the optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent ... ·...
TRANSCRIPT
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
4 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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-
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot5
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.
6 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot7
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)
8 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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)
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot9
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)).
10 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot11
(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).
12 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot13
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
14 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
(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-
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot15
(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
16 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
(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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot17
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
18 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
(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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot19
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
20 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot21
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
22 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot23
(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
24 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
(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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot25
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
26 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot27
(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-
28 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
(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.
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot29
(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
30 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot31
(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
32 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot33
(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-
34 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
(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).
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot35
(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
36 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot37
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
38 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot39
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-
40 S. Cherubini, J.-C. Robinet, A. Bottaro and P. De Palma
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
The optimal and near-optimal wave packet in a boundary layer and its ensuing turbulent spot41
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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