Håkan Wedin

Stefania Cherubini

Alessandro Bottaro

Ph.D H M Fransson

Thesis

Effect of plate permeability on nonlinear stability of the asymptotic suction boundary layer

Keywords: number(s): 47.27.Cn, 05.45.-a, 47.85.Dh

The nonlinear stability of the asymptotic suction boundary layer is studied numerically, searching for finite-amplitude solutions that bifurcate from the laminar flow state. By changing the boundary conditions for disturbances at the plate from the classical no-slip condition to more physically sound ones, the stability characteristics of the flow may change radically, both for the linearized as well as the nonlinear problem. The wall boundary condition takes into account the permeability K of the plate; for very low permeability, it is acceptable to impose the classical boundary condition ( K = 0). This leads to a Reynolds number of approximately Re c = 54 400 for the onset of linearly unstable waves, and close to Re g = 3200 for the emergence of nonlinear solutions [F.

I. INTRODUCTION

The motion of a fluid is generally characterized as being in a laminar or turbulent condition. The state in between those two, the transitional state, usually has large fluctuations in pressure and shear stress due to an alternation between laminar and turbulent flow. To achieve a dynamical system description of the turbulent transition, one needs insight on the behavior of exact coherent solutions to the Navier-Stokes equations, such as simple traveling waves (TWs) and complex periodic orbits. The study of finite-amplitude TWs provides a good illustration of how a flow may bifurcate from a well-ordered laminar flow to a nonlinear state. These waves have been shown to have some relevance to the actual transition process; for instance, simple two-dimensional Tollmien-Schlichting (TS) waves have proved to be a part of the early stage of the transition process in the Blasius boundary layer flow in a low-disturbance environment [START_REF] Matsubara | Disturbance growth in boundary layers subjected to free-stream turbulence[END_REF]. When they are of large enough amplitude, they go through a secondary instability that brings the flow to turbulence [START_REF] Herbert | Secondary instability of boundary layers[END_REF]. Secondly, the signature of three-dimensional (3D) traveling waves in Ref. [START_REF] Hof | Experimental observation of nonlinear traveling waves in turbulent pipe flow[END_REF] and the observation of a mechanism that sustains them in Ref. [START_REF] Hof | Turbulence regeneration in pipe flow at moderate Reynolds numbers[END_REF] has provided convincing evidence that numerically obtained exact coherent structures (ECSs) can be good candidates for obtaining a theoretically grounded view of turbulence. The regeneration cycle presented in Ref. [START_REF] Waleffe | On a self-sustaining process in shear flows[END_REF] can be used to discover self-sustained three-dimensional traveling waves to the incompressible Navier-Stokes equations [START_REF] Waleffe | Three-dimensional coherent states in plane shear flows[END_REF][START_REF] Waleffe | Exact coherent structures in channel flow[END_REF][START_REF] Waleffe | Homotopy of exact coherent structures in plane shear flows[END_REF]. TWs * hakanwedin@hotmail.com satisfying u t = -cu x , propagating at constant wave speed c without altering their shape, have now been discovered for many canonical shear flows. Examples of previous work include plane Poiseuille flow [START_REF] Ehrenstein | Three-dimensional wavelike equilibrium states in plane Poiseuille flow[END_REF], plane Couette flow [START_REF] Nagata | Three-dimensional finite-amplitude solutions in plane Couette flow: Bifurcation from infinity[END_REF][START_REF] Gibson | Visualizing the geometry of state space in plane Couette flow[END_REF], the Blasius boundary layer flow [START_REF] Rotenberry | Finite amplitude steady waves in the Blasius boundary layer[END_REF][START_REF] Wedin | Unstable flow structures in the Blasius boundary layer[END_REF], pipe flow [START_REF] Wedin | Exact coherent structures in pipe flow: travelling wave solutions[END_REF][START_REF] Faisst | Travelling waves in pipe flow[END_REF], and the asymptotic suction boundary layer flow [START_REF] Milinazzo | Finite-amplitude steady waves in plane viscous shear flows[END_REF]. The square duct configuration has been studied less, but it has been shown to possess traveling waves of various shapes [START_REF] Wedin | Three-dimensional traveling waves in a square duct[END_REF][START_REF] Wedin | Advances in Turbulence XII, Proceedings of the 12th EUROMECH European Turbulence Conference[END_REF][START_REF] Uhlmann | Traveling-waves consistent with turbulence-driven secondary flow in a square duct[END_REF][START_REF] Okino | A new nonlinear vortex state in square duct flow[END_REF]. The intriguing aspect of the TWs is their innate unstable nature (see, e.g., [START_REF] Okino | A new nonlinear vortex state in square duct flow[END_REF]), which renders them interesting flow objects to account for in a dynamical system description of turbulence. Although the global traveling waves appear to resemble turbulent flow structures, they do not capture the bursting flow phenomenon, i.e., the breakup and recreation of exact coherent structures, and they are not localized in the streamwise direction, as observed particulary in pipe flow. A more recent approach and the next step in the hierarchy of solutions, capable of revealing turbulent dynamics, are recurrent time-dependent periodic orbits that may be extracted from numerical simulations [START_REF] Kawahara | Periodic motion embedded in plane Couette turbulence: Regeneration cycle and burst[END_REF][START_REF] Kawahara | Laminarization of minimal plane Couette flow: Going beyond the basin of attraction of turbulence[END_REF][START_REF] Kawahara | Proceedings of the IUTAM Symposium on One Hundred Years of Boundary Layer Research[END_REF][START_REF] Cherubini | Edge states in a boundary layer[END_REF][START_REF] Duguet | Selfsustained localized structures in a boundary-layer flow[END_REF][START_REF] Kreilos | Edge states for the turbulence transition in the asymptotic suction boundary layer[END_REF][START_REF] Khapko | Localized edge states in the asymptotic suction boundary layer[END_REF][START_REF] Chandler | Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow[END_REF][START_REF] Avila | Streamwiselocalized solutions at the onset of turbulence in pipe flow[END_REF] or through a bifurcation from a global traveling wave [START_REF] Chantry | Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow[END_REF]. However, the entanglement of the stable and unstable manifold of the TWs, which can be viewed as saddle points in a reference moving with the base flow, creates the potential for subcritical transition and consequent turbulence sustainment. Thus, investigating the TWs dynamics might be crucial to unravel the nonlinear dynamics of shear flows that are prone to subcritical transition, such as the asymptotic suction boundary layer. In fact, although for this flow the critical Reynolds number appears to be very high, namely Re c ≈ 54 382 [START_REF] Hocking | Non-linear instability of the asymptotic suction velocity profile[END_REF], the amplitude thresholds for the subcritical transition are very low, and they scale as Re -2 , as recently found in Ref. [START_REF] Cherubini | Non linear optimals in the asymptotic suction boundary layer: Transition thresholds and symmetry breaking[END_REF] using a nonlinear optimization coupled with a bisection procedure in order to find the minimal-energy perturbation able to induce subcritical transition in such a flow. Such a subcritical transition scenario is characterized by inclined streamwise vortices and strongly bent streaks, able to rapidly bypass the asymptotic growth of TS waves. Previous investigations of TWs in the asymptotic suction boundary-layer flow have been performed in Ref. [START_REF] Milinazzo | Finite-amplitude steady waves in plane viscous shear flows[END_REF], neglecting the influence of the permeability of the flat plate on the development of the perturbations. This hypothesis, which has been used in most of the linear and nonlinear stability analyses concerning this flow, might have an effect on the determination of its stability thresholds, which appears to be overpredicted with respect to experimental observations [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layer[END_REF].

This paper presents two-dimensional traveling-wave solutions of the Navier-Stokes equations for the asymptotic suction boundary layer configuration. The original side of the results presented herein is the physically sound condition imposed for the disturbance velocity at the plate, which differs from the classical no-slip boundary condition used elsewhere. Linear stability results and nonlinear finite-amplitude solutions will be provided, focusing on the two-dimensional case, in order to provide a thorough comparison with the linear and nonlinear critical Reynolds numbers available in the literature. The paper is organized as follows: In Sec. II, we present the flow configuration and the equations. In Sec. III, the finite-amplitude solutions are presented together with a three-dimensional stability analysis and their role in phase space. Finally, conclusions can be found in Sec. IV.

II. DEFINITIONS

Two-dimensional nonlinear solutions in the asymptotic suction boundary layer have already been presented in [START_REF] Milinazzo | Finite-amplitude steady waves in plane viscous shear flows[END_REF], with vanishing disturbance velocity at the porous plate. The original aspect in the present work is the imposition of a condition for the velocity through the plate derived using Darcy's law; this leads to a drastic change in the stability characteristics of the laminar flow. The streamwise, vertical, and spanwise coordinates are represented by x, ŷ, and ẑ; the unit vectors are i, j, and k used for defining the velocity vector û = ûi + vj + ŵk referring to the streamwise, wall-normal, and spanwise component, respectively; p stands for pressure, ν and μ are the kinematic and dynamic viscosity, respectively, and time is t. The dimensional laminar base state with no pressure gradient imposed is Û( ŷ) = Û ( ŷ)i + V j. The solution of Û can be found in [START_REF] Schlichting | Boundary-layer Theory[END_REF] and is

Û ( ŷ) = Û∞ [1 -exp(-V0 ŷ/ν)], (1)
V = -V0 = const, (2)
where the wall-normal component V0 > 0 is constant over a large area as opposed to a mere local action of the suction. As in earlier studies, a uniform suction V0 is assumed to be maintained through the plate; in an experimental setup, the porous medium is composed of voids of finite size, hence the suction is not exactly uniform (see the sketch in Fig. 1). However, if the pore size and spacing between pores are sufficiently small, the wall-normal suction can be considered uniform over a representative elementary volume that is large on the pore scale. Thus, the flow components in Eqs. ( 1) and ( 2) characterize a type of boundary-layer flow that does not vary in the streamwise direction; this solution is approached when uniform suction is applied for a sufficiently long distance from the beginning of the suction zone, resulting in the analytical solution given in Eqs. ( 1) and ( 2).

The Navier-Stokes equations are nondimensionalized (thus the caret symbols are removed) using the streamwise freestream speed Û∞ at the outer edge of the computational domain ŷ = ŷ∞ , density ρ, and the displacement thickness δ * , so that the Reynolds number is defined as Re = Û∞ δ * /ν. The governing equations are the Navier-Stokes equations and the continuity equation:

u t + u • ∇u + ∇p - 1 Re ∇ 2 u = 0, ∇ • u = 0. (3)
To search for two-dimensional nonlinear solutions, we perform a bifurcation study in which we add a perturbation to the base state U(y), yielding the total flow u = U(y)i + V j + u , where is an amplitude to be determined. The nondimensional laminar flow is U (y) = 1e -y and V = -1/Re. Substituting for u in the Navier-Stokes and continuity equations, and subtracting the base flow equations, leads to the disturbance equations. Since the base flow depends only on y, we impose periodicity in x and t for u representing the solution as a traveling wave,

u = b u (b) (y)exp[ibα(x -ct)], (4)
where b = -NX, . . . , + NX, α is a real streamwise wave number, and c = c r + ic i is in general a complex eigenvalue to be determined (i = √ -1). The Fourier amplitude u (b) (y) is expressed in a Chebyshev series with NY + 1 polynomials T i , where u (-b) = u * (b) (y) ( * denotes complex conjugate). The above solution of u , for a monochromatic wave, grows/decays and oscillates in time with growth rate αc i and frequency αc r . The wall-normal flow domain is truncated from [0,∞) to the computational domain [0,y ∞ ] and then transformed to the interval [-1, + 1] in γ using the formula

y = a(1 + γ )/(b - γ ), where a = y i y ∞ /(y ∞ -2y i ), b = 1 + 2a/y ∞ ,
where y i is the distance from the plate where half of the collocation points sit. The equations are imposed on the Gauss-Lobatto grid distribution, γ k = cos (kπ/NY ), for k = 0,1,2, . . . ,NY . To ensure a uniform free-stream flow u(y = y ∞ ) = (1, -1/Re), also in the presence of finite-amplitude perturbations, we impose Fourier modes decaying exponentially as y → ∞. To comply with the uniform flow condition, we set v (b=0) = 0. At the plate, the streamwise disturbance velocity is assumed to vanish, i.e., u (t,x,y = 0) = 0; however, due to the fact that the wall is porous, it is not obvious that the v component of the perturbation should be zero, as customarily assumed in most linear and nonlinear studies [START_REF] Milinazzo | Finite-amplitude steady waves in plane viscous shear flows[END_REF][START_REF] Kreilos | Edge states for the turbulence transition in the asymptotic suction boundary layer[END_REF][START_REF] Khapko | Localized edge states in the asymptotic suction boundary layer[END_REF][START_REF] Gustavsson | Development of three-dimensional disturbances in boundary layers with suction[END_REF][START_REF] Fransson | Flow control of boundary layers and wakes[END_REF][START_REF] Fransson | Optimal linear growth in the asymptotic suction boundary layer[END_REF][START_REF] Levin | Transition thresholds in the asymptotic suction boundary layer[END_REF][START_REF] Davidsson | Stability and transition in the suction boundary layer and other shear flows[END_REF]. For low permeabilities, it is acceptable to represent the flow through the porous plate via Darcy's law, neglecting inertia, i.e., û = -( K/ μ) ∇ p, [START_REF] Waleffe | On a self-sustaining process in shear flows[END_REF] under the assumption that the porous medium is isotropic so that K is a scalar parameter. Typically, the permeability K ranges from 2 × 10 -17 m 2 for concrete to 10 -6 m 2 for some metal foams [START_REF] Innocentini | Permeability[END_REF], a function of the porosity and the structure of the medium. Following Gustavsson [START_REF] Gustavsson | Development of three-dimensional disturbances in boundary layers with suction[END_REF], we consider a dimensional flow given by (-V0 + v , P1 + p ) at ŷ = 0 and the unperturbed state (-V0 , P0 ) at ŷ = -ĥ, with ĥ the plate's thickness (see Fig. 1). In dimensionless terms, Darcy's law for the disturbance at y = 0 becomes

v = -ap , ( 6
)
with a = Û∞ K/(ν ĥ) the dimensionless permeability parameter. We limit the range of K to an upper value of 2 × 10 -9 m 2 for Darcy's law to be tenable (for larger permeabilities, the Forchheimer model is a better representation of the relation between the pressure gradient and the average velocity through the porous medium). Using the data of Fransson and Alfredsson [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layer[END_REF] ( Û∞ = 5 ms -1 , ν = 1.5 × 10 -5 m 2 s -1 , ĥ = 3.2 mm), we thus have 0 a 0.2; the gas permeability of the porous plastic plate used in [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layer[END_REF] has been measured to be 3.7 × 10 -12 m 2 , which translates to a dimensionless value a = 3.85 × 10 -4 , to be later used as a reference.

Subtracting the x derivative of the streamwise momentum equation from the y derivative of the normal momentum equation, both taken at the wall, using continuity and the relation v = -ap , we finally obtain the wall condition for v :

v xx -v yy 2a + v v yy - v yy + v yyy Re = 0, (7)
which, in a linearized setting, using the Fourier series in (4), translates to (1) (y) the first mode of the vertical velocity component in Eq. ( 4) and D = d/dy. The recent linear stability study by Pluvinage et al. [START_REF] Pluvinage | Instabilities in the asymptotic suction boundary layer over a permeable, compliant wall[END_REF] adopts a condition equivalent to the one given above, whereas the present nonlinear contribution employs the full equation [START_REF] Waleffe | Exact coherent structures in channel flow[END_REF].

a Re (D 3 + D 2 ) + 1 2 (D 2 + α 2 ) v = 0, with v = v
To give a measure of the amplitude of the nonlinear wave, we compute the energy E of the x-dependent part:

E = 1 2 2 2π/α 0 2π/β 0 y ∞ 0 [u 2 + v 2 + w 2 ]dX dy dz, ( 8
)
where X corresponds to the time-independent Galilean frame X = xc r t. For the two-dimensional TS waves, we have w = 0. Finally, the linear and nonlinear dynamics of the computed TWs has been analyzed by using direct numerical simulation (DNS) in a three-dimensional framework. A computational box with dimensions L x × L y × L z = 4π/α × y ∞ × 2π is considered, and y ∞ is the same as the one chosen for computing the TWs. In this computational domain, the Navier-Stokes equations, written in a perturbative formulation, are discretized and solved using a finite-difference fractional-step method with second-order accuracy in space and time [START_REF] Verzicco | A finite-difference scheme for the three-dimensional incompressible flows in cylindrical coordinates[END_REF]. In the wall-normal direction, the grid distribution given by the mapping of the Gauss-Lobatto points in the [0,y ∞ ] domain has been kept. Concerning the streamwise and spanwise directions, equally spaced grid points have been used. The number of grid points in each direction has been chosen after careful validation of the reference domain 20π × 40 × 2π . For instance, this domain has been discretized by a grid made by 201 × 121 × 61 grid points. Concerning the boundary conditions, at the wall the nondimensional Darcy law for the wall-normal perturbations given in Eq. ( 6) is imposed, whereas a zero perturbation condition is used for the streamwise and spanwise velocities as well as for all of the perturbation components at the upper boundary. In the spanwise and streamwise direction, periodicity is imposed for the three velocity components. For each value of Re, α, y ∞ , and a, the simulations have been initialized by superposing to the base flow the TW solution for the streamwise and wall-normal velocity components; the perturbation of this relative fixed point has been initialized by a random zeromean noise of amplitude 10 -8 for the spanwise velocity component, multiplied by an envelope function that smoothly brings it to zero toward the wall. This three-dimensional velocity component allows the most unstable mode of the TW to develop more quickly. Both the linear and nonlinear time evolution of the perturbations have been studied, the former by neglecting the perturbative nonlinear terms in the Navier-Stokes equations, the latter by considering the whole equations.

III. RESULTS

The core of the results presented herein revolves around the fact that modifying the wall conditions of v to the physically based slip condition [START_REF] Waleffe | Exact coherent structures in channel flow[END_REF] leads to a dramatic change of the stability of the flow. To bring the solutions into the nonlinear regime, we locate a vanishing imaginary part of the eigenvalue c (c i = 0) in parameter space resolving the linearized governing equations. We then search for a bifurcation to obtain a nonlinear wave with wave speed c r . For this, a FORTRAN subroutine utilizing a Newton-Raphson technique is used, developed in [START_REF] Rheinboldt | Algorithm 596: A program for a locally parameterized continuation process[END_REF][START_REF] Rheinboldt | A locally parameterized continuation process[END_REF]. For c = c r , a timeindependent problem can be considered in a Galilean frame of reference (or xc r t → X) moving with the wave speed c r . The solutions presented in Fig. 2 show the envelope of the perturbation energy E over Re for α = 0.154 and various values of the nondimensional permeability a. The solution with the classical vanishing disturbance velocities at the wall (i.e., a = 0) corresponds to the study presented in Ref. [START_REF] Milinazzo | Finite-amplitude steady waves in plane viscous shear flows[END_REF] and is close to the case a = 3.85 × 10 -4 of the wind tunnel experiments by Fransson and Alfredsson [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layer[END_REF]. For a = 0 it is remarkable that when reducing the Re from more than 50 000 to around 5000-6000, a very low perturbation energy FIG. 2. The energy E of the wavy part of the nonlinear solutions, see Eq. ( 8), mapped out in Re for a streamwise wave number of α = 0.154 and eight values of a. The classical wall condition a = 0 corresponds to a bifurcation at Re c ≈ 54 000 for α c = 0.1555 [START_REF] Milinazzo | Finite-amplitude steady waves in plane viscous shear flows[END_REF][START_REF] Fransson | Flow control of boundary layers and wakes[END_REF]. With increasing a, Re c is reduced. This is also the case for the nonlinear lower limit, cf. Table I. Proof of convergence is shown by the individual symbols; for a < 0.02 we use as a base resolution (NX,NY ) = (14,140) and confirmed with [START_REF] Wedin | Advances in Turbulence XII, Proceedings of the 12th EUROMECH European Turbulence Conference[END_REF]160). The a = 0.02 solution is mapped out using a truncation of (10,160) and confirmed with [START_REF] Wedin | Exact coherent structures in pipe flow: travelling wave solutions[END_REF]200), while the a = 0.08, 0.12 and 0.20 solutions are represented by a resolution of [START_REF] Wedin | Exact coherent structures in pipe flow: travelling wave solutions[END_REF]200) and confirmed with [START_REF] Wedin | Advances in Turbulence XII, Proceedings of the 12th EUROMECH European Turbulence Conference[END_REF]220). is sufficient to bring the flow to a new flow state, as shown in Fig. 2, in agreement with the results in Ref. [START_REF] Cherubini | Non linear optimals in the asymptotic suction boundary layer: Transition thresholds and symmetry breaking[END_REF]. When increasing the value of a, a striking change is observed in the bifurcation diagram; progressively lower values of the I shows the values of Re g for all a's presented in the plot. The inset shows the energy for a = 0.20 at Re = 305 and 320 using a truncation of (14,180) and 3178 using [START_REF] Rotenberry | Finite amplitude steady waves in the Blasius boundary layer[END_REF]240). The lower values of Re shown in the smaller plot are close to the limit Re g for a = 0.20 given in Table I. bifurcation point Re c for linear stability are found, and the same goes for the lower limit where alternative nonlinear states to the Navier-Stokes equations emerge, Re = Re g . Figure 2 shows that for a low nondimensional permeability a = 3.85 × 10 -4 , the use of the classical condition (a = 0) is reasonable for approximating the real flow. This is better illustrated by the comparable envelope in the perturbation energy over α in Fig. 3 for a = 3.85 × 10 -4 and a = 0. However, increasing a decreases the value of both Re c and Re g further away from the classical case of a = 0. The larger plot in Fig. 3 shows E of the nonlinear solutions mapped out in α as a function of a for Re = 3178. When a = 0, the envelope is very narrow since we are near Re g . Increasing a further, one sees that for a = 0.01 the solutions encompass a greater range in both amplitude and wave number. According to the inset in Fig. 3 for a = 0.20, at Re = 3178 the range of α's admitted by the Navier-Stokes equations is broadened to both larger and smaller values compared to the lower-a case in the main plot at the same Reynolds number. The a = 0.20 solution has only one branch at Re = 3178 (also seen in Fig. 2) as opposed to two for the solutions at lower values of a. Decreasing Re further, an upper and a lower branch appear for Re lower than the bifurcation point (where E = 0). Moreover, the inset in Fig. 3 shows the admitted streamwise wave numbers for a = 0.20 as a function of Re, down to values close to its Re g . Moving in α-Re parameter space, one finds the lower limits in Re of the linear and the finite amplitude solutions given in Table I. In particular, Re c changes from 54 379 (for a = 0) to 796 (for a = 0.2), whereas Re g changes from 3168 to 294. Notice that in the case with a = 0, the perturbation at the wall is set to zero, whereas homogeneous suction is still present in the base flow. Thus, the critical Reynolds numbers found for a = 0 cannot be compared to those of the Blasius boundary layer (i.e., Re c = 520 and Re g = 510 [START_REF] Rotenberry | Finite amplitude steady waves in the Blasius boundary layer[END_REF][START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]), but rather with the ones provided in the literature for the asymptotic suction boundary layer with no-slip boundary conditions for the disturbance velocity, namely Re c ≈ 54 400 and Re g ≈ 3200 [START_REF] Milinazzo | Finite-amplitude steady waves in plane viscous shear flows[END_REF][START_REF] Fransson | Flow control of boundary layers and wakes[END_REF]. Concerning the case with a = 3.85 × 10 -4 , one can notice that the predicted critical Reynolds number and streamwise wave number are in good agreement with the ones provided by Fransson and Alfredsson [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layer[END_REF] (i.e., Re c = 54 382 and α c = 0.1555).

Investigating the stability of these solutions is crucial to characterize their dynamics; thus, we have performed linearized DNSs initialized by several upper branch TW solutions selected for different a values at Re = 3178 and α = 0.2 (close to the saddle-node bifurcation for a = 0; see Fig. 3). The plot in Fig. 4 provides the time evolution of the energy integrated in the three-dimensional computational domain, showing that for t > 200, an asymptotically increasing trend exists for all of the considered solutions. These curves change only slightly for increasing spanwise domains, meaning that the unstable mode has a wavelength that is contained in the 2π domain. The main unstable mode is characterized by a low temporal frequency (ω ≈ 0.06 for a = 0.2) and a growth rate, σ , which decreases with an increase of a, as shown by the red triangles in the inset frame. This might seem surprising, since an increasing value of a has been previously found to increase the base flow instability; however, this behavior might be linked to the relative distance of the selected bifurcated solutions to the saddle-node bifurcation, as also observed in Ref. [START_REF] Wang | Lower branch coherent states in shear flows: Transition and control[END_REF] for Couette flow. We have performed simulations at different Reynolds numbers in order to establish the dependence of the unstable mode growth rate with respect to its distance (in terms of Re) to the saddle-node bifurcation. The results are displayed in Fig. 5, which shows σ as a function of Re for two values of a, namely a = 0 and 0.01: except for Reynolds numbers very close to the bifurcation point, σ scales with Re following a power law, close to Re -0.26 for a = 0 and Re -0.12 for a = 0.01; conversely, a steepest dependence on the Reynolds number, providing an asymptotic scaling of Re -0.48 , was observed for a lower branch fixed point in the Couette flow in Ref. [START_REF] Wang | Lower branch coherent states in shear flows: Transition and control[END_REF]. Thus, in the range of Reynolds number explored here, the TWs for the case of nonzero permeability are more stable than those found for a = 0, and this fact is of some relevance when assessing the stability of the ASBL flow. The most unstable modes together with the associated TWs are shown in Fig. 6 for the two selected values of a, Re = 3178 and α = 0.2. The TW superposed to the base flow solution, shown by the colored isocontours in the z = 2π plane, induces a streamwise modulation in the thickness of the asymptotic suction boundary-layer flow. One can observe that this modulation is stronger for a larger value of a (at the same Re), since the selected upper-branch TWs tend to have a larger amplitude for increasing a (see the envelopes in Fig. 3). The unstable mode is shown in the same figure by the isosurfaces of streamwise (red for positive and green for negative) and spanwise (white for positive and black for negative) perturbation. The unstable mode is localized in the streamwise region in which the TW induces a deceleration of the flow, and it allows a three-dimensionalization of the flow structures leading the flow to transition. One can also observe that, for a = 0.01, the unstable mode is more elongated in the streamwise direction than for a = 0, probably due to the presence of a nonzero wall-normal velocity at the wall (see the gray isocontours at the wall), which may tend to induce a delocalization of the disturbance.

The destabilization of the TWs and the nonlinear dynamics of the resulting flow structures have been further investigated by nonlinear DNS. Figure 7 provides the dynamics of the total perturbation of the base flow, u -U, for a = 0, Re = 3178, and α = 0.20, at four different times, t = 300, 350, 400, and 500. The black/white and the light blue surfaces indicate wall-normal and streamwise perturbations to the base flow, respectively, whereas the colored contours represent the streamwise instantaneous velocity on the plane z = 2π. At t = 300 (top left frame), the perturbation still appears to be two-dimensional, maintaining the shape of the TS-wave nonlinear solution, inducing a streamwise modulation on the asymptotic suction boundary layer profile. The fact that the flow field remains very close to the nonlinear TS wave for such a long time confirms that the TW solution is characterized by a high-dimensional stable manifold. One can notice also that, similarly to what has been observed experimentally by Fransson and Alfredsson [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layer[END_REF] for the asymptotic suction boundary layer in the presence of TS waves, the streamwise velocity perturbation has a large-amplitude peak very close to the wall, whereas the wall-normal one is weaker and peaks farther from the wall. However, at t ≈ 350 (top right frame), the flow structures are suddenly three-dimensionalized by the effect of the previously shown unstable mode, which induces a spanwise modulation of the perturbation. In particular, the streamwise perturbation to the base flow, which is localized in the vicinity of the wall, appears more affected by the unstable mode than the wall-normal one, which is placed at larger values of the wall-normal coordinate. At t ≈ 450 (bottom left frame), even the flow structures placed farther from the wall begin to be affected by the growing unstable wave, displaying strong spanwise oscillations. As a consequence, the TW experiences breakdown and very rapid transition to turbulence (see the bottom right frame at t = 500). The trajectory of the perturbation can also be observed on an E v -E w projection of the state space in Fig. 8 by the red (gray) line representing the case with zero permeability, where E w is the integrated energy of the spanwise component of the velocity perturbation, and E v is the wall-normal one. One can see that both E w and E v strongly increase only after the unstable mode is triggered (the small red dot represents the flow state at t = 300). Then the trajectory follows the unstable manifold of the TW solution (the middle-sized red dot representing the state at t = 400) very rapidly increasing in both E w and E v , and, after having reached a peak, it begins to relax toward a statistically steady state (see the larger red dot for t = 500). A similar picture can be observed for a nonzero permeability, the black curve showing the trajectory for a = 0.1. Note that a larger peak value of E w and E v is reached before transition to turbulence when an increased permeability is considered, resembling a bursting phenomenon, characterized by a large increase of both the energy and the dissipation [START_REF] Van Veen | Homoclinic tangle on the edge of shear turbulence[END_REF]. Furthermore, for a = 0.1, the transition occurs at larger times, as shown by the black small dot, which represents the flow state at t = 400, being still very close to the TW solution, as well as by the flow structures displayed in Fig. 9 for t = 450, 500, 550, and 600. The route to the transition is similar to the previous case, although it occurs in a larger time frame, and the streamwise perturbations appear to be more elongated in the streamwise direction. These results clearly point out the strong influence of the permeability of the flat plate on the nonlinear stability and transition to turbulence of the asymptotic suction boundary-layer flow.

IV. CONCLUSIONS

Finite-amplitude traveling wave (TW) solutions for the asymptotic suction boundary-layer flow are investigated, taking into account the permeability of the wall. Comparing with earlier results that set vanishing disturbance velocity components at the porous boundary, we show that both the linear stability limit Re c as well as the lower limit of the nonlinear solutions Re g change dramatically when the permeability K of the plate is accounted for, using Darcy's law. Increasing the plate permeability from zero, the linear stability limit Re c is lowered from around 54 400 to less than 800, corresponding to the largest K examined here. Moreover, the nonlinear limit Re g of the finite-amplitude solutions changes from 3168 to 294. For the range of permeabilities studied, this means that the asymptotic suction boundary layer has still a higher limit in Re c compared to the Blasius boundary layer [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]. However, looking at the nonlinear limit Re g , the solutions for the asymptotic suction case are far below the corresponding minimum point of the Blasius boundary layer [START_REF] Rotenberry | Finite amplitude steady waves in the Blasius boundary layer[END_REF]. Thus, wall suction appears to be stabilizing for low-amplitude perturbations, while finite-amplitude disturbances are more easily destabilized-compared to the Blasius case-provided the permeability of the porous wall, K, is sufficiently large. The stability of the traveling waves is also investigated, showing a low-frequency main unstable mode whose growth rate decreases with increasing permeability, with a power-law dependence on the Reynolds number. The nonlinear dynamics of the flow in the vicinity of the computed TWs is finally investigated by direct numerical simulations, showing that the flow remains for a large time close to the TW solution; transition, however, occurs very quickly once the unstable mode is triggered, inducing a strong increase of the spanwise and wall-normal perturbations of the base flow, resembling a bursting phenomenon. The unstable mode induces a three-dimensionalization of the TW, which rapidly leads to transition, providing a relevant scenario for subcritical transition due to TS waves.

In this work, it has been shown that imposing boundary conditions that take into account the permeability of the wall may strongly affect the bifurcation curves and the stability properties of two-dimensional TW solutions. This may explain why previous theoretical stability results, obtained by neglecting wall permeability, appear to overpredict the stability of the flow with respect to experimental observations [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layer[END_REF]. Moreover, the time evolution of the nonlinear TS waves computed here fits well the scenario of transition induced by TS waves in boundary-layer flows, where TS waves of sufficiently high amplitude (>1% of the mean streamwise velocity) are found to generate three-dimensional waves and vortices that induce the growth of chaotic hotspots [START_REF] Klingmann | Experiments on the stability of Tollmien-Schlichting waves[END_REF]. This transition scenario typical of boundary-layer flows may be strongly delayed by wall suction in the case a negligible wall permeability [START_REF] Fransson | On the disturbance growth in an asymptotic suction boundary layer[END_REF], since the critical Reynolds number for TS waves is found to strongly increase. However, in this work we have shown that, when wall permeability is accounted for, the critical Reynolds numbers decrease again, reaching values even lower than the ones typical of the Blasius boundary layer. Thus, the dynamics of the nonlinear TS waves can be considered representative of the transition scenarios that can be observed for moderate Reynolds numbers and non-negligible wall permeability. Nevertheless, it is well known that the asymptotic suction boundary layer can also experience bypass transition due to the formation and secondary instability of streamwise streaks and vortices, which are inherently threedimensional coherent structures. Thus, future works will aim at computing three-dimensional TW solutions in order to unravel the effect of the plate's permeability on the bypass transition scenario.

FIG. 1 .

 1 FIG. 1. A schematic of the plate showing the main idea behind the imposed boundary conditions at y = 0. Variables in capital letters refer to the base flow.

FIG. 3 .

 3 FIG. 3. (Color online) Perturbation energy E of the wavy part mapped out in α for a = 0 (22,100), 3.85 × 10 -4 (26,120), 5 × 10 -3(18,160), 0.01[START_REF] Wedin | Advances in Turbulence XII, Proceedings of the 12th EUROMECH European Turbulence Conference[END_REF]160), and 0.02[START_REF] Wedin | Exact coherent structures in pipe flow: travelling wave solutions[END_REF]200) at Re = 3178 (see the legends). The values in the parentheses after each value of a refer to the truncation (NX,NY ) used. The selected Re lies in the neighborhood of the Re g of the a = 0 solution as evidenced by its narrow envelope in α. TableIshows the values of Re g for all a's presented in the plot. The inset shows the energy for a = 0.20 at Re = 305 and 320 using a truncation of (14,180) and 3178 using[START_REF] Rotenberry | Finite amplitude steady waves in the Blasius boundary layer[END_REF]240). The lower values of Re shown in the smaller plot are close to the limit Re g for a = 0.20 given in TableI.

FIG. 4 . 26 FIG. 5 .

 4265 FIG. 4. (Color online) Energy growth extracted from a linearized three-dimensional DNS initialized with several nonlinear Tollmien-Schlichting waves. The solutions selected are the upper branch ones with α = 0.20 and Re = 3178 (see Fig.3). The inset shows the growth rate σ for several values of a ranging from 0 to 0.2, whereas the main plot shows the time evolution of the three-dimensional energy for a = 0, 0.01, 0.05, 0.1, and 0.2.

FIG. 6 .

 6 FIG. 6. (Color online) Colored isocontours of the streamwise instantaneous velocity on the plane z = 2π and isosurfaces of the streamwise (red and green) and the spanwise (black and white) perturbation to the TW extracted at t = 2000 from a linearized DNS initialized by the upper branch solution with α = 0.20, Re = 3178, a = 0 (left), and a = 0.01 (right). The gray isocontours in the right frame represent the wall-normal perturbation at the wall.

FIG. 7 .

 7 FIG. 7. (Color online) Colored isocontours of the streamwise instantaneous velocity on the plane z = 2π and isosurfaces of the streamwise (light blue) and the wall-normal (black and white) perturbation to the base flow extracted at t = 300, 350, 400, and 500 (from left top to right bottom) from a DNS initialized by the upper branch solution with α = 0.20, Re = 3178, and a = 0.

FIG. 8 .

 8 FIG. 8. (Color online) Flow trajectory plotted on an E v -E w projection of the phase space, extracted from a DNS initialized with the upper branch solutions with α = 0.20, Re = 3178, a = 0.0 [red (gray) line], and a = 0.1 (black line). The red dots indicate the state at t = 300, 400, and 500 (for increasing dot size), whereas the black ones indicate the state at t = 450, 550, and 650 (for increasing dot size).

FIG. 9 .

 9 FIG. 9. (Color online) Colored isocontours of the streamwise instantaneous velocity on the plane z = 2π and isosurfaces of the streamwise (light blue) and the wall-normal (black and white) perturbation to the base flow extracted at t = 450, 500, 550, and 650 (from left top to right bottom) from a DNS initialized by the upper branch solution with α = 0.20, Re = 3178, and a = 0.1. The gray isocontours at y = 0 represent the wall-normal perturbation at the wall.

TABLE I .

 I The critical Reynolds numbers Re c and Re g of the Tollmien-Schlichting waves for a few selected values of the nondimensional permeability a. Re c is the critical limit for linear instability, and Re g defines the emergence of nonlinear states. The corresponding limits of the Blasius boundary-layer flow (no suction) lie at Re c = 520 and Re g = 510[START_REF] Rotenberry | Finite amplitude steady waves in the Blasius boundary layer[END_REF][START_REF] Schmid | Stability and Transition in Shear Flows[END_REF].

	a	α c	Re c	α g	Re g
	0	0.1555	54 379	0.200	3168
	3.85 × 10 -4	0.155	53 171	0.205	3128
	5 × 10 -3	0.160	41 492	0.185	2693
	0.01	0.170	32 886	0.168	2324
	0.02	0.175	22 268	0.136	1795
	0.20	0.120	796	0.131	294

ACKNOWLEDGMENTS

The financial support of Marie Curie European Reintegration Grant (ERG) No. 268265 is gratefully acknowledged by H.W. The authors wish to thank Jens Fransson at the KTH Royal Institute of Technology for insightful comments, and the personnel at the library of Luleå University of Technology, Sweden, for efficient help and for their availability.


Mirko Farano

Stefania Cherubini

Jean-Christophe Robinet

Pietro De Palma

Hairpin-like optimal perturbations in plane Poiseuille flow

Keywords: instability, nonlinear instability, transition to turbulence

In this work it is shown that hairpin vortex structures can be the outcome of a nonlinear optimal growth process, in a similar way as streaky structures can be the result of a linear optimal growth mechanism. With this purpose, nonlinear optimizations based on a Lagrange multiplier technique coupled with a direct-adjoint iterative procedure are performed in a plane Poiseuille flow at subcritical values of the Reynolds number, aiming at quickly triggering nonlinear effects. Choosing a suitable time scale for such an optimization process, it is found that the initial optimal perturbation is composed of sweeps and ejections resulting in a hairpin vortex structure at the target time. These alternating sweeps and ejections create an inflectional instability occurring in a localized region away from the wall, generating the head of the primary and secondary hairpin structures, quickly inducing transition to turbulent flow. This result could explain why transitional and turbulent shear flows are characterized by a high density of hairpin vortices.

Introduction

Due to their long lifetime, coherent structures determine a large part of the mean properties of transitional and turbulent shear flows, and their generation, instability, and sustainment mechanisms may be useful to explain many details of the dynamics of such flows. Two main examples of coherent structures are streaks and hairpin vortices, which have been observed experimentally and numerically in channel flows, pipe flows, and boundary-layer flows [START_REF] Singer | Characteristics of a young turbulent spot[END_REF][START_REF] Matsubara | Disturbance growth in boundary layers subjected to free-stream turbulence[END_REF][START_REF] Adrian | Hairpin vortex organization in wall turbulence[END_REF][START_REF] Wu | Direct numerical simulation of turbulence in a nominally zero-pressuregradient flat-plate boundary layer[END_REF].

Streaky structures are observed at different scales in transitional and turbulent shear flows [START_REF] Brandt | Transition in boundary layers subject to free-stream turbulence[END_REF][START_REF] Hwang | Self-sustained process at large scales in turbulent channel flow[END_REF] and their origin seems now to be well understood. As conjectured by [START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF] some decades ago, elongated near-wall zones of low and high momentum are created by a mechanism of transient growth of the perturbations known as the lift-up effect, linked to the non-normality of the Navier-Stokes operator (Cherubini, Robinet & De Palma 2010b). Such a mechanism is based on the transport of the mean shear by rolls of streamwise vorticity. The quest for the origin of these coherent structures has stimulated the search for perturbations providing maximum amplification in a finite 'target time', defined as 'optimal perturbations'. By optimizing the perturbation energy in a linear framework, several authors have found initial optimal disturbances corresponding to streamwise-elongated rolls inducing streamwise streaks at target time [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF][START_REF] Luchini | Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations[END_REF]. The shape of these optimal perturbations corresponds well to the coherent structures found in transitional and turbulent flows in a low-to-moderate-disturbance environment [START_REF] Matsubara | Disturbance growth in boundary layers subjected to free-stream turbulence[END_REF]. Thus, streaky structures can be explained as the projection of flow disturbances onto the linear optimal flow structure [START_REF] Luchini | Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations[END_REF].

Also, the mechanism of formation of hairpin vortices has been extensively studied. [START_REF] Wu | Direct numerical simulation of turbulence in a nominally zero-pressuregradient flat-plate boundary layer[END_REF] have analysed the generation of hairpin structures from Λ-shaped vortex structures induced in a boundary layer by the receptivity process from large-amplitude free-stream turbulence; [START_REF] Acarlar | A study of hairpin vortices in a laminar boundary layer. Part 2. Hairpin vortices generated by fluid injection[END_REF] and [START_REF] Henningson | A mechanism for bypass transition from localized disturbances in wall-bounded shear flows[END_REF] have studied the formation of hairpin vortices from localized disturbances in boundary-layer and channel flows, respectively. [START_REF] Suponitsky | The generation of streaks and hairpin vortices from a localized vortex disturbance embedded in unbounded uniform shear flow[END_REF] have studied a simple model of interaction between a localized vortical disturbance and a uniform unbounded shear flow, showing that a small-amplitude initial disturbance always evolves into a streaky structure, whereas a large-amplitude one evolves into a hairpin vortex under some conditions. Furthermore, it is well established that hairpin vortices may appear as the consequence of the secondary instability and break-up of elongated streaks [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]). These and many other studies indicate that, unlike streaky structures, hairpin vortices are generated through nonlinear interactions [START_REF] Eitel-Amor | Hairpin vortices in turbulent boundary layers[END_REF]. The fact that they naturally arise in many transitional and turbulent shear flows, as the consequence of the instability of streaks or induced by other causes (such as roughness elements or flow injection at wall), suggests the existence of a strong energy growth mechanism triggered by nonlinearity. Whether the hairpin vortex might be recognized as an optimal flow structure in a nonlinear energy growth process is the question we want to address in the present paper.

With this purpose, we perform nonlinear optimizations in a simple parallel shear flow such as the plane Poiseuille flow. Nonlinear optimizations in parallel shear flows have been recently performed by [START_REF] Pringle | Using nonlinear transient growth to construct the minimal seed for shear flow turbulence[END_REF], [START_REF] Monokrousos | Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows[END_REF][START_REF] Pringle | Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos[END_REF], [START_REF] Rabin | Triggering turbulence efficiently in plane couette flow[END_REF], [START_REF] Cherubini | Nonlinear optimal perturbations in a couette flow: bursting and transition[END_REF] for pipe and Couette flow. In both cases, the nonlinear optimal perturbations were found to induce strongly bent streaks, but hairpin vortices were never observed. However, in those works, optimizations were performed for rather long target times, for which the lift-up mechanism dominates the dynamics. On the other hand, for instance, Karp & Cohen (2014) provide the description of a nonlinear mechanism generating hairpin vortices acting on a time scale which is one order of magnitude smaller. In order to focus our optimization analysis on the generation of the hairpin structure, we choose small target times (O(10h/U), h and U being the reference length and velocity, respectively) and finite-amplitude initial perturbations to rapidly trigger nonlinear effects. In particular, the chosen time scale is typical of the Orr mechanism, much smaller than the typical scale of the lift-up mechanism.

Hairpin-like optimal perturbations in plane Poiseuille flow

The paper is organized as follows: in § 2 we define the problem; then, in § 3, we discuss the results; finally, in § 4, an outlook is provided.

Problem formulation

The Navier-Stokes (NS) equations are solved to compute the flow between two parallel plates, choosing the non-dimensional variables so that half the distance between the plates is h = 1 and the centreline velocity of the laminar flow is U c = 1. Dirichlet boundary conditions for the three velocity components are imposed at the upper and lower walls of the computational box; whereas periodicity is prescribed in the streamwise and spanwise directions (denoted x and z, respectively, whereas y indicates the wall-normal direction). The streamwise, wall-normal, and spanwise dimensions of the computational domain are 2π, 2, and π, respectively. The NS equations are solved by a fractional-step method with second-order accuracy in space and time [START_REF] Verzicco | A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates[END_REF], using a staggered grid with 300 × 100 × 120 points, determined by a grid refinement study.

The optimization procedure aims at computing the velocity perturbation u = (u, v, w) T at t = 0 providing the maximum value of a chosen objective function at a given target time, T opt . The objective function is the ratio between the energy density at T opt and the initial (given) one (E(0) = E 0 ), the energy density being defined as:

E(t) = 1 2V V (u 2 + v 2 + w 2 )(t) dV (2.1)
where V is the volume of the computational domain.

The optimization problem is subject to partial differential constraints, namely the nonlinear NS equations in perturbative form. The constrained optimization is performed by a Lagrange multiplier technique coupled with a direct-adjoint iterative procedure using a gradient-based method. For further details about the optimization algorithm the reader is referred to Cherubini et al. (2010a[START_REF] Cherubini | The minimal seed of turbulent transition in the boundary layer[END_REF].

Results

We perform nonlinear optimizations of finite-amplitude three-dimensional perturbations for the plane Poiseuille flow, focusing on subcritical values of the Reynolds number, namely, Re = 2000, 3000, 4000, 5000. In order to find optimal perturbations rapidly triggering coherent structures generated by nonlinear effects, we focus on the time scale of the Orr mechanism, preventing the dominant effect of the lift-up mechanism which acts on a much larger time scale. [START_REF] Butler | Three-dimensional optimal perturbations in viscous shear flow[END_REF] found that the lift-up time scale for the Poiseuille flow at Re = 5000 is about equal to 380. On the other hand, performing several linear two-dimensional optimizations, we have found that the transient energy growth due to the Orr mechanism peaks at t ≈ 10 for all of the considered Reynolds numbers. Thus, T opt = 10 has been chosen for the following nonlinear three-dimensional optimizations. Moreover, using this short target time, we need a sufficiently large initial energy density in order to trigger nonlinear mechanisms. Therefore, several nonlinear optimizations have been performed increasing the value of E 0 until, quite suddenly, the initial optimal perturbation localizes in space and the energy gain increases with respect to a linear optimization. Such an increment of the energy gain with respect to the linear case is between 10 % and 30 % for the values of the Reynolds number considered in the present work. The value of E 0 has been successively bisected in order to determine It is noteworthy that the value of the nonlinearity threshold is greater than the minimal energy capable of inducing transition, as one would anticipate. For example, we have performed optimizations and bisections for T opt = 50 and Re = 4000, finding a transition energy threshold of about 2.5 × 10 -7 which is one order of magnitude lower than the energy corresponding to the nonlinearity threshold. On the other hand, the energy at the nonlinearity threshold is not very high since we have verified that the linear optimal perturbation rescaled with E 0th (see figure 1a) does not lead to transition. In order to evaluate the distance of the nonlinearity threshold from the laminar-turbulent boundary for each optimal perturbation, figure 1(a) also provides the rescaled energy level necessary for placing each optimal perturbation (computed at the nonlinearity threshold) on the laminar-turbulent boundary. Notice that the scaling law for the transition threshold (E(0) ∝ Re -2.35 , corresponding about to Re -1.2 for the amplitude) is not far from the value provided by the experimental analysis of [START_REF] Cohen | Aspects of linear and nonlinear instabilities leading to transition in pipe and channel flows[END_REF], who found a scaling law for the transition threshold amplitude of A ∝ Re -3/2 using wall-normal flow injection for inducing hairpin vortices.

Figure 2(a) shows the nonlinear optimal perturbation obtained for (Re, E 0 ) = (4000, 2 × 10 -6 ), composed at t = 0 of several thin tubes of counter-rotating vorticity alternated in the spanwise direction and fully localized in space. The vortices are characterized by a large streamwise vorticity component and are tilted against the base flow, confirming the presence of linear energy growth mechanisms such as the [START_REF] Orr | The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part ii: a viscous liquid[END_REF] and the lift-up [START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF] mechanisms. However, this optimal solution is strongly different from the linear one computed for the same flow due to: (i) its strong localization; (ii) its smaller wavelength in the spanwise direction; (iii) the presence of an inclination of the vortical structures with respect to the streamwise 2013), this nonlinear optimal structure is characterized by a symmetry with respect to a z = const. plane, which is maintained when the initial energy E 0 is increased.

We have verified that the optimal perturbation is characterized by this symmetry only for small target times; for target times typical of the lift-up mechanism, the optimal perturbation strongly resembles the one found in Couette flow [START_REF] Cherubini | Nonlinear optimal perturbations in a couette flow: bursting and transition[END_REF]. It appears that all of the velocity and vorticity components grow together until reaching transition in a relatively short time (at t ≈ 50). It is noteworthy that in other transition scenarios, such as oblique and streaks instability (see [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF]), the streamwise velocity experiences a much larger growth than the other components at short times. Snapshots of the time evolution of the vorticity (green) and of the negative streamwise velocity (black) are provided in figure 5. The initial vortex tubes are alternated in z and have opposite inclination with respect to the streamwise direction (see figure 5a), so that the downstream tilting due to the y two of these vortices in Hairpin-like optimal perturbations in plane Poiseuille flow a tube of spanwise vorticity resembling a hairpin head (see figure 5b,c). Patches of positive and negative streamwise velocity can be observed in the flow; for t T opt they appear rather localized in the streamwise direction (figure 5c) and they elongate in the streamwise direction, creating bent streaks only at times larger than the target time (figure 5d,e). It appears that the hairpin head is originated by a strong localized streamwise velocity defect [START_REF] Adrian | Hairpin vortex organization in wall turbulence[END_REF], which is already rather large at t = 0. This defect further increases its amplitude due to the Orr mechanism and to a modified lift-up mechanism (see [START_REF] Cherubini | The minimal seed of turbulent transition in the boundary layer[END_REF] driven by the initial vortex tubes, until inducing local inflection points in the instantaneous velocity profile.

The dynamics of the nonlinear optimal perturbation shares some important features with the dynamics of the finite-amplitude perturbations analysed by [START_REF] Suponitsky | The generation of streaks and hairpin vortices from a localized vortex disturbance embedded in unbounded uniform shear flow[END_REF]. In particular, they observed the formation of a single hairpin vortex from initial Gaussian vortices of maximum vorticity magnitude Ω max > 1 and streamwise characteristic length L < 5. Such constraints are satisfied by the nonlinear optimal perturbations obtained here which are characterized by Ω max ≈ 1.8 and L ≈ 2.5. Thus, for better analysing the dynamics of the nonlinear optimal perturbation, we have studied the evolution of the main vortical structures in time following the approach of [START_REF] Suponitsky | The generation of streaks and hairpin vortices from a localized vortex disturbance embedded in unbounded uniform shear flow[END_REF]. Let us consider the centre of the vortical structure (CVS), whose coordinates are defined by the following equation:

X i = V Ω 2 x i dV V Ω 2 dV , (3.1)
where Ω is the vorticity magnitude, the denominator is the enstrophy integral, and the index i = 1, 2, 3 represents Cartesian components. To identify the spatial orientation of the vortical structure centred at X i , we employ the tensor of enstrophy distribution (TED), defined as follows:

T ij = V Ω 2 (x i -X i )(x j -X j ) dV. (3.2)
Since T ij is a symmetric tensor, all its eigenvalues are real. The eigenvector associated with the largest eigenvalue identifies the direction of the principal axis of the vortex, along which the vortex has the largest extension. The insets in figure 6(a,b) show by red lines the principal axis of the vortical structures (represented in green by the Q criterion surfaces) extracted from the DNS at t = 5 and t = 10, respectively. As the flow structures are mainly aligned with the principal axis, we evaluate along this axis the distribution of the streamwise and wall-normal velocity components. The black lines in figure 6(a,b) show such distributions versus the abscissa s along the principal axis (s = 0 at the CVS) at two different times. At t = 5 (figure 6a) we can observe that the velocity components u and v mostly have opposite sign and are equal to zero for s = 0. This behaviour recalls a fundamental feature of eddy motion in wall turbulence; namely, fluctuations have higher probability of spending time in the second (Q2) and fourth (Q4) quadrants of the u-v plane [START_REF] Adrian | Hairpin vortex organization in wall turbulence[END_REF], inducing two kind of events: (i) ejections, characterized by negative streamwise fluctuations lifted away from the wall by positive wall-normal fluctuations; (ii) sweeps, characterized by positive streamwise fluctuations being transported toward the wall.

To confirm such a similarity, we have computed the spatial probability density function (PDF) of the perturbation velocity components at different times. shows the PDF values (in a logarithmic scale) of the streamwise and wall-normal components of the velocity perturbation, confirming the presence of ejections and sweeps already at t = 0 and up to the time of formation of the hairpin vortex. For instance, in figure 6(a) a sweep is observed upstream of the CVS (s < 0) followed by a strong ejection and a weak sweep downstream of the CVS (s > 0). At t ≈ 10, the hairpin head is formed downstream of the CVS (s > 0), in correspondence with an ejection, as shown in figure 6(b). This is not surprising since, as found in wall turbulence for a channel flow, the eddy associated with an ejection has a hairpin vortex shape [START_REF] Adrian | Hairpin vortex organization in wall turbulence[END_REF]. Comparing figure 6(a) and figure 6(b), one can observe that when the hairpin head is formed, the zone of ejection increases its intensity, while the sweep region enlarges in the streamwise direction, reducing its intensity.

When the evolution of the same initial perturbation is computed by a DNS based on the solution of the linearized NS equations, the velocity distribution for t > 5 is characterized by increasing values of u and much lower values of v, as shown by the red thin lines in figure 6. Therefore, in the linearized case, streaks with larger intensity will be formed due to the lift-up effect but they will not be associated with wall-normal velocities of opposite signs, rapidly damping the ejection event that is responsible for the lifting of the hairpin head, thus preventing the creation of the hairpin structure.

Figure 8 shows that the alternation of strong sweeps and ejections found at t = 0 is maintained at larger times, thus representing the basic element for the rapid hairpin formation. Moreover, in figure 8(b) one can observe that the hairpin head is placed right in the zone of interaction of the stronger ejection with the stronger sweep, indicating that the formation of the spanwise vortex connecting the initial quasi-streamwise vortices might be a consequence of an inflectional instability taking place in this interaction zone. This observation is in agreement with the minimal flow-element model proposed by [START_REF] Cohen | A minimal flow-elements model for the generation of packets of hairpin vortices in shear flows[END_REF], in which a wavy spanwise vortex sheet was necessary to provide the inflection points for creating hairpin vortices from streamwise counter-rotating vortex pairs. Figure 9(a,b) provides the instantaneous velocity and vorticity profiles at t = 10, computed solving the nonlinear and the linearized NS equations, respectively, extracted along a vertical axis passing through the hairpin head obtained in the nonlinear case. In figure 9(a) one can observe an inflection point, located at the ordinate corresponding to the peak of vorticity coinciding with the hairpin head (at y ≈ 0.7), in the outer zone of the velocity profile; whereas, in the near-wall region, at y ≈ 0.15, one can observe the deformation of the velocity profile induced by the formation of a negative streak. Inflection points are observed in the linearized case as well (figure 9b), but are placed closer to the wall (y ≈ 0.4), consistent with a shear layer instability induced by a streak. In the nonlinear case, the vorticity peak moves in time towards the centre of the channel, reaching in 10 time units an amplitude almost twice that of the linearized case. When the head of the hairpin is pushed by the wall-normal perturbation toward the centre of the channel, it is advected by the base flow at a higher velocity with respect to the part near the wall, stretching the whole vortical structure in the streamwise direction. Then, due to conservation of circulation, stretching provides a further growth of the vorticity, sustaining the hairpin vortex and allowing secondary hairpin structures to be created [START_REF] Adrian | Hairpin vortex organization in wall turbulence[END_REF].

The time evolution of the hairpin structure is shown in figure 10, providing DNS snapshots showing isosurfaces of the Q-criterion (green), isolines of the streamwise velocity disturbance (red for positive and black for negative), and contours of the wall-normal velocity (white for positive and black for negative). One can notice the alternation of patches of u and v with different signs, spread all over the domain, and their effect on the lifting and stretching of the hairpin structure. Nonlinear effects are crucial to sustain the alternation of the u and v perturbation components as well as the vortical structures. This can be clearly inferred by comparing figure 10 with figure 11, showing the time evolution of the nonlinear optimal perturbation obtained by a linearized DNS.

Outlook

Nonlinear optimal perturbations having the shape of hairpin vortices at target time have been computed for plane Poiseuille flow, using small target times and finite initial energies. The corresponding initial optimal perturbations are localized in space and composed of spanwise-alternated thin vorticity tubes inclined with respect to the streamwise direction and placed around regions of large streamwise and wall-normal perturbations of opposite sign, resembling localized sweeps and ejections.

The streamwise alternation of sweeps and ejections induces strong streamwise velocity defects which generate an inflection point in the velocity profile. Inflectional instability thus creates a spanwise vorticity tube that is lifted and stretched, generating the hairpin head. Transition is triggered suddenly and occurs in a very localized region, already inducing the formation of a hairpin structure at target time (t = 10). The initial flow structure able to create a hairpin vortex in a very short time (t ≈ 10), rapidly leading the flow to turbulence, may be considered as a hairpin precursor, characterized by alternated sweeps and ejections of energy varying with Re -2.36 . It appears that, when nonlinear effects are damped, the wall-normal velocity component is not maintained in time, while the streamwise component greatly increases due to the lift-up mechanism, hampering the creation of the hairpin vortex and the subsequent fast transition to turbulence.

In this work we have shown that, for plane Poiseuille flow, a suitable combination of localized sweeps and ejections is capable of maximizing the energy growth in a short time interval, generating a hairpin structure transitioning towards turbulence. Thus, hairpin vortex structures can be the outcome of a nonlinear optimal growth process, in a similar way as streaky structures are linked to a linear optimal growth mechanism. This nonlinear optimal growth process could explain why the final stages of transition to turbulence and turbulent shear flows are characterized by a high density of hairpin structures. Future work will aim at extending these findings to different shear flows, as well as to noisy or turbulent environments.

FIGURE 1 .

 1 FIGURE 1. Scaling law for the initial energy (squares), the transition threshold (circles) (a), and the energy gain (b) for the short-time nonlinear optimal perturbation at the nonlinearity threshold.

FIGURE 2 .

 2 FIGURE 2. Isosurfaces of the Q-criterion for the nonlinear optimal perturbation obtained with Re = 4000, T opt = 10, and E 0 = 2 × 10 -6 : (a) t = 0, Q = 0.01 (Q max = 0.086; red/green surfaces for positive/negative streamwise vorticity values) and (b) t = 10, Q = 0.2 (Q max = 3.88).

FIGURE 3 .

 3 FIGURE 3. Isosurfaces of the three velocity components: (a,d) u, (b,e) v, (c,f ) w (light grey for positive and black for negative values, u, v, w = ±0.01) of the nonlinear optimal perturbation for T opt = 10: (a,b,c) E 0 = 2 × 10 -6 , Re = 4000; and (d,e,f ) E 0 = 1 × 10 -5 , Re = 2000. direction. In particular, figure 2(a) shows three pairs of vortices, all of them having different streamwise inclination and length, characterized by a streamwise vorticity of alternating sign. One can observe that the two inner pairs of vortices at the centre of the structure are very close to each other. When such a structure evolves in time, a strong interaction of the vortices occurs immediately after the tilting in the streamwise direction. Therefore, a 'hot spot' is created, leading to the formation of the hairpin vortex at target time t = T opt = 10, as shown in figure 2(b). It is also noteworthy that, unlike other parallel shear flows, such as Couette flow[START_REF] Cherubini | Nonlinear optimal perturbations in a couette flow: bursting and transition[END_REF], this nonlinear optimal structure is characterized by a symmetry with respect to a z = const. plane, which is maintained when the initial energy E 0 is increased. We have verified that the optimal perturbation is characterized by this symmetry only for small target times; for target times typical of the lift-up mechanism, the optimal perturbation strongly resembles the one found in Couette flow[START_REF] Cherubini | Nonlinear optimal perturbations in a couette flow: bursting and transition[END_REF]. Figure3(a-c) clearly shows the symmetry of the streamwise and wall-normal velocity components, whereas the spanwise component is antisymmetric with respect to a z-constant axis, recalling the structure of varicose perturbations(Schmid &

FIGURE 4 .FIGURE 5 .

 45 FIGURE 4. Time evolution of the energy gain and velocity (a) and vorticity (b) r.m.s. values for the nonlinear optimal perturbation with Re = 4000, E 0 = 2 × 10 -6 , and T opt = 10.

FIGURE 6 .FIGURE 7 .

 67 FIGURE 6. Velocity profile along the principal axis of the TED extracted at two different times from the DNS (black) and the linearized DNS (red), initialized by the nonlinear optimal at (Re, E 0 ) = (4000, 2 × 10 -6 ). The insets show the Q-criterion surfaces ((a) Q = 0.2 with Q max = 1.03; (b) Q = 0.3 with Q max = 3.9; green) and the principal axis (red lines) at two different times. The dot indicates the CVS. (a) t = 5, (b) t = 10. 0.050

FIGURE 8 .

 8 FIGURE 8. Wall-normal and streamwise components of the velocity vector disturbance along the principal axis of the TED for the nonlinear optimal with E 0 = 2 × 10 -6 : (a) t = 0, (b) t = 14. Isosurface of the Q-criterion ((a) Q = 0.01 with Q max = 0.086, (b) Q = 4 with Q max 12.05).

FIGURE 9 .FIGURE 10 .

 910 FIGURE 9. Streamwise velocity (solid lines) and vorticity magnitude (dashed lines) profiles at t = 5 along a wall-normal line passing through the hairpin head computed by the DNS (a) and by the linearized DNS (b), initialized with the nonlinear optimal for E 0 = 2 × 10 -6 . Circles indicate inflection points.

FIGURE 11 .

 11 FIGURE 11. Same as figure 10 but for a linearized DNS. Q max = 0.79, 0.25, 0.05 for (a) to (c), respectively. (a) t = 5, (b) t = 10, (c) t = 15.

† Email address for correspondence: s.cherubini@gmail.com