the Power 6 of the IDRIS, France.