the optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent ... ·...

Under consideration for publication in J. Fluid Mech. 1

The optimal and near-optimal wave packet

in a boundary layer and its ensuing

turbulent spot

S. CHERUBINI1,2, J. -C. ROBINET2,A. BOTTARO 3 AND P. DE PALMA1

1DIMEG, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy

2SINUMEF Laboratory Arts et Metiers ParisTech, 151, Bd. de l’Hopital, 75013 Paris, France

3DICAT, University of Genova, Via Montallegro 1, 16145 Genova, Italy

(Received ?? and in revised form ??)

The three-dimensional global optimal dynamics of a flat-plate boundary layer is studied

by means of an adjoint-based optimization. The spatially localized optimal initial per-

turbation is found to be characterized by counter-rotating vortices with a x-modulated

structure, which produces, at optimal time, streamwise streaks alternated in the x di-

rection. Such an optimal perturbation is found to be qualitatively similar to the su-

perposition of local optimal perturbations computed at different wave numbers in the

streamwise direction, meaning that perturbations with non-zero streamwise wave num-

ber have a role in the transient dynamics of a boundary layer. For extended spanwise

domains, a near-optimal three-dimensional perturbation has been extracted during the

optimization process; it is localized also in the spanwise direction, resulting in a wave

packet of elongated disturbances modulated in the spanwise and streamwise direction.

By means of DNS the non-linear evolution of the optimal and near-optimal perturbation

is investigated, and is found to induce transition at lower levels of the initial energy than

local optimal and suboptimal perturbations. Moreover, it has been observed that transi-

2 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma

tion occurs in a localized zone of the flow close to the centre of the packet, by means of a

mechanism including at the same time features of both quasi-sinuous and quasi-varicose

streaks breakdown. In the same zone a hairpin vortex is observed before transition; it

has an active role in the breakdown of the streaks and results in a turbulent spot which

spreads out in the boundary layer.

1. Introduction

Hydrodynamic stability of wall-bounded shear flows has seen spectacular progress since

the early years of the twentieth century (Orr (1907); Sommerfeld (1908)), theoretical

progress which has gone hand-in-hand with experimental and numerical developments

aimed at unraveling some of the mysteries of transition to turbulence. On the one hand,

the small perturbation analysis of fluid flows has acquired and improved upon, via the

development of global modes concepts - tools of plasma physics (Briggs (1964); Bers

(1983)), and it is now understood that the impulse response of a channel or boundary

layer flow is a convected wave packet which may decay, grow algebraically, or exponen-

tially (as function of the Reynolds number, Re) while travelling downstream at the group

velocity. The response to an impulse forcing contains all frequencies and wavenumbers,

and this justifies much of the interest behind experimental works onto the evolution of

wave packets in boundary layers (Gaster & Grant (1975); Breuer & Landahl (1990); Co-

hen et al. (1991); Breuer et al. (1997)). If the regime is subcritical (i.e. all eigenmodes are

damped), only those disturbances with wavenumbers/frequencies for which non-normal

growth is allowed might display some amplification. Non-normal growth arises from the

constructive interference of damped eigenmodes which are nearly anti-parallel, and typi-

cally results in the creation of streamwise elongated flow structures (Schmid & Henning-

The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot3

son (2001)). If growth is sufficient, non-linear effects might kick in, and the recent theory

of exact coherent structures (ECS) (Waleffe (1998, 2003)) has demonstrated some effec-

tiveness in describing late stages of transition. The more convincing evidence for this has

come from the circular pipe flow: ECS in the form of streamwise travelling waves have

been identified in a pipe (Faisst & Eckhardt (2003); Wedin & Kerswell (2004)) which

have close qualitative and quantitative similarities to the transient flow structures within

a puff (Hof et al. (2004)). Exact coherent structures are however unstable states (at least

those computed so far for the simple parallel flows examined), and the flow trajectory will

ultimately evolve towards other states in phase space. Puffs are localized, downstream

travelling flow structures within a laminar field, sustained by energy provided by the

neighbouring laminar motion at the upstream end of the puff. They survive as long as

hairpin vortices, present within them, manage to generate new hairpins. When this sus-

tainment mechanism fails the puffs rapidly decay within several integral time scales, and

this has been related to the intermittent character of transition in wall-bounded shear

flows, at low Re. At larger values of the Reynolds number, an impulse perturbation of

sufficiently large amplitude typically results in bypass transition, by generating a tur-

bulent spot, which rapidly grows and spreads, leading the flow - in a frame of reference

moving at the spot velocity - towards the fully turbulent state.

Since the early observations by Emmons (1951), many studies have been dedicated to

the interior structure of turbulent spots, their shapes, spreading rates, and the mech-

anism of their rapid growth (Wygnanski et al. (1976); Perry et al. (1981); Gad-el Hak

et al. (1981); Chambers & Thomas (1983); Barrow et al. (1984); Henningson et al. (1987);

Sankaran et al. (1988); Lundbladh & Johansson (1991); Bakchinov et al. (1992); Henning-

son et al. (1993); Singer (1996); Matsubara & Alfredsson (2001)). Nothing has however

been done so far to identify the initial, localized, states which most easily bring the flow


on the verge of turbulent transition via the formation of a spot. In other words, since

in most practical cases boundary layers undergo transition by receptively selecting and

amplifying exogenous disturbances, such as those arising from the presence of localized

roughness elements or gaps on the wall, it makes sense to inquire on the initial - spatially

localized - flow patterns which most easily amplify and cause breakdown. Work along

similar lines has been conducted since the eighties, when the concept of optimal perturba-

tion was introduced, generating much hope to fill the gap in the understanding of bypass

transition. In most cases, optimal perturbations - within a linearized framework - have

been found to consist of pairs of counter-rotating streamwise vortices, capable to elicit

streamwise streaks by the lift-up effect (Landahl (1980, 1990)). Such a mechanism is very

seductive, since it brings into play some of the main ingredients present in transitional

flow fields (streaks and vortices, with the notable exception of the travelling waves), but

has the limitation of focusing onto a single wavenumber/frequency at a time, plus that

of not considering non-linear effects. When a direct simulation is performed to assess

the effectiveness of linear optimal perturbations in triggering transition, the outcome is

rather disappointing (Biau & Bottaro (2009)), and suboptimal disturbances are found to

be much more efficient than optimals.

In this work a new attempt is made to identify initial disturbances capable to provoke

breakdown to turbulence effectively in a boundary layer. We aim to optimize not sim-

ply an initial state (at x = 0 or t = 0) characterized by a single wavenumber and/or

frequency, but a wave packet, localized in the streamwise direction (and eventually also

of limited spanwise extent). Such a wave packet exists at t = 0 , and is formed by the

superposition of small amplitude monochromatic waves. The procedure to find such a

packet is that of classical optimization theory. To assess whether the optimal localized

flow state thus computed is efficient in provoking breakdown, direct numerical simula-


tions are then performed, highlighting the importance of non-linear effect which lie at

the heart of the initiation of a turbulent spot.

The paper is organized as follows. In the first section, after the definition of the prob-

lem, the numerical tools are briefly described, namely the three-dimensional direct-adjoint

optimization method and the global eigenvalue model. In the second section, a thorough

discussion of the results of the linear optimization analysis is provided, focusing on the

streamwise and the spanwise dynamics. Then, the results of non-linear simulations are

presented, and some features of the transition mechanism are explained. Finally, conclu-

sions are drawn.

2. Problem formulation

2.1. Problem formulation

The behaviour of a three-dimensional incompressible flow over a flat plate is governed

by the Navier–Stokes equations:

ut + (u · 5)u = −5 π +1

Re52 u

5 · u = 0,

(2.1)

where u is the velocity vector and π is the pressure. Dimensionless variables are defined

with respect to the inflow displacement thickness δ∗ and the freestream velocity U∞, so

that the Reynolds number is Re = U∞δ∗/ν, where ν is the kinematic viscosity. Different

rectangular computational domains are employed, the reference one having Lx = 400,

Ly = 20 and Lz = 2Z = 10.5, x, y and z being the streamwise, wall-normal and spanwise

directions, respectively. The inlet is placed at xin = 200 displacement thickness units from

the leading edge of the bottom wall, whereas the outlet of the reference domain is placed

at xout = 600. Most of the computations have been carried out at Re = 610.


2.2. Three-dimensional direct-adjoint optimization

The linear behaviour of a perturbation q′ = (u, v, w, p)T evolving in a laminar incom-

pressible flow past a flat plate is studied by employing the governing equations linearized

around the two-dimensional steady state Q = (U, V, 0, P )T :

ux + vy + wz = 0,

ut + (uU)x + Uyv + V uy + px −uxx

Re− uyy

Re− uzz

Re= 0,

vt + Uvx + (vV )y + uVx + py −vxx

Re− vyy

Re− vzz

Re= 0,

wt + Uwx + V wy + pz −wxx

Re− wyy

Re− wzz

Re= 0.

(2.2)

In order to close the system, a zero perturbation condition is chosen for the three velocity

components at the x and y boundaries, whereas periodicity of the perturbation is imposed

on the spanwise direction. The outlet zero perturbation condition is enforced by means

of a fringe region, which lets the perturbation at the exit boundary vanish smoothly. The

fringe region, whose characteristics will be discussed in subsection 2.4, is implemented

by adding onto the equations a forcing term in a limited region with x > xout.

In order to identify the perturbation at t = 0 which is able to produce the largest

disturbance growth at any given T , a Lagrange multiplier technique is used. Let us

define the disturbance energy density as:

E(t) =∫ Z

−Z

∫ Ly

o

∫ xout

xin

(u2 + v2 + w2

)dxdydz; (2.3)

the objective function of the procedure is the energy of perturbations at time t = T , i.e.

E(T ). The Lagrange multiplier technique consists in seeking extrema of the augmented


functional L with respect to every independent variable. Such a functional is written as:

L = =+∫ Z

−Z

∫ Ly

o

∫ xout

xin

∫ T

0

a (ux + vy + wz) dtdxdydz

+∫ Z

−Z

∫ Ly

o

∫ xout

xin

∫ T

0

b

(ut + (uU)x + Uyv + V uy + px −

uxx + uyy + uzz

Re

)dtdxdydz

+∫ Z

−Z

∫ Ly

o

∫ xout

xin

∫ T

0

c

(vt + Uvx + (vV )y + uVx + py −

vxx + vyy + vzz

Re

)dtdxdydz

+∫ Z

−Z

∫ Ly

o

∫ xout

xin

∫ T

0

d

(wt + Uwx + V wy + pz −

wxx + wyy + wzz

Re

)dtdxdydz

+λ0 [E(0)− E0] ,

(2.4)

where the linearized Navier-Stokes equations (2.2) and the value of the energy at t = 0

have been imposed as constraints, and a, b, c, d are the Lagrange multipliers. Integrating

by parts and setting to zero the first variation of L with respect to u, v, w, p allows us

recover the adjoint equations:

bt + bxU + (bV )y − cVx + ax +bxx

Re+

byy

Re+

bzz

Re= 0,

ct + (cU)x + cyV − bUy + ay +cxx

Re+

cyy

Re+

czz

Re= 0,

dt + (dU)x + (dV )y + az +dxx

Re+

dyy

Re+

dzz

Re= 0,

bx + cy + dz = 0,

(2.5)

where q† = (a, b, c, d)T is now identified as the adjoint vector. By using the boundary

conditions of the direct problem, one obtains:

b = 0 , c = 0 , d = 0 , for y = 0 and y = Ly,

b = 0 , c = 0 , d = 0 , for x = xin and x = xout,

(2.6)

and the optimality and compatibility conditions, respectively:

−b + 2λ0u = 0 , −c + 2λ0v = 0 , −d + 2λ0w = 0 , for t = 0, (2.7)


b + 2u = 0 , c + 2v = 0 , d + 2w = 0 , for t = T. (2.8)

The equations and conditions derived above are solved by means of an iterative ap-

proach. The direct and adjoint equations are parabolic in the forward and backward

direction, respectively, and must be integrated successively until convergence is reached.

The optimization procedure for a chosen objective time T can be summarized as follows:

(a) An initial guess is taken for the initial condition q0 at t = 0, with an associated

initial energy E0.

(b) The direct problem (2.2) is integrated from t = 0 to t = T .

(c) At t = T the initial condition for the adjoint problem is provided by the compati-

bility condition (2.8).

(d) The adjoint problem (2.5) is integrated backward in time from t = T to t = 0,

starting from the initial condition of step (c).

(e) At t = 0 the optimality condition (2.7) determines the new initial condition q0

for the direct problem and the Lagrange multiplier λ0 is chosen in order to satisfy the

constraint E(0) = E0.

(f) The objective function = is evaluated, in order to assess if its variation between

two consecutive iterations is smaller than a chosen threshold. In such a case the loop is

stopped, otherwise the procedure is restarted from step (b).

2.3. Global eigenvalue analysis

The same objective function has been optimized by a global model described in Alizard

& Robinet (2007), where the perturbation is supposed to be characterized by only one

wavelength in the spanwise direction, so that q(x, y, z, t) = q(x, y, t)exp (iβz). Such a

perturbation is decomposed in temporal modes as:

q(x, y, t) =N∑

k=1

κ0k qk(x, y) exp (−iωkt) , (2.9)


where N is the total number of modes, qk are the eigenvectors, ωk are the eigenmodes

(complex frequency), and κ0k represents the initial energy of each mode. A substitution of

such a decomposition in equation (2.1) and a successive linearization lead to the following

eigenvalue problem

(A− iωkB) qk = 0, k = 1, . . . , N. (2.10)

where matrix A and B are defined in the Appendix. In order to optimize the objective

function (2.3) for an initial perturbation of given energy, we define the maximum energy

gain obtainable at time t over all possible initial conditions u0 as:

G (t) = maxu0 6=0

E (t)E (0)

. (2.11)

By decomposing the perturbation into the eigenmode basis (2.9), it is possible to rewrite

equation (2.11) in the following form:

G(t) = ||F exp(−itΛ)F−1||22, (2.12)

where Λ is the diagonal matrix of the eigenvalues ωk, and F is the Cholesky factor of

the energy matrix M of components

Mij =∫ Ly

o

∫ xout

xin

(u∗i uj + v∗i vj + w∗i wj) dxdy, i, j = 1, . . . , N, (2.13)

where the superscript “∗” denotes the complex-conjugate. Finally, the maximum ampli-

fication at time t and the corresponding optimal initial condition, u0, are computed by

a singular value decomposition of the matrix F exp(−itΛ)F−1 (Schmid & Henningson

(2001)).


2.4. Numerical methods

Both linearized and non-linear Navier–Stokes equations are integrated by a fractional

step method using a staggered grid (Verzicco & Orlandi (1996)). The viscous terms are

discretized in time using an implicit Crank–Nicolson scheme, whereas an explicit third-

order-accurate Runge–Kutta scheme is employed for the non-linear terms. A second-

order-accurate centered space discretization has been used. The same method has been

used for the integration of the adjoint operator (2.5).

A fringe region of Lf = 90 displacement thickness units has been employed for the

computations on the reference domain (Lx = 400, Ly = 20, Lz = 10.5), whereas for

different domain length, the dimension of the fringe region has been scaled proportionally

to Lx. The forcing function applied in the fringe region has been defined as:

f =

A

1 + e(x1−xoutx−xout

− x1−xoutx1−x )

for xout < x < x1

A for x1 < x < x2

A

1 + e(

xout+Lf−x2x−x2

−xout+Lf−x2xout+Lf−x )

for x2 < x < xout + Lf

(2.14)

where x1 and x2 are placed at the abscissae xout + Lf/3 and xout + 2Lf/3, respectively,

and A = 100. For linear computations, the reference computational domain has been

discretized by a 501 × 150 × 41 Cartesian grid stretched in the wall-normal direction

(the height of the first cell close to the wall is equal to 0.1, whereas the length of the

cell in the x and z direction is respectively 0.98 and 0.25). In order to make sure that

such grid is sufficiently fine to accurately describe the linear dynamics of the considered

flow, computations have been carried out with a 801 × 200 × 61 grid, and the results

have been found to be essentially unchanged. Non-linear simulations have been carried

out on a yet finer grid (801 × 200 × 121) chosen in order to assure that the turbulence

at small scales would be well resolved. Moreover, in order to allow the development of


(a) (b)

Figure 1. (a) Envelope of the energy gain computed by direct-adjoint method (black squares)

and global model (solid line) at Re = 610 and Lx = 400; (b) normalized variation e of the

objective function versus the number of iterations at T = 247 represented in a logarithmic scale.

turbulent structures in the considered flow, the reference domain has been increased of 90

displacement thickness units in the streamwise direction. Finally, in order to reduce the

computational efforts, in the non-linear simulations the fringe region has been replaced

by a non reflective convective boundary condition (Bottaro (1990)).

Concerning the global model, the problem (2.10) is discretized with a Chebyshev/

Chebyshev collocation spectral method employing N = 1100 modes, and is solved with

a shift-and-invert Arnoldi algorithm using the ARPACK library (Lehoucq et al. (1997)),

the residual being reduced to 10−12. At the upper boundary and at the inlet a zero

perturbation condition is imposed, whereas at the outflow a homogeneous Neumann

condition is prescribed. The modes are discretized using Nx = 230 collocation points in

the x-direction and Ny = 47 collocation points in y-direction. A study of the sensitivity

of the global model here considered with respect to grid resolution and domain lengths

has been carried out in Alizard & Robinet (2007).


3. Results

3.1. The linear optimal dynamics

Direct-adjoint computations have been carried out at Re = 610, for a domain with di-

mensions Lx = 400, Ly = 20, Lz = 10.5, where the value of Lz has been chosen in order to

obtain the largest amplification, as shown in Section 3.1.2. The optimal energy gain, G(t),

has been found to reach a maximum of about 736 approximately at time tmax ≈ 247, as

shown by the square data points in figure 1 (a). Such a value is greater than that found

by a local parallel approach at the same Reynolds number. Figure 1 (b) provides the nor-

malized variation of the objective function, e = (E(t)(n)−E(t)(n−1))/E(t)(n), versus the

number of iterations n for a direct-adjoint computation performed at the optimal time.

The algorithm is able to reach very quickly a level of convergence of about 10−4, whereas

for an increasing number of iterations the convergence rate reduces, so that about 80

iterations are needed in order to reach a level of convergence of 10−5. Provided that

only minor differences have been observed between the solutions corresponding to the

two convergence levels 10−4 and 10−5, the convergence level 10−4 has been considered

satisfactory for all the computations presented in this section. In order to validate such

results, the optimal energy gain has been computed also by the global model of section

2.3. The resulting curve has been found to follow qualitatively the data obtained by the

direct-adjoint method, although the maximum G(t) is found to be lower than the one

previously computed, as shown in figure 1 (a) by the solid line. Such a discrepancy could

be the consequence of an imperfect convergence of the optimization due to the limited

number of modes (N = 1100) taken into account when computing equation (2.12), which

could not be increased in the present computations due to constraints on memory oc-

cupation related to the storage of matrix (2.10). Another possible reason of discrepancy

could be the difference in outflow conditions between the two approaches. However, the


Figure 2. Optimal perturbation on the plane z− y at t = 0, x = 450 for Re = 610. The vectors

represent the v and w components whereas the colours are relative to the streamwise velocity.

two results are quite close to one another and the difference at the peak point is less

than 4%. To further ensure that the solution found is indeed optimal, a direct-adjoint

computation has been carried out by adding a fringe region also at the inlet boundary.

The optimal G(t) value found differs by less than 1.5% from that computed previously

and the optimal disturbance is unchanged.

In order to get some insight onto the amplification mechanism, the evolution of the

optimal perturbation in time is analyzed. In figure 2 the optimal perturbation at time

t = 0 on the plane z − y for a given streamwise position, x = 450, is depicted. The

optimal spatially localized initial disturbance is characterized by a counter-rotating vor-

tex pair in the z − y plane. In previous works, several authors (see Farrell 1988; Butler

& Farrell 1992; Luchini 2000; Schmid 2000; Corbett & Bottaro 2000) have optimized

locally the time amplification in terms of energy of a perturbation characterized by a

given wave number in both the streamwise and spanwise direction (denoted respectively

α and β). Those authors have found that the local optimal perturbation is characterized


(a)

(b)

(c)

Figure 3. Iso-surfaces of the streamwise (a), wall-normal (b) and spanwise (c) velocity compo-

nents of the optimal perturbation for Re = 610 at t = 0 and for a longitudinal domain length

Lx = 800. Light surfaces indicate positive velocities, dark ones are relative to negative velocities.

by a counter-rotating vortex pair in the z − y plane, with no modulation in the stream-

wise direction (α = 0), which amplifies itself in time by means of the lift-up mechanism

(Landahl (1980)). In the present case, where the perturbation has no given wavelength,

a modulation is found in the x direction, the perturbation being composed by upstream-


(a)

(b)

(c)

Figure 4. Iso-surfaces of the streamwise (a), wall-normal (b) and spanwise (c) velocity com-

ponents of the optimal perturbation for Re = 610 at t = tmax and for a longitudinal domain

length Lx = 800. Light surfaces indicate positive velocities, dark ones negative velocities.

elongated structures with velocity components of alternating signs also in the x−y plane,

as shown by the iso-surfaces of the streamwise, wall-normal and spanwise velocity in fig-

ure 3 (a), (b) and (c), respectively. The time evolution of such an optimal solution shows

that the perturbation tilts downstream via the Orr mechanism (Schmid & Henningson


(2001)), while being amplified by the lift-up mechanism, resulting at the optimal time

in streaky structures again with alternating-sign velocity components in the x direction,

as displayed in figure 4. Indeed, the amplification of the initial perturbation is also due

to the Orr mechanism and to the spatial growth due to the non-parallelism of the flow.

In order to estimate such effects, a two-dimensional direct-adjoint optimization has been

performed using a domain with Lx = 400 and Ly = 20, for which the lift-up mecha-

nism is inhibited. The energy gain has been found to reach only a maximum value of

80 at time t ≈ 650, which is very low with respect to the maximum amplification of the

three-dimensional case, proving that both the Orr mechanism and the convective spatial

growth have a secondary role in the three-dimensional optimal dynamics at Re = 610.

The same results have been reproduced by means of the global model.

Furthermore, in order to investigate the influence of the longitudinal domain length

on the transient amplification, two computations have been performed with Lx2 = 800

and Lx3 = 1200, which are equal to two and three times the length of the reference

domain, Lx1 = 400, respectively. The two domains with lengths Lx2 and Lx3 have been

discretized by 1001 and 1501 points in the streamwise direction, respectively. As shown in

figure 5, the optimal energy gain reaches a peak of 1250 at tmax ≈ 490 for Lx2 and 1720 at

tmax ≈ 821 for Lx3, increasing of about 500 and 1000 units with respect to the reference

domain value. Such a strong increase of the energy gain with the longitudinal length of

the domain is mostly due to a combined effect of the Orr mechanism and of the spatial

amplification, linked to the non-parallelism of the flow. Indeed, a two-dimensional direct-

adjoint optimization performed using Lx2 = 800 predicts again an increase of about

500 units for the G(t) peak with respect to the value found for the reference domain

with Lx1 = 400. The effects of the variation of the domain length on the shape of the


Figure 5. Envelope of the energy gain computed by direct-adjoint method at Re = 610 for

three longitudinal domain lengths, Lx1 = 400 (squares), Lx2 = 800 (triangles), Lx3 = 1200

(diamonds).

optimal perturbation are shown in figure 6, where the optimal solutions are provided

for Lx1 and Lx2 using dashed and solid lines in the x − y plane, respectively. One can

observe in figure 6 (a) that using a longer domain the optimal perturbation at t = 0 has

a higher inclination with respect to the streamwise direction, so that the Orr mechanism

accounts for a greater amplification. Moreover, as shown in figure 6 (b), the perturbation

at t = tmax corresponding to Lx2 is more tilted in the streamwise direction, a combined

effect of the lift-up mechanism with the convective amplification of the flow.

3.1.1. Analysis of the dynamics in the streamwise direction

The relation between the global optimal and the local optimal perturbations at differ-

ent α is now analyzed so as to investigate the origin of the x modulation. Following the

method in Corbett & Bottaro (2000), a local optimization has been performed on the

inlet velocity profile for 160 values of α varying in the range −0.4, 0.4. In order to allow

a meaningful comparison, the highest wave number, 0.4, has been chosen to be greater

than the highest streamwise wave number recovered by spatial Fourier transform of the

global optimal perturbation. The same criterion has been used to choose the smallest


(a)

(b)

Figure 6. Contours of the optimal perturbation streamwise velocity component for Lx1

(dashed contours) and Lx2 (solid contours) at t = 0 (a) and t = tmax (b) for Re = 610.

wave number, 0.005. A three dimensional perturbation has been reconstructed as a su-

perposition of local single-wavenumber optimal (at α = 0) and suboptimal (at α 6= 0)

solutions computed at the objective time tmax. The three-dimensional disturbance at

t = 0 has been obtained by normalizing and superposing such solutions, whereas, at

tmax, they have been weighted by the value of their amplification. Such a reconstruction

is able to qualitatively reproduce at different times the x-alternated packets of counter-

rotating vortices as well as the streak-like x-alternated structures, as shown in figure 7,

demonstrating that the three-dimensional dynamics of a boundary layer is characterized

by the superposition of modes with zero and non-zero streamwise wave number. Such a

result is in contrast with the local theory, which predicts that the optimal dynamic of a


Figure 7. Iso-surfaces and y− z slice of the streamwise velocity component of the perturbation

computed for t = tmax by a superposition of local suboptimals for β = 0.6 and Re = 610.

boundary layer flow is dominated by zero-α waves.

In order to quantify the presence of non-zero streamwise wave numbers in the three-

dimensional dynamics of the considered flow, we now study the variation with time of the

most amplified streamwise wave number identified in the optimal perturbation. Figure

8 shows that for the three streamwise domain lengths considered here the characteristic

streamwise wave number (αc), defined as the most amplified wave number recovered in

the optimal perturbation by means of a spatial Fourier transform, is rather high at small

times, and decreases with time. Nevertheless, the αc seems to converge to an asymptotic

value different from zero which does not vary with the domain length. However, even if

there is no evidence that for a boundary layer of infinite longitudinal extent, αc would

not converge to zero at an infinite time, such an hypothetical optimal perturbation would

not play a role in transition, which occurs at finite and even small times. Moreover, it is

noteworthy that, for small times, αc increases with the length of the domain at a given

time, meaning that the presence of non-zero α waves in the optimal perturbation is not

a consequence of the small longitudinal size of the reference domain, but is a feature

of the optimal three-dimensional perturbation. The analysis of the variation of αc with


time has been also performed for the three-dimensional perturbation reconstructed as

a superposition of local optimals and suboptimals. As shown in figure 9, the αc curves

obtained from the global optimal perturbation and the superposition of local optimal and

suboptimal solutions, for Lx = 800, have been found to be very close for all times but the

smallest (t = 41), meaning that such a reconstruction is able to predict the streamwise

wave number dominating the optimal dynamics of the considered flow.

Finally, in order to verify the generality of the optimal dynamics described here, the

direct-adjoint optimization has been carried out also for two smaller Reynolds numbers,

Re = 300 and Re = 150. Both the values of the Reynolds number have been chosen

with the aim of keeping the entire flow, from the inlet to the outlet, locally stable with

respect to Tollmien-Schlicting modes, in order to ensure that the x-modulation of the

optimal perturbation is not due to the interaction of local optimals with such modes.

Indeed, for all the values of Re, the optimal perturbation has been found modulated in

the x-direction, although a decrease of the characteristic streamwise wave number αc has

been observed for decreasing Reynolds numbers. Even for a very low Reynolds number

value such as Re = 150, αc maintains a value which is different from zero at all times,

confirming the generality of such a behaviour.

3.1.2. Analysis of the dynamics in the spanwise direction

The effects of the spanwise dimension on the optimal dynamics have been analyzed by

performing the direct-adjoint optimization for several spanwise domain lengths. Figure 11

(a) shows the behaviour of the maximum optimal energy gain versus the spanwise mini-

mum wave number, βL = (2π)/Lz, computed by means of the global model (squares) and

by direct-adjoint iterations (circles). It is worth to point out that, concerning the global

model, the wave number of the optimal perturbation is prescribed, β = βL, whereas


Figure 8. Variation of the most amplified longitudinal wave number αc with time at

Re = 610, for Lx1 = 400 (squares), Lx2 = 800 (diamonds) and Lx3 = 1200 (circles).

Figure 9. Variation of the most amplified longitudinal wave number αc with time at Re = 610,

as computed by direct-adjoint method for Lx = 800 (diamonds) and by superposition of local

optimals (squares).

in the direct-adjoint computation no wavelength is imposed, so that βL represents just

the minimum wave number (the maximum wavelength) found by the procedure. Indeed,

looking at figure 11 (a) one can notice a discrepancy between the predictions of the two

methods. The global model predicts a well defined peak for βL = 0.6, hereafter called

βLopt, corresponding to Lz = 10.5, which is very close to the optimal wave number com-

puted locally by Corbett & Bottaro (2000). In the direct-adjoint optimization for high


Figure 10. Variation of the most amplified longitudinal wave number αc with time for different

values of the Reynolds number: Re = 610 (squares), Re = 300 (diamonds) and Re = 150

(circles), and β = 0.6.

values of βL the optimal amplification peaks match those computed previously by the

global model, whereas a plateau is found for low values of βL. A similar behaviour is

shown in figure 11 (b) for the value of the optimal time. In particular, for large spanwise

domain lengths, a plateau in the time of maximum growth is found by the direct-adjoint

optimization, whereas the global model predicts a large increase of the optimal time. It is

worth to notice that, for low values of the spanwise wavenumber, the single-wavenumber

dynamics is close to the two-dimensional one, which is characterized by larger optimal

times (for example, tmax = 650 for a two-dimensional domain with Lx = 400, Ly = 20

and Re = 610), thus explaining the higher values of the optimal time recovered by the

global model for low βL.

Concerning the results obtained using the direct-adjoint, the shape of the optimal per-

turbation in the spanwise direction is analyzed for several βL. As shown in figure 12

for βL = 0.1, the perturbation resulting in the highest energy gain is characterized by

six wavelengths of the optimal disturbance found for βLopt = 0.6. Therefore, at optimal

time the dynamics as well as the value of the G(t) exactly match those at βLopt . On


(a) (b)

Figure 11. Variation of the peak of the optimal energy gain (a) and of the optimal time (b) with

the transversal wave number βL as computed by the global model (squares) and by direct-adjoint

method (circles), where βL = (2π)/Lz.

Figure 12. Spanwise shape of the optimal perturbation streamwise velocity component at

x = 400 and y = 1 for βL = 0.6 (dashed line) and βL = 0.1 (solid line).

the other hand, a suboptimal dynamics is recovered by the single-spanwise-wavenumber

global model, explaining the lower value of the G(t) shown in figure 11.

The effects of the spanwise domain length on the streamwise shape of the optimal

perturbation are now analyzed. Figure 13 (a) shows the behaviour in time of the char-

acteristic wave number, αc, for several values of βL larger than the optimal one. The

values of αc are found to decrease when βL is increased, but they show a trend which is

not far from that displayed at βLopt (solid line). Indeed, they are found to slowly con-

verge to a value different from zero, which is very close to that achieved asymptotically

when βLopt= 0.6. Figure 13 (b) shows the variation of αc with time for two βLs smaller


(a) (b)

Figure 13. Variation of the most amplified longitudinal wave number αc with time at Re = 610,

as computed by direct-adjoint method for βL = 0.6 (squares), βL = 0.8 (triangles), βL = 1.0

(diamonds), βL = 1.3 (circles) (a) and for βL = 0.6 (squares), βL = 0.1 (diamonds), βL = 0.035

(circles).

Figure 14. Contours of the optimal perturbation streamwise velocity component for βL = 0.1

(gray contours) and βL = 0.6 (dashed and solid lines for positive and negative perturbations

respectively) on an x− z plane for y = 1 at t = 0.

than optimal, which are more representative of realistic cases. One of the considered βL

(βL = 0.035) has been chosen to be incommensurate with the optimal one, for com-

pleteness. All curves are found to almost overlap the optimal curve, meaning that for a

sufficiently large domain the streamwise shape of the optimal perturbation is very close


Figure 15. Normalized variation of the objective function with respect to the number of di-

rect/adjoint iterations represented in a logarithmic scale at T = 247 and βL = 0.1 for a compu-

tation initialized by an artificial wave packet (solid line) and a single-wavenumber initial guess

(dashed line)

to the β-optimal one, even when the spanwise domain length is not an exact multiple of

the optimal one. As a result, it is possible to conclude that the three-dimensional opti-

mal perturbation in a boundary layer would be characterized, for large spanwise domain

lengths, by streamwise elongated structures alternated in the x and z directions with an

angle Θopt = arctan(αopt/βopt) (Θopt ≈ 4.5 for Lx = 400). An x− z slice of the optimal

perturbation for βL = 0.1, plotted in figure 14, shows the Θopt inclination of the optimal

disturbances at t = 0.

The results analyzed here have shown that the optimal three-dimensional perturbation

in a boundary layer flow, although being localized and characterized by a superposition

of frequencies in the streamwise direction, results to be a single-spanwise-wavenumber

perturbation extending over the whole spanwise domain. It could be argued that such a


disturbance would not be very realistic; indeed, in nature as well as in an experimental

framework, disturbances are most likely characterized by a wide spectrum of frequencies,

and not by a monochromatic signal, and are often localized in wave packets, for example

when they are caused by a localized roughness element or a by gap on the wall.

In order to have some clues onto the effect of small localized disturbances, an artificial

wave packet has been built for the two largest spanwise domain lengths (corresponding

to βL = 0.1 and βL = 0.035) by multiplying the optimal single-wavenumber perturba-

tion times an envelope of the form exp−(z2/Lz). Such an artificial wave packet has been

used as initial configuration for the optimization process, to investigate in which way the

optimization algorithm alters the initial guess in the course of the iterations, inducing it

to spread out in the whole domain. In figure 15 the residual of the objective function is

shown versus the number of iterations for two cases, the first initialized by the artificial

wave packet (solid line), the second initialized by a single-spanwise-wavenumber pertur-

bation (dashed line) for βL = 0.1. One could observe that in the first case the convergence

is slower, and that both curves experience a marked change in slope. To explain such a

behaviour, the partially optimized perturbation has been extracted at different levels of

convergence, e. Figure 16 shows such intermediate solutions at e1 = 10−3 and e2 = 10−4,

as well as the wave packet used as initial guess. Analyzing such intermediate solutions it is

possible to notice that at e1 the perturbation is still spanwise localized, although its shape

has changed. In particular, the streak-like structures at the edge of the wave packet are

inclined with respect to the z axis, resulting in oblique waves bordering the wave packet.

Such a tilting is probably due to the superposition of different modes in the spanwise

direction. Indeed, Fourier transforms in z of the perturbation at different streamwise lo-

cations have shown that such a perturbation is composed by a superposition of different


(a)

(b)

(c)

Figure 16. Contours of the perturbation streamwise velocity component for the initial guess

(a), for the intermediate solution converged at e1 = 10−3 (b), and at e2 = 10−4 (b).

modes, although the spectrum remains centered around the optimal wave number, as dis-

played in figure 17 (a). On the other hand, at the convergence level e2 the perturbation is

spread out in the whole domain, although a single-spanwise-wavenumer perturbation is

still not recovered. Furthermore, both solutions, although strongly different in the span-


(a) (b)

Figure 17. Fourier transform in z at y = 1 and x = 400 of the streamwise velocity component

of the temporary solution at the level of convergence e1 = 10−3 for βL = 0.1 (a) and βL = 0.035

(b).

wise direction, are very much amplified, reaching a value of G which differs by less than

1% from the optimal one (G(t)opt = 736, G(t)e1 = 728, G(t)e2 = 734). Thus, it is possible

to conclude that several near-optimal perturbations exist, intermediate solutions of the

optimization process, characterized by different shapes in the spanwise direction, and

producing quasi-optimal amplifications in the disturbance energy (the difference being

less than 1%). It is worth to point out that such a result was not unexpected, since the lin-

earized Navier-Stokes operator is self-adjoint in the spanwise direction for the considered

flow. Indeed, no transient energy growth could be produced in the spanwise direction, so

that the influence of the spanwise shape of the perturbation on the energy gain is weak

with respect to the streamwise and wall-normal ones. In fact, the curve of the optimal

energy gain with respect to the spanwise wavenumber is quite flat close to the peak, as

displayed in figure 11 in the range from βL = 0.5 to βL = 0.8, so that a perturbation

composed by a superposition of such wavenumbers (like the near-optimal wave packets at

e1 for βL = 0.1 and βL = 0.035 whose spectra are shown in figure 17 (a) and (b)) could

induce an energy gain very close to the optimal one. Moreover, it is worth to point out

that the self-adjoint character of the Navier-Stokes operator in the spanwise direction

may also explain the reduction of the rate of convergence observed in figure 15 after level

e1, when the largest residual adjustments of the solution occur in the spanwise direction.


(a) (b)

Figure 18. Iso-surfaces of the near-optimal perturbation streamwise velocity component at

the convergence level e1 = 10−3 for Lz = 180 at the time instants t = 0 (a) and t = tmax (b).

In the following section the non-linear evolution of a near-optimal wave packet is stud-

ied, using the intermediate solution extracted at e1 = 10−3. Figure 18 (a) and (b) show

such a state at t = 0 and t = tmax, respectively, for βL = 0.035. This perturbation is far

more interesting than the optimal one when large spanwise domain lengths are consid-

ered, being characterized by the superposition of modes localized in both the streamwise

and spanwise direction, and amplifying its energy up to values which are very close to

optimal.

3.2. The non-linear dynamics

Although the three-dimensional linear optimization has assessed that the initial pertur-

bation resulting in the largest energy amplification in a boundary layer flow is in the

form of an array of elongated structures, alternated in the x and z direction following

an optimal angle Θopt, it is not straightforward that such a perturbation would also be

able to effectively induce transition. Therefore, it is worth to investigate the non-linear


evolution of the global optimal solution.

In the last decades, different routes to transition have been postulated and studied,

based on linear asymptotic stability analysis (K- and H-type transition, see Klebanoff

et al. (1962); Kachanov & Levchenko (1984); Rist & Fasel (1995)), local optimal growth

analysis (by-pass transition, see Westin et al. (1994); Andersson et al. (1999); Jacobs

& Henningson (1999) as well as oblique transition, see Elofsson & Alfredsson (1998);

Berlin et al. (1994)). Recent studies by Nagata (1990, 1997); Waleffe (1998); Faisst &

Eckhardt (2003); Wedin & Kerswell (2004); Eckhardt et al. (2007) have outlined the pos-

sibility that transition could be related to some non-linear equilibrium solutions of the

Navier-Stokes equations. Nevertheless, as previously mentioned, it is still unclear which

type of perturbation is the most suited to excite such unstable states, taking the flow

on the edge of chaos. Indeed, a weak connection has been found between the local linear

optimal perturbations and the occurrence of travelling waves in a flow. In a recent study

on the flow in a duct of square cross-section, Biau et al. (2008) have argued that single-

wavenumber suboptimal perturbations, which are in the form of streamwise-modulated

waves, are more effective in inducing transition than the zero-streamwise-wavenumber

local optimal ones. This suggests that rapidly growing travelling wave packets could play

an active role in the route to chaos. Therefore, an investigation of the non-linear evo-

lution of the global optimal wave packet and of its possible role in transition is of interest.

In order to investigate the capability of the three-dimensional optimal perturbation to

induce transition in a boundary-layer flow, simulations have been performed in which the

initial optimal disturbance has been superposed to the base flow with several given initial

energies. The optimal transversal domain length Lz = 10.5 and a streamwise length of


(a) (b) (c)

Figure 19. Mean skin friction coefficient at T = 700 of the considered flow perturbed with

the global three-dimensional optimal (solid line), the local optimal with α = 0 (dashed line)

and the suboptimal with α = βLopt (dashed-pointed line), with three different initial energies,

E0a = 0.5 (a), E0b = 2 (b) and E0c = 10 (c). The lowest and highest dotted lines represent

the theoretical values of the mean skin friction coefficient for a laminar and turbulent boundary

layer, respectively.

Lx = 400 have been chosen. To allow a comparison with some of the most probable

transition mechanisms already known in the literature, the DNS has been performed

also for the evolution of a local optimal (with α = 0) and a suboptimal (with α = βLopt)

perturbation superposed to the base flow, using the same initial energies chosen for the

global optimal case. Figure 19 (a), (b) and (c) show the mean skin friction coefficient ob-

tained in the simulations initialized with the three different initial energies, E0(a) = 0.5,

E0(b) = 2 and E0(c) = 10, respectively. The lowest and highest dotted lines in figures 19

represent the theoretical trend of the laminar and turbulent skin friction coefficient in a

boundary layer, whereas the solid, dashed and dashed-dotted lines represent the mean

skin friction coefficient obtained in the simulations initialized using the global optimal,

the local optimal and the local suboptimal, respectively. The value of the energy E0(a)

has been found to be the lowest one able to induce transition in the flow perturbed

by the global optimal disturbance. Indeed, figure 19 (a) shows that in such a case the

mean skin friction (solid line) which initially follows the theoretical laminar value, rises


up towards the turbulent value. On the other hand, the skin friction coefficient curves

relative to the local optimal and suboptimal case lay on the theoretical laminar curve

for such a low value of the energy, meaning that the global optimal disturbance is more

effective than the local ones in inducing chaotic behaviour in the flow. Figure 19 (b) and

(c) show that the suboptimal x-modulated perturbation begins to induce transition for

the energy value E0(b), whereas the zero-streamwise wave number perturbed flow experi-

ences transition only for an initial disturbance energy of value E0(c). It is worth to notice

that such an initial energy value results in a streak amplitude prior to transition of 28%

of the freestream velocity value, which is close to the threshold amplitude identified by

Andersson et al. (2001) for sinuous breakdown of streaks. Moreover, such results confirm

those by Biau et al. (2008), and point out the effectiveness of suboptimal perturbations

in inducing transition.

In order to generalize the result to larger and more realistic domain lengths, the non-

linear evolution of the near-optimal wave packet discussed in the previous section, has

been investigated. The perturbation has been extracted at five instants of time from a

DNS in which the initial energy is E0 = 0.5. The largest spanwise domain length has

been chosen, i.e. Lz = 180. As in the previous case, the flow has been found to experience

transition for such initial energy when perturbed at t = 0 with the near-optimal wave

packet. Figure 20 (a) shows the perturbation on an x − z plane at t = 0, the angle Θ

which is highlighted in the figure being equal to the one obtained in the optimal case

(Θopt ≈ 4.5). As a reference for the non-linear results, in figure 20 (b) one can observe

the result of the linear evolution, obtained by a linearized DNS, of such a wave packet

at the optimal time tmax = 247. The wave packet seems to be convected downstream

without remarkable structural changes. It is also possible to notice that the angle Θ in


(a)

(b)

Figure 20. Contours of the near-optimal perturbation streamwise velocity component for

βL = 0.035 on an x− z plane for y = 1 at t = 0 (a) and its linear evolution at t = tmax (b).

figures 20 (a) and (b) is unchanged.

Figure 21 (a) provides a snapshot at t = 160 of the non-linear evolution of the near-

optimal packet, which is convected downstream, amplifying itself and saturating its en-

ergy. Indeed, one can notice an enlargement of the positive streamwise velocity structures

in the spanwise direction, whereas their streamwise modulation seems almost unchanged.

At t = 220 (see figure 21 (b)) the streaky structures partially merge, and two kinks ap-

pear at the leading edge of the most amplified streak, affecting the streamwise modulation

of the wave packet. To better analyze such a stage of the streaks breakdown, the wall-

normal and spanwise components of velocity have been also extracted at the same time.

As shown in figure 22 (a), two kinks are observed also in the wall-normal perturbation

velocity, which are alternated in the streamwise direction and symmetric with respect to

the z = 0 axis. On the other hand, an array of spanwise-antisymmetric transversal veloc-


(a)

(b)

(c)

(d)

(e)

Figure 21. Contours of the evolution of the near-optimal perturbation streamwise velocity

component for βL = 0.035 and y = 1 at five instants of time, T = 160 (a), T = 220 (b), T = 250

(c), T = 330 (d) and T = 420 (e).


(a) (b)

(c)

Figure 22. Contours of the wall-normal (a), spanwise (b) and streamwise (c) perturbation

velocity components for βL = 0.035 and y = 1 at T = 220. The dotted line is the z = 0

axis, whereas solid and dashed lines in (c) represent respectively positive and negative spanwise

velocity contours.

ity packets alternated in the longitudinal direction are observed, which are depicted in

figure 22 (b). Such a pattern is similar to that observed in the case of quasi-varicose streak

breakdown (see Brandt et al. (2004)), which is likely to occur when a low-speed streak

interacts with a high velocity one incoming in its front. A similar scenario is recovered

in the present case, as shown in figure 22 (c) for the high and low-speed streaks labeled


as A and B. Looking at figure 22 (c), one could observe that the spanwise perturbations

affect not only the streaks A and B on the z = 0 axis, but also the high-speed ones on the

two sides of the z = 0 axis (labeled as C and D). Indeed, such spanwise perturbations,

although anti-symmetric with respect to the z = 0 axis, could be considered symmetric

with respect to the middle axis of C and D. As a consequence, they are able to induce

spanwise oscillations on such streaks, resulting in a pattern which is typical of quasi-

sinuous streak breakdown (see Brandt et al. (2004)), and is likely to occur when low and

high-speed regions interact on a side. Thus, it can be concluded that both the scenarios

of quasi-sinuous and quasi-varicose breakdown could be identified in the present case due

to the staggered arrangement of the streaks in the spanwise and streamwise directions,

so that both front and side interactions between fast and slow velocity regions take place

simultaneously. As a result, at least four streaks experience breakdown at the same time,

due to the strong inflections of the velocity profiles in their interaction zones, explaining

the efficiency of the global optimal and near-optimal perturbation in inducing transition.

Prior to breakdown, a hairpin vortex is clearly identified in the interaction zone of the

streaks labeled as A, B, C, and D, and is preceded upstream by a pair of quasi-streamwise

vortices, as shown in figure 23. The vortical structures shown in this figure have been

identified by the positive values of the quantity

Q =12(||Ω||2 − ||S||2), (3.1)

Ω being the rate-of-strain tensor and S the vorticity tensor, which is known as the

Q-criterion (Hunt et al. (1988)). As shown in the literature (see Adrian (2007) for a

complete review), at the interior of the hairpin a low momentum region is present, which

corresponds here to the low-speed streak B. The streak breakdown could be thus con-

nected to the presence of an hairpin in the hearth of the wave packet, which manages to


Figure 23. Iso-surfaces of the streamwise velocity, (black represents negative perturbations

whereas dark grey indicates positive perturbations) and of the vortical structures identified by

the Q-criterion (represented by the light grey surfaces) at T = 180.

create new hairpins, as anticipated by the presence of a small arch placed upstream of

the hairpin in figure 23.

Due to streak breakdown, close to the linearly optimal time (at t = 250) the most

amplified elongated structures in the middle of the wave packet have already experienced

transition, cf. figure 21 (c) showing the presence of several subarmonics in both the

streamwise and spanwise direction. As shown in figure 21 (d), the turbulent region spreads

out in the spanwise and streamwise direction leading the nearest streaky structures to

transition. Finally, in figure 21 (e) the perturbation at t = 420 is depicted; it shows

that for a sufficiently large time the linear wave packet has totally disappeared and

the disturbance has taken the form of a localized turbulent spot. In order to investigate


Figure 24. Contours of the perturbation streamwise velocity component for βL = 0.035 and

z = 0 at T = 330.

the physical character of such a structure, the spanwise rate of spread of the spot has

been measured, defined as the angle at its virtual origin between its plane of symmetry

and its mean boundary, identified by using the criterion of the 2% freestream velocity

contours used by Wygnanski et al. (1976). Such an angle is very close to that measured by

Wygnanski et al. (1976) for a boundary layer flow, namely about 9. It is worth to notice

that this value is about twice the optimal inclination of the initial wave packet (Θopt =

4.5), meaning that turbulence spreads out more quickly in the spanwise direction than

in the streamwise one. Moreover, figure 24 shows that the turbulent region presents a

leading edge pointing downstream and a calmed region trailing behind the chaotic zone,

which are both typical features of a turbulent spot (Schubauer & Klebanoff (1955)).

4. Conclusions and Outlook

A direct-adjoint three dimensional optimization procedure has been employed to com-

pute the spatially localized perturbation which is capable to induce the maximum growth

of the disturbance energy in a flat-plate boundary layer. The optimal initial perturba-

tion is characterized by a pair of counter-rotating vortices with a x-modulated streamwise

component opposing to the mean flow, resulting at optimal time in streak-like structures


alternated in the x direction. Such a result has been verified by means of a linearized

global model in which the perturbation is characterized by a single wavenumber in the

spanwise direction. Moreover, such an optimal perturbation has been found to be qual-

itatively similar to the superposition of single-wavenumber local optimal perturbations

computed for different wave numbers in the streamwise direction, meaning that not only

zero streamwise wave number perturbations have a role in the transient dynamics of a

boundary layer.

For sufficiently large domains, the optimal wave packet has been found to maintain

the same modulation in the spanwise and streamwise direction, meaning that the per-

turbation inducing the maximum energy amplification in a boundary layer flow is always

characterized by an array of streak-like structures alternated following a given angle Θopt.

Nevertheless, due to the fact that a single-spanwise-wavenumber perturbation extended

in the whole domain is not easily recovered in a realistic case, a near-optimal localized

perturbation characterized by a large spectrum of frequencies has been looked for. By

means of a partial optimization, a near-optimal three-dimensional perturbation has been

recovered, which is localized also in the spanwise direction, resulting in a wave packet of

elongated disturbances modulated in the spanwise and streamwise direction.

The capability of the localized optimal perturbations to induce transition has been

investigated by means of DNS. In order to allow a comparison with the transition mech-

anisms already known in the literature, the DNS has also been performed considering

a local optimal (with α = 0) and a suboptimal (with α = βL) perturbation, respec-

tively, which have been superposed to the base flow with different energy levels. Such

non-linear computations have shown that the global optimal disturbance is able to in-


duce transition for lower levels of the initial energy than local optimal and suboptimal

perturbations. Moreover, simulations have been carried out of the non-linear evolution

of the near-optimal wave packet, which has been found to result in a turbulent spot

spreading out in the boundary layer. In particular, transition occurs in a localized zone

of the flow close to the center of the packet, by means of a mechanism including features

of both quasi-sinuous and quasi-varicose breakdown. In fact, it has been found that in

this zone the streaks are able to interact on their side as well as on their front, due to

their alternated arrangement, so that more than one streak experience transition at the

same time, explaining the efficiency of the optimal perturbation (as well as the near-

optimal one) in inducing a chaotic behaviour in the considered flow. Moreover, a hairpin

vortex emerges in the region of interaction of the streaks prior to transition, and it is

believed to have a role in the breakdown of the streaky structures in the hearth of the

wave packet. At larger time a turbulent spot spreading out in the boundary layer is found.

The optimal and near-optimal wave packets computed by means of the three-dimensional

direct-adjoint optimization could thus represent the linear precursor of the turbulent spot,

and the transition mechanism investigated here is an optimal path to transition via lo-

calized disturbances. Recent studies (see Nagata (1990, 1997); Waleffe (1998); Faisst &

Eckhardt (2003); Wedin & Kerswell (2004); Eckhardt et al. (2007)) have suggested that

travelling wave packets could play a key role in the description of the paths towards

chaotic motion. In order to verify the existence of a perturbation able to lead the flow

efficiently to a chaotic state by means of some purely non-linear mechanism, a three-

dimensional non-linear global optimization is currently being pursued.

Some computations have been performed on the NEC-SX8 and the IBM Power 6 of

the IDRIS, France. The authors would like to thank R. Verzicco for providing the DNS


code; F. Alizard for providing the code for the global eigenvalue analysis; M. Napolitano

for making possible the cooperation between the laboratories in which this work has been

developed; and C. Pringle for interesting discussions.

Appendix A. Global eigenvalue analysis operators

The global instability analysis is based on the following eigenvalue problem

(A− iωkB) qk = 0, k = 1, . . . , N. (A 1)

where the operators A and B are defined as follows:

B =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 0

, (A 2)

A =

C1 − C2 +∂U

∂x

∂U

∂y0

∂

∂x∂V

∂xC1 − C2 +

∂V

∂y0

∂

∂y

0 0 C1 − C2 iβ

∂

∂x

∂

∂yiβ 0

(A 3)

with the term C1 = U∂/∂x + V ∂/∂y which represents the effect of the advection of the

perturbation by the base flow and C2 =(∂2/∂x2 + ∂2/∂y2 − β2

)/Re which models the

viscous diffusion effects.

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