On the breakdown of boundary layer streaks

Abstract

A scenario of transition to turbulence likely to occur during the development of
natural disturbances in a flat-plate boundary layer is studied. The perturbations at
the leading edge of the flat plate that show the highest potential for transient energy
amplification consist of streamwise aligned vortices. Due to the lift-up mechanism
these optimal disturbances lead to elongated streamwise streaks downstream, with
significant spanwise modulation. Direct numerical simulations are used to follow the
nonlinear evolution of these streaks and to verify secondary instability calculations.
The theory is based on a linear Floquet expansion and focuses on the temporal,
inviscid instability of these flow structures. The procedure requires integration in the
complex plane, in the coordinate direction normal to the wall, to properly identify
neutral modes belonging to the discrete spectrum. The streak critical amplitude,
beyond which streamwise travelling waves are excited, is about 26% of the free-stream
velocity. The sinuous instability mode (either the fundamental or the subharmonic,
depending on the streak amplitude) represents the most dangerous disturbance.
Varicose waves are more stable, and are characterized by a critical amplitude of about
37%. Stability calculations of streamwise streaks employing the shape assumption,
carried out in a parallel investigation, are compared to the results obtained here
using the nonlinearly modified mean fields; the need to consider a base flow which
includes mean flow modification and harmonics of the fundamental streak is clearly
demonstrated.

Full text

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Study of generation, growth and breakdown of
streamwise streaks in a Blasius boundary layer.
by
Luca Brandt
May 2001
Technical Reports from
Royal Institute of Technology
Department of Mechanics
SE-100 44 Stockholm, Sweden

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Typsatt i AMS-L
A
T
E
X.
Akademisk avhandling som med tillst˚and av Kungliga Tekniska H¨ogskolan i
Stockholm framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie
licentiatexamen onsdagen den 6:e juni 2001 kl 13.15 i sal E3, Huvudbyggnaden,
Kungliga Tekniska H¨ogskolan, Osquars Backe 14, Stockholm.
c
Luca Brandt 2001
Kopiecenter, Stockholm 2001

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Study of generation, growth and breakdown of streamwise
streaks in a Blasius boundary layer.
Luca Brandt 2001
Department of Mechanics, Royal Institute of Technology
SE-100 44 Stockholm, Sweden.
Abstract
Transition from laminar to turbulent flow has been traditionally studied
in terms of exponentially growing eigensolutions to the linearized disturbance
equations. However, experimental findings show that transition may occur also
for parameters combinations such that these eigensolutions are damped. An al-
ternative non-modal growth mechanism has been recently identified, also based
on the linear approximation. This consists of the transient growth of streamwise
elongated disturbances, mainly in the streamwise velocity component, called
streaks. If the streak amplitude reaches a threshold value, secondary instabili-
ties can take place and provoke transition. This scenario is most likely to occur
in boundary layer flows subject to high levels of free-stream turbulence and is
the object of this thesis. Different stages of the process are isolated and studied
with different approaches, considering the boundary layer flow over a flat plate.
The receptivity to free-stream disturbances has been studied through a weakly
non-linear model which allows to disentangle the features involved in the gener-
ation of streaks. It is shown that the non-linear interaction of oblique waves in
the free-stream is able to induce strong streamwise vortices inside the bound-
ary layer, which, in turn, generate streaks by the lift-up effect. The growth of
steady streaks is followed by means of Direct Numerical Simulation. After the
streaks have reached a finite amplitude, they saturate and a new laminar flow,
characterized by a strong spanwise modulation is established. Using Floquet
theory, the instability of these streaks is studied to determine the features of
their breakdown. The streak critical amplitude, beyond which unstable waves
are excited, is 26% of the free-stream velocity. The instability appears as span-
wise (sinuous-type) oscillations of the streak. The late stages of the transition,
originating from this type of secondary instability, are also studied. We found
that the main structures observed during the transition process consist of elon-
gated quasi-streamwise vortices located on the flanks of the low speed streak.
Vortices of alternating sign are overlapping in the streamwise direction in a
staggered pattern.
Descriptors: Fluid mechanics, laminar-turbulent transition, boundary layer
flow, transient growth, streamwise streaks, lift-up effect, receptivity, free-stream
turbulence, nonlinear mechanism, streak instability, secondary instability, Di-
rect Numerical Simulation.

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Preface
This thesis considers the study of the generation, growth and breakdown of
streamwise streaks in a zero pressure gradient boundary layer. The thesis is
based on and contains the following papers
Paper 1. Brandt, L., Henningson, D. S., & Ponziani D. 2001 Weakly
non-linear analysis of boundary layer receptivity to free-stream disturbances.
submitted to Phisics of Fluids .
Paper 2. Andersson, P., Brandt, L., Bottaro, A. & Henningson,
D. S. 2001 On the breakdown of boundary layer streaks. Journal of Fluid
Mechanics,428, pp. 29-60.
Paper 3. Brandt, L. & Henningson, D. S. 2001 Transition of streamwise
streaks in zero pressure gradient boundary layers. To appear in Proceedings of
the second international symposium on Turbulence and Shear Flow Phenomena
(TSFP2 ), Stockholm, June 2001.
The papers are re-set in the present thesis format.

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PREFACE v
Division of work between authors
The Direct Numerical Simulations were performed with a numerical code
already in use mainly for transitional research, developed originally by Anders
Lundbladh and Dan Henningson (DH). It is based on a pseudo-spectral tech-
nique and has been further developed by Luca Brandt (LB) for generating new
inflow conditions and extracting flow quantities needed during the work.
The numerical implementation of the perturbation model presented in Pa-
per 1 was done in collaboration between LB and Donatella Ponziani (DP). The
writing was done by LB and DP with great help from DH.
The DNS data and secondary instability calculations presented in Paper
2 were done by LB, who also collaborated in the writing process. The theory
and the writing was done by Paul Andersson, Alessandro Bottaro and DH.
The DNS in Paper 3 was performed by LB. The writing was done by LB
with help from DH.

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Contents
Preface iv
Chapter 1. Introduction 1
Chapter 2. Transition in zero pressure gradient boundary layers 4
2.1. Natural transition 4
2.2. By–pass transition 5
2.2.1. Receptivity 7
2.2.2. Disturbance growth 8
2.2.3. Breakdown 8
Chapter 3. Conclusions and outlo ok 10
Acknowledgment 14
Bibliography 15
Paper 1. Weakly non-linear analysis of boundary layer receptivity
to free-stream disturbances 21
Paper 2. On the breakdown of boundary layer streaks 51
Paper 3. Transition of streamwise streaks in zero pressure
gradient boundary layers 93
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CHAPTER 1
Introduction
The motion of a fluid is usually defined as laminar or turbulent.Alaminar
flow is an ordered, predictable and layered flow (from Latin “lamina”: layer,
sheet, leaf) as opposed to the chaotic, swirly and fluctuating turbulent flow.
In a laminar flow the velocity gradients and the shear stresses are smaller;
consequently the drag force over the surface of a vehicle is much lower than in
a turbulent flow. One of the major challenges in aircraft design is in fact to
obtain a laminar flow over the wings to reduce the friction in order to save fuel.
On the other hand a turbulent flow provides an excellent mixing in the flow
because of the chaotic motion of the fluid particles, and it is therefore required
in chemical reactors or combustion engines.
In real applications, as the velocity of the fluid or the physical dimension
limiting the flow increase, a laminar motion cannot be sustained; the pertur-
bations inevitably present within the flow are amplified and the flow evolves
into a turbulent state. This phenomenon is called transition.
Transition and its triggering mechanisms are today not fully understood,
even though the first studies on this field dates back to the end of the nine-
teenth century. The very first piece of work is traditionally considered the
classical experiment of Osborne Reynolds in 1883 performed at the hydraulics
laboratory of the Engineering Department at Manchester University. Reynolds
studied the flow inside a glass tube injecting ink at the centerline of the pipe
inlet. If the flow stayed laminar, he could observe a straight colored line inside
the tube. When transition occurred, the straight line became irregular and
the ink diffused all over the pipe section. He found that the value of a non
dimensional parameter, later called Reynolds number, Re =Ur
ν,whereUis
the bulk velocity, rthepiperadiusandνthe kinematic viscosity, governed the
passage from the laminar to the turbulent state. Reynolds stated quite clearly,
however, that there is no a single critical value of the parameter Re,above
which the flow becomes unstable and transition may occur; the whole matter
is much more complicated. He also noted the sensitivity of the transition to
disturbances in the flow before entering the tube.
The knowledge of why, where and how a flow becomes turbulent is of
great practical importance in almost all the application involving flows either
internal or external; at the present state models able to predict transition onset
are available only for simple specific cases. In gas turbines, where a turbulent
free stream is present, the flow inside the boundary layer over the surface of
1

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21. INTRODUCTION
y
v
u
x
wz
U
U∞
Figure 1.1. Boundary layer flow with free-stream velocity
U∞. The velocity has components u,vand win the coordinate
system x,yand z.
a blade is transitional for 50 −80% of the chord length. Wall shear stresses
and heat transfer rates are increased during transition and a correct design of
the thermal and shear loads on the turbine blades must take into account the
features of the transitional process.
The present thesis deals with transition in the simplified case of the bound-
ary layer over a flat plate subject to a uniform oncoming flow. The friction at
the wall will slow down the fluid particles; due to viscosity the velocity of the
flow will vary from the free stream value a distance above the wall (boundary
layer thickness) to zero at the plate surface, with the thickness growing as the
flow evolves downstream, see figure 1.1. This flow is also referred to as Blasius
boundary layer after the scientist who, under certain assumptions, solved the
governing fluid dynamics equations (Navier–Stokes equations) for this particu-
lar configuration. This is probably the most simple configuration, but still helps
us to gain some physical insight in the transition process. It has been in fact
observed that independently of the background disturbances and environment
the flow eventually becomes turbulent further downstream. The background
environment determines, however, the route the transition process will follow
and the location of its onset. Other effects present in real applications such
as curvature of the surface or pressure gradients, which give an accelerating or
decelerating flow outside the boundary layer, will not be considered.
The transition process may be divided into three sages: receptivity, dis-
turbance growth and breakdown. In the receptivity stage the disturbance is
initiated inside the boundary layer. This is the most difficult phase of the full
transition process to predict because it requires the knowledge of the ambient
disturbance environment, which is stochastic in real applications. The main
sources of perturbations are free stream turbulence, free stream vortical dist-
urbances, acoustic waves and surface roughness. Once a small disturbance is
introduced, it may grow or decay according to the stability characteristics of
the flow. Examining the equation for the evolution of the kinetic energy of

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1. INTRODUCTION 3
the perturbation (Reynolds–Orr equation), a strong statement can be made on
the non linear effects: the non linear terms redistribute energy among different
frequencies and scales of the flow but have no net effect on the instantaneous
growth rate of the energy. This implies that linear growth mechanisms are
responsible for the energy of a disturbance of any amplitude to increase (Hen-
ningson 1996). After the perturbation has reached a finite amplitude, it often
saturates and a new, more complicated, laminar flow is established. This new
steady or quasi-steady state is usually unstable; this instability is referred to
as “secondary”, to differentiate it from the “primary” growth mechanism re-
sponsible for the formation of the new unstable flow pattern. It is at this stage
that the final non linear breakdown begins. It is followed by other symmetry
breaking instability and non linear generation of the multitude of scales and
frequencies typical of a turbulent flow. The breakdown stage is usually more
rapid and characterized by larger growth rates of the perturbation compared
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CHAPTER 2
Transition in zero pressure gradient boundary layers
2.1. Natural transition
Historically, the first approach to transition was the analysis of the stability
of a flow. Equations for the evolution of a disturbance, linearized around a
mean velocity profile were first derived by Lord Rayleigh (1880) for an inviscid
flow; later Orr (1907) and Sommerfeld (1908) included the effects of viscosity,
deriving independently what we today call the Orr–Sommerfeld equation. As-
suming a wave–like form of the velocity perturbation and Fourier transforming
the equation, it reduces to an eigenvalue problem for exponentially growing
or decaying disturbances. The first solutions for unstable waves, traveling in
the direction of the flow (two-dimensional waves), were presented by Tollmien
(1929) and Schlichting (1933). The existence of such solutions (TS-waves) was
experimentally proofed by Schubauer & Skramstad in 1947.
About at the same time, Squire’s theorem (1933), stating that two dimen-
sional waves are the first to become unstable, directed the early studies on
stability towards two-dimensional perturbations. The stability of such eigen-
modes of the Orr–Sommerfeld problem depends on their wavelength, frequency,
and on the Reynolds number, defined for a boundary later flow as Re =U∞δ
ν,
with δthe boundary layer thickness. Since δis increasing in the downstream
direction, see figure 1.1, the Reynolds number varies and the TS-waves growth
rate is also function of the downstream position along the plate. The classi-
cal stability theory assumes that the boundary layer has a constant thickness,
the so called paral lel flow assumption, implying that the base flow is char-
acterized only by its streamwise velocity component which varies only in the
wall normal direction. The stability of a disturbance is evaluated for different
Reynolds numbers, mimicking the downstream evolution of the Blasius flow.
This proofed to be a reasonable approximation, even if different models have
been now developed to include the boundary layer growth in the stability cal-
culations (see for example the Parabolized Stability Equations introduced by
Herbert & Bertolotti 1987).
If an amplified Tollmien–Schlichting wave grows above an amplitude in
urms of 1% of the free-stream velocity, the flow become susceptible to sec-
ondary instability. Klebanoff, Tidstrom & Sargent (1962) observed that three-
dimensional perturbations, which are present in any natural flow, were strongly
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2.2. BY–PASS TRANSITION 5
amplified. The three-dimensional structure of the flow was characterized by re-
gions alternating in the spanwise direction of enhanced and diminished pertur-
bation velocity amplitudes, denoted by them “peaks and valleys”. The spanwise
scale of the new pattern was of the same order of the streamwise wavelength
of the TS-waves and the velocity time signal showed the appearance of high
frequency disturbance spikes at the peak position. This transition scenario
was later denoted as K-type after Klebanoff but also fundamental since the
frequency of the secondary, spanwise periodic, fluctuations is the same as the
one of the TS-waves. In the non-linear stages of the K-type scenario, rows of
“Λ-shaped” vortices, aligned in the steamwise directions, have been observed.
An other scenario was also observed, first by Kachanov, Kozlov & Levchenko
(1977). This is denoted N-type after Novosibirsk, where the experiments were
carried out or H-type after Herbert, who performed a theoretical analysis of
the secondary instability of TS-waves (Herbert 1983). In this scenario, the
frequency of the secondary instability mode is half the one of the TS-waves
and, thus, this is also known as subharmonic breakdown. “Λ-shaped” vortices
are present also in this case, but they are arranged in a staggered pattern.
Experiments and computations reveals that the N-type scenario is the first to
be induced, when small three-dimensional perturbations are forced in the flow.
Transition originating from exponentially growing eigenfunctions is usually
called classical or natural transition. This is observed in natural flows only
if the background turbulence is very small; as a rule of thumb it is usually
assumed that natural transition occurs for free-stream turbulence levels less
than 1%. For higher values, the disturbances inside the boundary layer are
large enough that other mechanisms play an important role and the natural
scenario is bypassed.
2.2. By–pass transition
In 1969 Morkovin coined the expression “bypass transition”, noting that “we
can bypass the TS-mechanism altogether”. In fact, experiments reveal that
many flows, including channel and boundary layer flows, may undergo tran-
sition for Reynolds numbers well below the critical ones from linear stability
theory. The first convincing explanation for this was proposed by Ellingsen
& Palm (1975). They considered, in the inviscid case, an initial disturbance
independent of the streamwise coordinate in a shear layer and showed that the
streamwise velocity component may grow linearly in time, producing alternat-
ing low- and high-velocity streaks. Hultgren & Gustavsson (1981) considered
the temporal evolution of a three-dimensional disturbance in a boundary layer
and found that in a viscous flow the initial growth is followed by a viscous
decay (transient growth).
Landahl (1980) proposed a physical explanation for this growth. A wall
normal displacement of a fluid element in a shear layer will cause a perturba-
tion in the streamwise velocity, since the fluid particle will initially retain its
horizontal momentum. It was observed that weak pairs of quasi streamwise

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62. TRANSITION IN BOUNDARY LAYERS
counter rotating vortices are able to lift up fluid with low velocity from the
wall and bring high speed fluid towards the wall, and so they are the most
effective in forcing streamwise oriented streaks of high and low streamwise ve-
locity, alternating in the spanwise direction. This mechanism is denoted lift-up
effect and it is inherently a three-dimensional phenomenon. Some insight in it
may also be gained from the equation for the wall normal vorticity of the per-
turbation (Squire equation), which is proportional to the streamwise velocity
for streamwise independent disturbances. The equation is, in fact, forced by a
term due to the interaction between the spanwise variation of the wall normal
velocity perturbation and the mean shear of the base flow.
From a mathematical point of view, it is now clear that since the linearized
Navier–Stokes operator is non-normal for many flow cases (e.g. shear flows), a
significant transient growth may occur before the subsequent exponential be-
havior (see Schmid & Henningson 2001). Such growth is larger for disturbances
mainly periodic in the spanwise direction, that is with low frequency or stream-
wise wavenumbers; it can exist for sub-critical values of the Reynolds number
and it is the underlying mechanism in bypass transition phenomena.
For real applications, the most interesting case in which disturbances orig-
inating from non-modal growth are responsible for transition, is probably in
the presence of free-stream turbulence. Inside the boundary layer the turbu-
lence is highly damped, but low frequency oscillations, associated with long
streaky structures, appear. As the streaks grow downstream, they breakdown
into regions of intense randomized flow, turbulent spots. The leading edge of
these spots travels at nearly the free-stream velocity, while the trailing edge
at about half of the speed; thus a spot grows in size and merges with other
spots until the flow is completely turbulent. This scenario is usually observed
for higher levels of free-stream turbulence and transition occurs at Reynolds
numbers lower than in case of natural transition.
An other case where transient growth plays an important role is in the
so called oblique transition. In this scenario, streamwise aligned vortices are
generated by non-linear interaction between a pair of oblique waves with equal
angle but opposite sign in the flow direction. These vortices, in turn, induce
streamwise streaks, which may grow over a certain amplitude and become un-
stable, initiating the breakdown to a turbulent flow. Oblique transition has
been studied in detail both numerically and experimentally by Berlin, Wiegel
& Henningson (1999).
Transition to turbulence may, thus, follow different routes, according to
the disturbance environment. In general, as soon as streamwise vortices are
present in the flow, strong streamwise streaks are created, and the breakdown to
turbulence occurs through their growth and breakdown. In this thesis, bypass
transition is analyzed, isolating the different stages of the process and studying
them with different approaches. These will be shortly introduced in the next
sections.

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2.2. BY–PASS TRANSITION 7
2.2.1. Receptivity
The occurrence of streamwise elongated structures in boundary layers subject
to free-stream turbulence was first identified by Klebanoff (1971) in terms of
low frequency oscillations in hot wire signals, caused by slow spanwise oscilla-
tions of the streaks. Kendall (1985) denoted these disturbances as Klebanoff
modes. He also observed streamwise elongated structures with narrow span-
wise scales, with the maximum of the streamwise velocity perturbation located
in the middle of the boundary layer. The appearance of streaks has been iden-
tified as the dominant mechanism in transition in boundary layers subject to
free-stream turbulence (see Matsubara & Alfredsson 2001). However, a num-
ber of important parameters affect the receptivity of the boundary layer; not
only the level of free-stream turbulence, but also its spatial scales, its energy
spectrum, the degree of isotropy and homogeneity play an important role. In
fact, as observed by Westin et al. (1994), different experiments with apparently
similar conditions can disagree on the location and extent of transition.
From a theoretical point of view, Bertolotti (1997) has considered dist-
urbances which are free-stream modes, periodic in all directions. He found
receptivity to modes with zero streamwise wavenumbers and has shown that
the growth is connected to the theories of non modal growth. Andersson,
Berggreen & Henningson (1999) and Luchini (2000) used an optimization tech-
nique to determine which disturbance present at the leading edge will give the
largest disturbance in the boundary layer. They also found a pair of counter ro-
tating streamwise vortices as the most effective in streak’s generation. Besides
these linear models for receptivity, Berlin & Henningson (1999) have proposed
a non-linear mechanism. Numerical simulations have shown that oblique waves
in the free-steam can interact to generate streamwise vortices and subsequent
streaks inside the boundary layer.
Here we develop a theoretical analysis with the aim to isolate the features
involved in the generation of streamwise streaks. We consider a weakly non-
linear model based on a perturbation expansion of the disturbance truncated
at second order. The perturbation equations are derived directly from the
Navier–Stokes equations, superimposing a perturbation field to the base flow,
the Blasius profile. This is assumed of constant thickness so that the problem
may be transformed in Fourier space in the directions parallel to the wall. A
single, decoupled equation is then obtained for each wavenumber pair (α, β)
of the perturbation velocity field, with αand βthe streamwise and spanwise
wavenumber respectively. It is shown that both the problem for the first order
disturbance velocity and for the second order correction are governed by the
Orr-Sommerfeld/Squire operator, the one describing the linear evolution of
three-dimensional disturbances in the flow. The solution for the first order
perturbation velocity field is induced by an external forcing of small amplitude
. At second order, O(2), the problem is forced by non-linear interactions
of first order solutions with wavenumber pairs such that their combination is
equal to the wave-vector of the perturbation we want to study. In particular,

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82. TRANSITION IN BOUNDARY LAYERS
we study the long time response of the system to a couple of oblique modes
oscillating with a given frequency ω. The oblique modes are associated to
(α, ±β) wavenumbers and their quadratic interactions produce perturbations
with (0,±2β) wavenumbers that correspond, in physical terms, to streamwise
elongated structures.
2.2.2. Disturbance growth
Data from different experiments in boundary layers subject to free-stream tur-
bulence show that the growth of the maximum urms of the low frequency
perturbation is initially linear with the Reynolds number based on the local
displacement thickness. Thereafter the streaks reach a maximum amplitude
and then saturate. The linear analysis of Andersson et al. (1999) also shows
that optimal disturbances (streamwise streaks) grow linearly with the down-
stream distance.
Here, Direct Numerical Simulations (DNS), that is numerical solutions of
the governing equations without any simplifying assumptions, are used to fol-
low the non-linear saturation of the optimally growing streaks in a spatially
evolving boundary layer. The complete velocity vector field from the linear re-
sults by Andersson et al. (1999) is used as input close to the leading edge and
the downstream non-linear development is monitored for different initial ampli-
tudes of the perturbation. The numerical code used is described in Lundbladh
et al. (1999); it uses a pseudo-spectral algorithm to solve the three dimen-
sional, time-dependent, incompressible Navier–Stokes equations. In a spectral
method the solution is approximated by an expansion in smooth functions,
trigonometric functions and Chebyshev polynomials in our case. The algo-
rithm is defined pseudo-spectral because the multiplications in the non-linear
terms are performed in physical space, to avoid the evaluation of convolution
sums. The transformation between physical and spectral space can be per-
formed efficiently using Fast Fourier Transforms algorithms and this allows
for efficient implementations of these methods. Due to the fast convergence
rate of the spectral approximation of a function, the spectral methods have
higher accuracy compared to finite-element of finite difference approximations.
However, they are limited to applications in simple geometry.
2.2.3. Breakdown
Very carefully controlled experiments on the breakdown of streaks in chan-
nel flow were conducted by Elofsson, Kawakami & Alfredsson (1999). They
generated elongated streamwise streaky structures by applying wall suction,
and triggered a secondary instability by the use of earphones. They observed
that the growth rate of the secondary instability modes was unaffected by a
change of the Reynolds number of their flow and that the instability appeared
as spanwise (sinuous-type) oscillations of the streaks in cross-stream planes.
To determine the characteristic features of streak breakdown, we study the
temporal, inviscid secondary instability of the saturated streak calculated by

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2.2. BY–PASS TRANSITION 9
means of DNS. With temporal stability we indicate the unstable disturbances
are assumed to grow in time, rather than in a space as it would be more natural
in comparison with laboratory experiments.
The linear secondary stability calculations are carried out on the basis of
the boundary layer approximation, i.e. the mean field to leading order will
consist only of the streamwise velocity component U. Such a mean field varies
on a slow streamwise scale, whereas the secondary instability varies rapidly in
the streamwise direction x, as observed in flow visualizations (see Alfredsson
& Matsubara 1996 as example). Therefore we will assume a parallel mean
flow, with perturbation mode shapes dependent only on the cross-stream coor-
dinates.
The equations governing the stability of the streak are obtained by sub-
stituting U+u,whereu(x, y, z, t)=(u, v, w) is the perturbation velocity and U
is the streaky velocity profile, into the Navier–Stokes equations and dropping
non-linear terms in the perturbation. If viscosity is neglected it is possible to
find an uncoupled equation for the pressure. This is expanded in an infinite
sum of Fourier modes and only perturbation quantities consisting of a single
wave component in the streamwise direction are considered. Since the base flow
is periodic in the spanwise directions, Floquet theory can be used to express
the solution as
p(x, y, z , t)=Real{eiα(x−ct)
∞

k=−∞
ˆpk(y)ei(k+γ)βz},
where αis the real streamwise wavenumber and c=cr+iciis the phase speed.
Here βis the spanwise wavenumber of the primary disturbance field and γis
the (real) Floquet exponent. Because of symmetry of the streaks it is sufficient
to study values of γbetween zero and one half, with γ= 0 corresponding to
afundamental instability mode, and γ=0.5 corresponding to a subharmonic
mode. Here fundamental and subharmonic refer to the spanwise periodicity of
the modes: in the fundamental instability, perturbations have the same period-
icity of the streaks, while in the subharmonic case, the spanwise wavenumber is
half the one of the streaks. The most commonly used definitions of sinuous or
varicose modes of instability are adopted with reference to the visual appear-
ance of the motion of the low-speed streaks; the symmetries of the subharmonic
sinuous/varicose fluctuations of the low-speed streaks are always associated to
staggered (x) varicose/sinuous oscillations of the high-speed streaks.
The following highly non-linear stages of streak’s breakdown are also stud-
ied in this thesis, again using direct numerical simulations. These computa-
tions require now many more points and computer resources since the flow is
approaching a turbulent state and many scales have to been well resolved to
correctly follow the flow evolution. In particular, the performed analysis is
devoted to the identification of the large flow structures characteristic of this
type of transition.

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CHAPTER 3
Conclusions and outlook
In the present work different aspects of the transition process in a Blasius
boundary layer have been isolated and analyzed to identify and better under-
stand the mechanisms involved in transition induced by free-stream turbulence.
As said in the previous chapter, experimental findings show that, in this sce-
nario, the process is characterized by the occurrence and successive breakdown
of streamwise elongated streaks.
We have investigated how free-stream disturbances affect a laminar bound-
ary layer. In particular, we have analyzed the receptivity to oblique waves in
the free-stream and to free-stream turbulence–like disturbances. In both cases,
the generation of strong streamwise velocity component of the disturbance in
streamwise–independent modes is the dominant feature. The underlying mech-
anism can be reduced to a two-step process, first the non linear generation of
streamwise vorticity forced by the interactions between disturbances in the
free-stream and then the linear formation of streaks by the lift–up effect. A
remarkable result is that the importance of the two steps in the process is com-
parable. In fact the analysis of the equations derived applying a weakly non
linear model shows that an amplification of O(Re) occurs both for the genera-
tion of streamwise vortices and, afterwards, for the formation of streaks induced
by the interaction of the wall normal disturbance velocity and the shear of the
Blasius flow.
In the second paper presented in this thesis the non linear downstream
evolution of streaks is followed by means of direct numerical simulations, us-
ing as inflow condition close to the leading edge of the flat plate the optimal
perturbations derived in Andersson et al. (1999) in a linear contest. A new
mean field characterized by a strong spanwise modulation is established and its
secondary stability is studied. The importance of considering a base flow which
includes mean flow modification and harmonics of the fundamental streak is
demonstrated. The streak critical amplitude, defined as one half of the dif-
ference between the highest and lowest streamwise velocity in a cross stream
plane, beyond which disturbances are amplified is about 26% of the free-stream
velocity. This value is larger than the typical ones of 1 −2% characteristic of
the secondary instability of Tollmien-Schlichting waves. The sinuous instability
is clearly the most dangerous one resulting in harmonic spanwise oscillations
of the low speed region.
10

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3. CONCLUSIONS AND OUTLOOK 11
Also using DNS, the breakdown to a turbulent flow resulting from the sin-
uous secondary instability of streaks is studied. The late stages of the process
are investigated and flow structures identified. The main structures observed
during the transition process consist of elongated quasi-streamwise vortices lo-
cated on the flanks of the low speed streak. Vortices of alternating sign are
overlapping in the streamwise direction in a staggered pattern. They are differ-
ent from the case of transition initiated by Tollmien-Schlichting waves and their
secondary instability (see Rist & Fasel 1995 as example) or by-pass transition
initiated by oblique waves (Berlin et al. 1999). In these latter two scenar-
ios Λ-vortices with strong shear layer on top, streamwise vortices deforming
the mean flow and inflectional velocity profiles are observed. The present case
shows analogies with streak instability and breakdown found in the near wall
region of a turbulent boundary layer (see Schoppa & Hussain 1997 or Kawahara
et al. 1998).
In this first half of my graduate studies, some parts of the phenomena
observed and considered relevant in transition in boundary layer subject to
free-stream turbulence have been analyzed in detail using simple flow config-
urations and known repeatable disturbances. In the receptivity study or in
the secondary stability calculations limiting assumptions have also been made,
but still some interesting results, summarized above, are believed to have been
obtained. However, in a recent paper by Jacobs & Durbin (2001), full direct nu-
merical simulations of a transitional Blasius boundary layer under free-stream
turbulence have been presented. The authors write that no evidence of sinu-
ous, or other prefatory streak instability, is observed in their simulations, even
though they appear in a number of flow visualizations (Matsubara & Alfreds-
son 2001 as example). Perturbations enter the boundary layer in form of long
and intense “backward jets” (corresponding to what we call low-speed streaks)
that are lifted in the outer part of the boundary layer. They speculate that
backward jets are a link between free-stream eddies and the boundary layer.
The breakdown is on isolated jets, originating turbulent patches and spots.
Preliminary DNS of laminar-turbulent transition in a boundary layer sub-
ject to free-stream turbulence have been performed using the spectral code also
here at KTH (Schlatter 2001). The next step in this research project will then
be devoted to the detailed study of this transition scenario through extensive
numerical computations. The goal is to try to identify secondary instabilities
or other mechanisms responsible for the formation of turbulent spots. More-
over the influence of the free-stream turbulence intensity and characteristic
length scales on the location of transition will be investigated. Figures 3.1
and 3.2 show some preliminary results. The simulations are reasonably well
resolved, but the limited dimensions of the computational domain may influ-
ence the streak growth and the consequently location and frequency of spot’s
appearance. Anyway strong streamwise streaks, turbulent spots and a turbu-
lent region at the end of the computational domain may be clearly seen; also
periodic oscillations of single streaks before spot formation are identified.

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12 3. CONCLUSIONS AND OUTLOOK
Figure 3.1. Visualization of streaks and the spot formation
showing the streamwise velocity uof the disturbance. Reδ∗
0=
300, x/δ∗
0=[0,900], z/δ∗
0=[−25,25], y/δ∗
0=2.5. Dark
areas indicate lower and light areas higher speed. The velocity
ranges from about −35%,35% of the local mean velocity. The
difference between to frames is 75 non-dimensional time units.

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3. CONCLUSIONS AND OUTLOOK 13
Figure 3.2. Visualization of the spot formation showing the
amplitude of the normal velocity v.Reδ∗
0= 300, x/δ∗
0=
[0,900], z/δ∗
0=[−25,25], y/δ∗
0=2.5. Dark areas indicate
lower and light areas higher speed, whereas grey indicates
zero velocity. The velocity ranges from −28%,28% of the
free-stream velocity. The difference between to frames is 75
non-dimensional time units.

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Acknowledgment
First of all, I want to thank my supervisor, Professor Dan Henningson, for
giving me the opportunity to study and work in the fascinating world of fluid
mechanics. He has always been available and helpful.
I also want to thank all the people I worked with during these years: Paul
Andersson, Alessandro Bottaro, Donatella Ponziani and Philipp Schlatter; I
surely learned from all of them.
People at the Department provided a nice atmosphere and I especially
enjoyed innebandy very much. I also want to name some of my friends and
colleagues in Stockholm: Janne Pralits, Markus H¨ogberg, Martin Skote, Arnim
Br¨uger, Bj¨orn Lindgren, M˚artensson Gustaf, Francois Gurniki, MarcoC, Clau-
dio Altafini and Christophe Duwig.
Grazie a mia moglie Giulia per tanto, ai miei genitori e al piccolo Lorenzo.
14

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Bibliography
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Proc. Transitional Boundary Layers in Aeronautics. (ed. R. A. W. M. Henkes &
J. L. van Ingen), pp. 373-386, Royal Netherlands Academy of Arts and Sciences.
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Andersson, P.,Berggren, M. &Henningson, D. 1999 Optimal disturbances and
bypass transition in boundary layers. Phys. Fluids.11 (1), 134–150.
Berlin, S. &Henningson, D.S. 1999 A nonlinear mechanism for receptivity of
free-stream disturbances Phys. Fluids 11 , (12), 3749–3760.
Berlin, S., Wiegel, M. & Henningson, D. S. 1999 Numerical and experimental
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16 BIBLIOGRAPHY
Kawahara, G., Jim´
enez, J., Uhlmann, M. & Pinelli, A. 1998 The instability
of streaks in near-wall turbulence. Center for Turbulence Research, Annual Re-
search Briefs 1998, 155-170.
Kendall, J. M. 1985 Experimental study of disturbances produced in a pre-
transitional laminar boundary layer by weak free-stream turbulence. AIAA Pa-
per 85-1695.
Klebanoff, P. S. 1971 Effect of free-stream turbulence on the laminar boundary
layer. Bull. Am. Phys. Soc. 10, 1323.
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Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear
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Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer
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Lundbladh, A., Berlin, S., Skote, M., Hildings, C., Choi, J., Kim, J. & Hen-
ningson, D. S. 1999 An efficient spectral method for simulation of incompress-
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tute of Technology, Stockholm, Sweden
Matsubara, M., Alfredsson, P.H. 2001 Disturbance growth in boundary layers
subjected to free stream turbulence J. Fluid Mech.,430, , 149-168.
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11, 57-70.
Reynolds, O. 1883 Phil.Trans.R.Soc.Lond.174, 935-982.
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect
liquid and of a viscous liquid. Part I: A perfect liquid. Part II: A viscous liquid.
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Rist,U.&Fasel,H.1995 Direct numerical simulation of controlled transition in a
flat-plate boundary layer. J. Fluid Mech.,298, 211-248.
Schlatter, P. 2001 Direct numerical simulation of laminar-turbulent transition in
boundary layer subject to free-stream turbulence Diploma Thesis, Department
of Mechanics, Royal Institute of Technology, Stockholm, Sweden
Schlichting, H. 1933 Berechnung der Anfachung kleiner St¨orungen bei der Platten-
str¨omung. ZAMM 13, 171-174.
Schmid,P.J.&Henningson, D.S. 2001 Stability and Transition in Shear Flows.
Springer.
Schoppa, W. & Hussain, F. 1997 Genesis and dynamics of coherent structures in
near-wall turbulence: a new look. In Self-Sustaining Mechanisms of Wall Turbu-
lence. (ed. R. L. Panton), pp. 385-422. Computational Mechanics Publications,
Southampton.
Schubauer,G.B.&Skramstad,H.F. 1947 Laminar boundary layer oscillations
and the stability of laminar flow. J. Aero. Sci. 14, 69-78.
Sommerfeld, A. 1908 Ein Beitrag zur hydrodynamischen Erkl¨arung der turbulenten
Fl¨ussigkeitbewegungen. Atti. del 4. Congr. Internat. dei Mat. III, pp 116-124,
Roma.
Squire, H. B. 1933 On the stability of three-dimensional disturbances of viscous
fluid flow between parallel walls. Proc.Roy.Soc.Lond.Ser.A142, 621-628.

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Tollmien, W. 1929 ¨
Uber die Entstehung der Turbulenz. Nachr. Ges. Wiss.
G¨ottingen 21-24, (English translation NACA TM 609, 1931).
Westin,K.J.A.,Boiko, A. V.,Klingmann, B. G. B.,Kozlov, V. V. &Al-
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193–218.

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Paper 1
1

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Weakly non-linear analysis of boundary layer
receptivity to free-stream disturbances
By Luca Brandt†, Dan S. Henningson†, and Donatella
Ponziani‡1
†Department of Mechanics, Royal Institute of Technology (KTH), S-100 44
Stockholm, Sweden
‡Department of Mechanics and Aeronautics, University of Rome ”La Sapienza”,18
Via Eudossiana, 00184 Rome, Italy
The intent of the present paper is to study the receptivity of a zero pressure
gradient boundary layer to free-stream disturbances with the aim to isolate the
essential features involved in the generation of streamwise streaks. A weakly
non-linear formulation based on a perturbation expansion in the amplitude
of the disturbance truncated at second order is used. It is shown that the
perturbation model provide an efficient tool able to disentangle the sequence of
events in the receptivity process. Two types of solutions are investigated: the
first case amounts to the receptivity to oblique waves oscillating in the free-
stream, the second the receptivity to free-stream turbulence-like disturbances,
represented as a superposition of modes of the continuous spectrum of the
Orr–Sommerfeld and Squire operators. A scaling property of the governing
equations with the Reynolds number is also shown to be valid.
1. Introduction
The objective of the present work is the study of the stability of the boundary
layer subjected to free-stream disturbances. From a theoretical point of view,
boundary layer stability has mainly been analyzed in terms of the eigensolutions
of the Orr-Sommerfeld, Squire equations that reduces the study to exponen-
tially growing disturbances. Experimental findings show that transition is also
characterized by the occurrence of streamwise elongated structures which are
very different from the exponentially growing perturbations. These streamwise
structures (or streaks) were first identified by Klebanoff (1971) in terms of low
frequency oscillations in hot wire signals caused by low spanwise oscillations of
the streaks (Kendall 1985; Westin et al. 1994) and are commonly referred to
as Klebanoff-modes.
Further analysis of the Orr-Sommerfeld, Squire equations (Gustavsson
1991; Butler & Farrel 1992; Reddy & Henningson 1993; Trefethen et al. 1993)
1Authors listed in alphab etical order
21

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22 L. Brandt, D.S. Henningson & D. Ponziani
have confirmed that disturbances other than exponentially growing perturba-
tions may lead to boundary layer instability. From a mathematical point of
view this is due to the non–normality of the Orr-Sommerfeld, Squire operator.
The physical mechanism behind this linear mechanism is the lift-up induced by
streamwise vortices that interact with the boundary layer shear thus generating
streaks in the streamwise velocity component. Transition due to these types of
disturbances is generally called by-pass transition.
The understanding and prediction of transition require the knowledge of
how a disturbance can enter and interact with the boundary layer, commonly
referred to as receptivity of the boundary layer. The disturbances are often
characterized as either acoustic or vortical disturbances convected by the free-
stream. Both types of disturbances have been investigated by asymptotic meth-
ods and a summary of the results can be found in the reviews by Goldstein &
Hultgren (1989) and Kerschen (1990). Bertolotti (1997) has assumed distur-
bances which are free-stream modes, periodic in all directions, and has studied
the boundary layer receptivity in a “linear region” excluding the leading edge.
He has found receptivity to modes with zero streamwise wavenumber and have
shown that the growth is most likely connected to the theories of non-modal
growth. To answer the question of which disturbance present at the leading
edge gives the largest disturbance in the boundary layer at a certain down-
stream position, Andersson et al. (1999) and Luchini (2000) have used an
optimization technique adapted from optimal-control theory. The disturbances
they found were also streamwise vortices that caused the growth of streaks, and
both the wall normal disturbance shape and growth rates agreed well with the
findings of Bertolotti (1997).
Berlin & Henningson (1999) have carried out numerical experiments on
how simple vortical free-stream disturbances interact with a laminar boundary
layer, and have identified a linear and a new non-linear receptivity mechanism.
The non-linear one was found to force streaks inside the boundary layer similar
to those found in experiments on free-stream turbulence and it worked equally
well for streamwise and oblique free-stream disturbances. The boundary layer
response caused by the non-linear mechanism was, depending on the initial
disturbance energy, comparable to that of the linear mechanism, which was
only efficient for streamwise disturbances.
In the present work we develop a theoretical analysis with the aim to isolate
the features involved in the generation of streamwise streaks in flows subjected
to free-stream turbulence. We consider a weakly non-linear model based on
a perturbation expansion in terms of the amplitude of the disturbance, trun-
cated at second order. The model, originally developed in a previous work
for the case of Poiseuille flow (Ponziani 2000; Ponziani et al. 2000), is here
extended to boundary layer flows. This implies the inclusion of the continuous
spectrum eigenfunctions in the representation of the first and the second order
solutions. To validate the model we first investigate a receptivity mechanisms
in a boundary layer imposing a localized disturbance both in the boundary

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Boundary layer receptivity to free-stream disturbances 23
layer and in the free-stream. In particular, we study the long time response
of the system to a couple of oblique modes oscillating with a given frequency
ω. For this case the linearized stability equations are driven at first order
by the external disturbance and at second order by the quadratic interactions
between first order terms. For the type of disturbance considered, the pres-
ence of oblique waves generates streamwise vortices which, in turn, induce
the formation of streaks inside the boundary layer. The oblique modes are
associated to (α, ±β) wavenumbers and their quadratic interactions produce
(0,2β) wavenumbers that correspond, in physical terms, to an elongated vorti-
cal structure, i.e. streamwise counter-rotating vortices. The results show that
the generation of streamwise vorticity, which is a non-linear mechanism, and
its subsequent lift-up can indeed be recovered through the weakly non-linear
formulation. The theory is validated through comparison with direct numerical
simulations.
The model is also applied to investigate the response of the boundary layer
to continuous spectrum modes. The latter are fundamental for the understand-
ing of the interaction between free-stream vortical eddies and the boundary
layer since they reduce to simple sines and cosines in the free-stream and can
easily be used to represent a free-stream turbulence spectrum. By using con-
tinuous modes, which are solutions of the linear problem, the model reduces to
solve a second order equation where the forcing is given by the weakly non-linear
interactions between continuous modes. An extensive parametric study is car-
ried out to analyze the interaction between Orr-Sommerfeld as well as Squire
modes, in particular considering the effect of the disturbance wavenumbers. A
scaling property of the resolvent of the Orr–Sommerfeld and Squire problem
with the Reynolds number is shown to be valid for the results obtained.
2. The perturbation model
In the following we define the streamwise, wall normal and spanwise directions
as x, y and z, respectively, with velocity perturbation u=(u, v, w). All vari-
ables are made dimensionless with respect to the constant displacement thick-
ness δ∗
0and the free-stream velocity U∞(time is made non dimensional with
respect to δ∗
0/U∞). The perturbation equations are derived directly from the
Navier-Stokes equations where we have superimposed a perturbation field to
the base flow, namely the Blasius profile. In order to impose periodic boundary
conditions in the directions parallel to the wall we assume a parallel base flow
and we consider no-slip boundary conditions and solenoidal initial conditions.
We study the evolution of a disturbance in a boundary layer over a flat
plate via perturbation theory by expanding the relevant variables in terms of
the amplitude of the disturbance 
u=u(0) +u
(1) +2u(2) +....
(1)
p=p(0) +p
(1) +2p(2) +....

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24 L. Brandt, D.S. Henningson & D. Ponziani
where u(0),p
(0) is the given base flow, while remaining terms are unknowns
to be determined by the perturbation analysis. We consider a general case in
which the perturbation equations are forced by an external forcing
F(x, y, z , t)=F(1)(x, y, z, t),
with a given initial condition.
Substituting the expansion (1), truncated at second order, into the Navier-
Stokes equations and collecting terms of like powers in , one obtains the gov-
erning equations for the first and the second order. These equations, can be
rewritten in the normal velocity v(j), normal vorticity η(j)(j=1,2) formula-
tion, thus obtaining the following Orr-Sommerfeld, Squire system
[( ∂
∂t +u(0) ∂
∂x)∆ −D2u(0) ∂
∂x −1
Re ∆2]v(j)=N(j)
v+F(j)
v(2)
[∂
∂t +u(0) ∂
∂x −1
Re ∆]η(j)+Du(0) ∂v
(j)
∂z =N(j)
η+F(j)
η(3)
where
N(j)
v=−[( ∂2
∂x
2+∂2
∂z
2)S(j)
2−∂2
∂x∂yS(j)
1−∂2
∂y∂zS(j)
3](4)
N(j)
η=−(∂
∂zS(j)
1−∂
∂xS(j)
3)(5)
with
S(j)
1=∂
∂xu(j−1)u(j−1) +∂
∂yu(j−1)v(j−1) +∂
∂zu(j−1)w(j−1) (6)
S(j)
2=∂
∂xu(j−1)v(j−1) +∂
∂yv(j−1)v(j−1) +∂
∂zv(j−1)w(j−1) (7)
S(j)
3=∂
∂xu(j−1)w(j−1) +∂
∂yv(j−1)w(j−1) +∂
∂zw(j−1)w(j−1) (8)
The first and second order equations have constant coefficients with respect to
the streamwise and spanwise directions, hence we consider the Fourier trans-
form in the (x, z) plane by making the following form assumption for the solu-
tion q(j)=(v(j),η
(j))T
q(j)(x, y, z , t)=
m
n
ˆq(j)
mn(y, t)ei(αmx+βnz)
and likewise for the external forcing. The wave numbers are defined as follows
αm=m2π/Lx(9)
βn=n2π/Lz(10)
k2
mn =α2
m+β2
n(11)
where Lxand Lzare, respectively, the streamwise and spanwise lengths of the
periodic domain. Hereafter, for reading convenience, the subscript mand nare

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Boundary layer receptivity to free-stream disturbances 25
omitted and we refer to the equations for the individual wave number (αm,β
n)
as (α, β). The resulting equations in matrix form read
(∂
∂t ˆ
M−ˆ
L)ˆq(1) =ˆ
Pˆ
F, (12)
(∂
∂t ˆ
M−ˆ
L)ˆq(2) =ˆ
P
k+p=m
l+q=n
[ˆ
N(ˆu(1)
kl ˆu(1)T
pq )]
T(13)
where
ˆ
L=
LOS 0
CL
SQ 
,ˆ
M=

k2−D20
01

,
ˆ
N=

iα
D
iβ 
,ˆ
P=
−iαD k
2−iβD
−iβ 0iα 
.
ˆ
Lis the linear operator that defines the classical Orr-Sommerfeld, Squire prob-
lem
LOS =iαu
(0)(D2−k2)−iαD
2u(0) −1
Re(D2−k2)2
C=−iβDu
(0) (14)
LSQ =−iαu
(0) +1
Re(D2−k2).
In this investigation we consider two different types of solutions. In the first
case the system of equations at first order is forced by an external force that
we assume pulsating with a given frequency ω
ˆ
F(1) =ˆ
f(y)eiωt +ˆ
f∗(y)e−iωt ,
where the ∗indicates the complex conjugate. At second order, the problem is
forced by the non-linear interactions of first order terms
ˆ
T=ˆ
P
k+p=¯m
l+q=¯n
[N(ˆu(1)
kl ˆu(1)T
pq )]
T.(15)
In the second case, we assume that the solution for the first order is given by a
continuous spectrum mode representation, and we solve only the second order
problem.
The initial conditions are
ˆq(j)(t=0)=ˆq0,j=1,2.(16)
With regard to the boundary conditions we enforce no-slip conditions
ˆv(j)=Dˆv(j)=ˆη(j)= 0 (17)

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26 L. Brandt, D.S. Henningson & D. Ponziani
while we assume boundedness in the free-stream. The remaining velocities are
recovered by
ˆu(j)=i
k2(αDˆv(j)−βˆη(j)) (18)
ˆw(j)=i
k2(βDˆv(j)+αˆη(j)) (19)
2.1. The solution to the forced problem
We like here to consider the harmonically forced problem described by the sys-
tem of Eqs. (12), (13) whose solution can be split into two parts (see Ponziani
et al. 2000): one representing the long time asymptotic solution ˆq(j)Land the
other describing the initial transient behavior ˆq(j)T,
ˆq(j)=ˆq(j)T+ˆq(j)L,j=1,2.(20)
First we consider the equations for the long time behavior at first order
(±iω ˆ
M−ˆ
L)ˆq(1)L=ˆ
Pˆ
F, (21)
ˆv(1)L=Dˆv(1)L=ˆη(1)L=0,y=0,y=y∞
whose long time response to the harmonic forcing is given by
ˆq(1)L
±ω=(±iω ˆ
M−ˆ
L)−1ˆ
Pˆ
f(y)e±iωt
.(22)
The equations that describe the transient at first order are given by
∂
∂t ˆ
Mˆq(1)T=ˆ
Lˆq(1)T
ˆq(1)T=−ˆq(1)L,t= 0 (23)
ˆv(1)T=Dˆv(1)T=η(1)T=0,y=0,y=y∞
Equations (21), (23) provide a complete description of the harmonic forced
linear problem; the solution of Eq. (23) is obtained as described in a later
section.
With regard to the second order solution, the structure of the quadratic in-
teraction term implies that several frequency components are excited at second
order. As for the first order problem we can split the governing equations into
two parts that describe the long time and the transient behavior. With regard
to the former, we point out that at first order the asymptotic solution in time is
characterized by given frequencies ±ω, which implies that only the zero and 2ω
frequency components are excited at second order. However, as demonstrated
by Trefethen et al. (1993), the maximum response of a system occurs for α=0
and ω= 0; hence we reduce our analysis to the most effective part, that is the
one associated to zero frequency and zero streamwise wavenumber
−ˆ
Lˆq(2)L
0=ˆ
TL
0(24)

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Boundary layer receptivity to free-stream disturbances 27
Here the terms ˆ
TL
0represent the convolution sum in (15) where only the contri-
bution with zero frequency is considered. Observe that this procedure can also
be applied to the solution corresponding to the continuous spectrum modes.
Indeed, if a first order solution is represented as a continuous spectrum mode,
it is still characterized by a given frequency that corresponds to the real part
of the associated eigenvalue.
The equations that describe the transient behavior at second order accounts
for different forcing terms that arise from the self interactions between first
order transient solutions and the quadratic interactions between the transient
solution and the long time solutions.
2.2. Scaling of forced solution
For the forced problem it is possible to show a Reynolds number dependence
for the norm of the resolvent. Let us introduce a new set of variables to rescale
the Orr-Sommerfeld, Squire problem as in Gustavsson (1991); Reddy & Hen-
ningson (1993); Kreiss et al. (1994)
t∗=t/Re, s∗=sRe, ˆv∗=ˆv/ β Re, ˆη∗=ˆη; (25)
with the new scaling we can rewrite (14) as
L∗
OS =iαReu
(0)(D2−k2)−iαReD
2u(0) −(D2−k2)2
C∗=−iDu
(0) (26)
L∗
SQ =−iαReu
(0) +(D2−k2).
The scaled equations exhibit a dependency only on the two parameters, αRe
and k2, rather than α, β, Re as in the original Orr-Sommerfeld, Squire equa-
tions. In the new variables the resolvent can be written as
(sI −ˆ
L)−1E=Re(s∗I−ˆ
L∗E∗(27)
where Eis the energy norm with respect to the original variables and E∗is the
energy norm with respect to the scaled ones. It is possible to show, see Kreiss et
al. (1994), that for αRe = 0 (that corresponds to the maximum response of the
system) the norm of the resolvent (s∗I−ˆ
L∗E∗scales as the Reynolds number.
Hence, Eq. (27) implies that the norm of the original resolvent (sI −ˆ
L)−1E
scales as the square of the Reynolds number.
Further, it is possible to show (see e.g. Kreiss et al. 1994) that if we
consider Reynolds number independent forcing the amplitude of the response
in the original unscaled problem is O(Re)forvand O(Re2)forη.
2.3. The initial value problem
In accounting for the transient solution it is worth making some observa-
tions. Since the eigenfunctions of the eigenvalue problem associated to the
Orr-Sommerfeld, Squire system form a complete set, see DiPrima (1969)
and Salwen & Grosch (1981), we can expand the perturbation solution ˆq(i)T
(i=1,2) as a superposition of modes. For Blasius boundary layer flow, the

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28 L. Brandt, D.S. Henningson & D. Ponziani
domain is semi-bounded and the spectrum has a continuous and a discrete
part, see Grosch & Salwen (1978). These authors have shown that in this
case the solution can be expanded in a sum over the discrete modes and in
an integration over the continuous spectrum. This analysis can be simplified
using a discrete representation of the continuous spectrum by cutting the upper
unbounded domain at a given y∞. Although the eigenvalues differ from the
exact representation of the continuous spectrum, particularly as the decay rate
increases, their sum has been found to describe correctly the solution to the
initial value problem, see Butler & Farrel (1992). Observe that formally, it is
possible to expand the solution using integrals over the continuous spectrum.
However, the added computational complexity, without any significant gain in
accuracy, justifies the use of the present simpler formulation.
With regard to the selection of a set of functions that are orthogonal to
the set of Orr-Sommerfeld, Squire eigenfunctions, we exploit the orthogonality
relation between the eigenfunctions of the Orr-Sommerfeld, Squire system (˜q)
and those of the adjoint Orr-Sommerfeld, Squire problem (˜q+). From the
definition of adjoint, it is easy to show that the eigenvalues of the adjoint are
the complex conjugate to the eigenvalue of the of the Orr-Sommerfeld, Squire
system. This leads to the orthogonality condition
(ˆ
M˜qj,˜q+
k)=Cδ
jk (28)
where δjk is the Kronecker symbol and C a constant that normalizes the eigen-
functions and that needs to be determined. Hence, for the initial value problem,
we can exploit the completeness of the Orr-Sommerfeld, Squire eigenmodes for
bounded flows to recover ˆq(i)T
ˆv(i)T
ˆη(i)T=
l
Kl˜vl
˜ηP
le−iλ
OS
lt+
j
Bj0
˜ηje−iλ
Sq
jt(29)
where (λOS
l,˜vl)and(λSq
j,˜ηj), respectively, are the eigenvalues and eigenvectors
of the non-normal LOS operator, and the homogeneous LSQ operator and ˜ηP
l
is the solution of the Squire problem forced by the Orr-Sommerfeld eigenfunc-
tions. The coefficient Kland Bjare determined from a given initial condition
(ˆv0,ˆη0) according to (28)
Kl=1
2k2y∞
0˜
ξl
0Hk2−D20
01
ˆv0
ˆη0dy (30)
Bj=1
2k2y∞
0˜
ξP
j
˜
ζjHk2−D20
01
ˆv0
ˆη0dy (31)

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Boundary layer receptivity to free-stream disturbances 29
where
˜
ξ
0,˜
ξP
˜
ζ.(32)
are the modes of the adjoint system, see Schmid & Henningson (2001).
2.4. Continuous spectrum modes
The Orr-Sommerfeld eigenvalue problem in a semi-bounded domain is charac-
terized by a continuous and a discrete spectrum. The discrete modes decay
exponentially with the distance from the wall, while the modes of the continu-
ous spectrum are nearly sinusoidal, whereby the free-stream disturbances can
be expanded as a superposition of continuous modes. Since they are associ-
ated to stable eigenvalues, they are not relevant for the classical linear stability
analysis; however they are fundamental for the understanding of the interac-
tion between free-stream vortical eddies and the boundary layer. In order to
determine the eigenfunctions of the continuous spectrum we consider first the
Orr-Sommerfeld equations for a small 3-D disturbance with no-slip boundary
conditions at the wall ˜v(0) = D˜v(0) = 0 and boundedness at y→∞.Inpar-
ticular in the free-stream the mean flow is constant (i.e. u(0) =1asy/δ∗>3)
and the Orr-Sommerfeld equation reduces to
(D2−k2)2˜v−iαRe{(1 −c)(D2−k2)}˜v= 0 (33)
where cis the phase velocity. The above equation admits the following solution,
see Grosch & Salwen (1978),
˜v=Ae
iγy +Be
−iγy +Ce
−ky,y→∞ (34)
where
k2+γ2+iαRe(1 −c)=0.
From this, an analytical expression for the eigenvalues is derived
c=1−i(1 + γ2
k2)k2
αRe (35)
where γrepresents the wave number in the wall-normal direction and assumes
any positive real value.
From a numerical point of view the crucial point is to enforce the bound-
edness of the eigenfunctions at y→∞. We follow the method introduced
by Jacobs & Durbin (1998) to recover the correct behavior of the solution in
the free-stream solving the equation as a two-point boundary value problem
using the spectral collocation method based on Chebyshev polynomial.
We need a total of four boundary conditions: the first two are the no slip at
the wall. The arbitrary normalization is ˜v(y∞)=1,wherey∞is the maximum
value o f yin the wall-normal direction. The condition of boundedness as y→∞

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30 L. Brandt, D.S. Henningson & D. Ponziani
is converted to a numerical condition at two specific values of y. In fact Eq. (34)
implies
D2˜v+γ2˜v=C(k2+γ2)e−ky (36)
in the free-stream. The missing boundary condition is derived evaluating rela-
tion (36) at two different point in the free-stream y1,y
2
(D2˜v+γ2˜v)y1
(D2˜v+γ2˜v)y2
=ek(y2−y1).(37)
A similar procedure is used to determine the continuous modes of the Squire
equation. However in this case the free-stream behavior of the solution is given
only by the two complex exponentials. Hence, from a numerical point of view
it suffices to enforce the arbitrary normalization condition ˜η(y∞)=1.
2.5. The numerical method
The temporal eigenvalue systems and the forced problems derived in the previ-
ous sections are solved numerically using a spectral collocation method based
on Chebychev polynomials. In particular, we consider the truncated Chebychev
expansion
φ(η)=
N

n=0
¯
φnTn(η),
where
Tn(η)=cos(narccos(η)) (38)
is the Chebychev polynomial of degree ndefined in the interval −1≤η≤1,
and the discretization points are the Gauss–Lobatto collocation points,
ηj=cosπj
N,j=0,1,... ,N,
that is, the extrema of the Nth-order Chebyshev polynomial TNplus the
endpoints of the interval. The calculations are performed using at least 301
Chebyshev collocation points in y. The wall-normal domain varies in the range
(0,y
∞), with y∞well outside the boundary layer (typically y∞= 50). The
Chebyshev interval −1≤η≤1 is transformed into the computational domain
0≤y≤y∞by the use of the mapping
y=y∞
1−η
2.(39)
The unknown functions ˆq=ˆq(y) are then approximated by
ˆqN(y)=
N

n=0
¯qnTn(η),
The Chebyshev coefficients ¯qn,n=0, ... ,Nare determined by requiring
the different equations derived from (12),(13) to hold for ˆqNat the collocation
points yj,j=p,... ,N−p,withp= 2 for the fourth order Orr–Sommerfeld

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Boundary layer receptivity to free-stream disturbances 31
equation and p= 1 for the second order Squire equation. The boundary con-
ditions are enforced by adding the equations
N

n=0
¯qnTn(0) =
N

n=0
¯qnTn(y∞)=0,
and the two additional conditions for the Orr–Sommerfeld problem
N

n=0
¯qnDTn(0) =
N

n=0
¯qnDTn(y∞)=0,
where DTndenotes the y-derivative of the n-th Chebyshev polynomial.
3. Receptivity to localized forcing
3.1. Disturbance generation and parameter settings
In order to trigger the formation of streamwise streaks in the boundary layer
we consider the response of the system to a couple of oblique waves. This is
similar to the investigations of Berlin & Henningson (1999), although here we
are able to understand the mechanism in more detail since we use an analytical
formulation. In this section we also compare our analytical results to direct
numerical simulations of the type presented by Berlin & Henningson (1999).
The oblique waves are generated by an harmonic localized wall–normal volume
force given by
F=f(y)cos(αx)cos(βz)ejωt (40)
with
f(y)= 1
√2πσe−(y−y0)2
2σ2
In our computations we chose Re = 400, and (α, ±β)=(0.2,±0.2). We analyze
two cases: for the first one the forcing is in the boundary layer (y0=2.2), for
the second the forcing is in the free-stream (y0=8.). The results presented
here correspond to the latter case with σ=0.5.
The formation of streamwise streaks in the boundary layer is initiated by
two oblique waves characterized by wave numbers (α, ±β)=(0.2,±0.2). In
the linear long time response of the system to the external forcing, there is no
evidence of streaks generation. However, if one accounts for the second order
interactions (in particular those that force the wave number (0,2β)) it is easy
to observe that the second order correction corresponds to a system of strong
streamwise longitudinal vortices in the boundary layer. These results are in
agreement with the work of Berlin & Henningson (1999) where the generation
of streaks in the boundary layer is triggered by the non-linear evolution of two
oblique waves.

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32 L. Brandt, D.S. Henningson & D. Ponziani
0 2 4 6 8 10 12 14 16 18 20
−1
−0.5
0
0.5
1
0 2 4 6 8 10 12 14 16 18 20
−2
−1
0
1
0 2 4 6 8 10 12 14 16 18 20
−1
−0.5
0
0.5
1
ˆv(1)
ˆu(1)
ˆw(1)
y
Figure 1. Velocity components of the linear perturbation ve-
locity for the oblique wave with (α, β )=(0.2,0.2). Forced
problem for Re = 400, y∞= 20, y0=8,σ=0.5, ω=0.2
and t= 100. DNS result: + real part, ✁imaginary part.
Perturbation model: - - -, real part; —–, imaginary part .
3.2. Comparison to DNS data: linear and non-linear case
In order to validate the perturbation model and its capability to select the
most effective interactions as a second order correction, we compare our results
with direct numerical simulations of the forced evolution problem and an initial
value problem. The DNS code, reported in Lundbladh et al. (1999), is used to
solve the temporal problem for a parallel Blasius base flow. For a quantitative
comparison we analyze the DNS results in terms of an amplitude expansion, so
as to isolate the linear, quadratic and cubic part of the solution, see Henningson
et al. (1993). We in fact run the same case with three different small amplitude
disturbances. Different Fourier modes are then extracted and compared with
the results obtained using the perturbation model.
We first consider the velocity field at early times, where the problem is
governed by Eq. (23) and the initial value problem is solved as a superposition
of the discretized eigenmodes. Comparisons of the three velocity component
for the Fourier mode (α, β)=(0.2,0.2) are shown in Fig. 1 at time t= 100.
The good agreement confirms the validity of the discrete representation of the
continuous spectrum.
With regards to the asymptotic time behavior, the numerical simulations
are run to time t= 50000 and the response is compared with the perturba-
tion results. Figure 2 depicts the velocity components associated to the mode

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Boundary layer receptivity to free-stream disturbances 33
0 5 10 15 20 25
−20
−10
0
10
0 5 10 15 20 25
−30
−20
−10
0
10
0 5 10 15 20 25
−20
−10
0
10
ˆv(1)
ˆu(1)
ˆw(1)
y
Figure 2. Velocity components of the linear perturbation ve-
locity for the oblique wave with (α, β )=(0.2,0.2). Forced
problem for Re = 400, y∞= 25, y0=8,σ=0.5, ω=0.2
and t= 50000. DNS result: - - -, real part; —–, imaginary
part (thick lines). Perturbation model: - - - real part; —–,
imaginary part (thin lines).
(α, β)=(0.2,0.2). The small differences observed in the figures are probably
due to the accumulation of truncation errors in the DNS after such a long time
integration. For the same problem, the second order correction with (0,2β)and
ω= 0 is displayed in Fig. 3. The formation of streamwise streaks in the longi-
tudinal component is clearly seen. A similar result (not reported) is observed
in the case of localized forcing inside the boundary layer (y0=2.2, σ=0.4).
However, in the latter case, the streaks exhibit an amplitude smaller than the
previous case (about one third).
Let us consider the transient part of the solution. The time evolution of the
energy of the response of the forced problems corresponding to two different
wave numbers ((0.2,0.2), (1., 1.)) and for the same values of y0and σ(y0=8,
σ=0.5) is shown in Fig. 4. The figure shows that the energy of the high wave
number disturbances attains its asymptotic value on a scale that is one order
of magnitude less than the one associated to the short wave number (both for
the first and the second order corrections).

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34 L. Brandt, D.S. Henningson & D. Ponziani
0 5 10 15 20 25
−2
0
2
4
6x 106
0 5 10 15 20 25
−2
0
2
4x 105
0 5 10 15 20 25
−2
0
2
4x 105
ˆv(2)
ˆu(2)
ˆw(2)
y
Figure 3. Velocity components of the second order pertur-
bation velocity for (α, β)=(0,0.4). Forced problem for
Re = 400, y∞= 25, y0=8,σ=0.5andt= 50000. DNS
result: - - -, real part; —–, imaginary part (thick lines). Per-
turbation model: - - -, real part; —–, imaginary part (thin
lines).
4. Role of Continuous Spectra in the Receptivity Mechanism
4.1. Reynolds number scaling
As shown in section 2.1, it is possible to prove that the resolvent, which gov-
erns the solution to the forced Orr-Sommerfeld, Squire system, is O(Re2). For
the second problem we address, the forcing is given by non-linear interactions
between continuous spectrum modes, and we analyze the forced solution at
different values of the Reynolds number. In Fig. 5 we compare the continu-
ous spectrum modes of the Orr-Sommerfeld operator for α=1,β=0.2and
γ=0.628 at two different Reynolds numbers (Re = 300 and Re = 500). We
observe that the modes differ only at the edge of the boundary layer and these
differences are small. Hence, we assume that the forcing term is Re indepen-
dent. This assumption is confirmed for large Reynolds number by the findings
of Jacobs & Durbin (1998), who have shown that the penetration depth of the
modes is proportional to (αRe)−0.13 which implies that for large values of Re
this depth becomes smaller and smaller.
In Fig. 6 we report the plot of the maximum of the streamwise second order
velocity (normalized by the first order energy and by the square of Re)asa
function of the streamwise and spanwise wave numbers for a given value of γ,

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Boundary layer receptivity to free-stream disturbances 35
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
0
0.2
0.4
0.6
0.8
1
E(α,β)
E∞(α,β)
E(0,2β)
E∞(0,2β)
a)
b)
t
Figure 4. Time evolution of the energy normalized with re-
spect to its asymptotic value. a) first order correction; b),
second order correction; —–, (α, β)=(1,1);---,(α, β )=
(0.2,0.2).
note that the second order distribution is displayed with reference to the (α,
β) values of the corresponding first order terms. The figure clearly confirms
the scaling in the energy norm found in section 2.1. The results show that the
maximum amplitude is obtained for α≈2andβ≈0.15. Further computations
(not reported) verify the validity of the scaling down to Re ≈100. However,
for lower values of the Reynolds number the maximum response is obtained
for values of βless than the one associated to Re > 100. Thus, it suffices to
investigate the forced results only at one Reynolds number, since they can be
subsequently scaled to arbitrary Re > 100.
4.2. Non-linear interaction and linear forcing
In a previous work it has been demonstrated that nonlinearities play a fun-
damental role in boundary layer receptivity, see Berlin & Henningson (1999).
In the present work we use a model based on a perturbation expansion in
the amplitude of the disturbance truncated at second order to single out the
mechanisms at work during the generation of streamwise streaks in flat plate
boundary layers subject to free-stream turbulence.
Since it is possible to rather well represent free-stream turbulence as a
superposition of modes associated to the continuous spectrum, see for exam-
ple Jacobs & Durbin (2000), we simplify the problem analyzing the weakly

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36 L. Brandt, D.S. Henningson & D. Ponziani
0 5 10 15 20 25 30 35 40 45 50
−1.5
−1
−0.5
0
0.5
1
1.5
˜v
y
Figure 5. Distribution of the Orr-Sommerfeld eigenfunction
vs y for α=1,β=0.2: —–, ˜vfor Re = 500; - - -, ˜v
for Re = 300; ···,Re(˜v)forRe = 300; ·-·,Im(˜v)for
Re = 300.
non-linear response of the system to a single pair of oblique continuous spec-
trum modes.
First we analyze the problem associated to the Orr-Sommerfeld continuous
spectrum modes. The first order solution corresponds to two eigenmodes of
the continuous spectrum with wave numbers (α, β, γ )and(α, −β,γ)whose
damping rate is set to zero. At second order we account for the quadratic
interactions between the two continuous spectrum modes and we focus our
attention on the (0,2β) contribution. The transient part of the solution is
neglected and the asymptotic time response is analyzed. We observe that the
second order forcing to the (0,2β) modes induces strong streamwise vorticity,
that in turn forces the formation of streaks inside the boundary layer by the
linear lift-up mechanism. This two-step process, first the non-linear generation
of streamwise vortices and then the linear forcing of the streamwise streaks,
is completely captured by the weakly non-linear model. Figure 7 shows that
the non-linear forcing of the Orr-Sommerfeld, Squire system (TL
0,shownin
Fig. 7(a)) induces second order spanwise and normal to wall velocities (see
Fig. 7(b)). As a consequence, streamwise vorticity is produced which then
creates streamwise streaks (see Fig. 7(c)) through the forcing of the Squire
equation due to the coupling term (which is approximately ten times larger
than the corresponding second order forcing, compare Figs. 7(a) and (d)).
The same analysis has been carried out also considering the second order
forcing induced by Squire continuous spectrum modes. The results show that

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Boundary layer receptivity to free-stream disturbances 37
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
β
α
Figure 6. Forcing induced by Orr-Sommerfeld continuous
spectrum modes for γ=0.27. Contour levels of the maxi-
mum of the streamwise second order velocity (normalized by
the first order energy and Re2). Note that the second order
distribution is displayed with reference to the (α, β)valuesof
the corresponding first order terms. Maximum value 0.1056,
contour spacing 0.011.
the same physical mechanism is induced and the amplitude of the generated
streaks is comparable for the two different classes of modes.
4.3. Parametric study
In order to find the most effective interactions between continuous spectrum
modes we carried out a parametric study varying the wave numbers in the range
0.1<α<2,0.05 <β<1and0.25 <γ<20.9 at a given Reynolds number.
In Figs. 8 and 9 we report, respectively, the results corresponding to forcing
induced by Orr-Sommerfeld and Squire continuous spectrum modes. The fig-
ures depict the maximum amplifications of the streamwise velocity component
normalized with respect to the energy density Eof the non-linearly interacting
modes. In particular, we plot
A(α, β)=maxγ
u(2)(α, β, γ)
E,
B(α, γ)=maxβ
u(2)(α, β, γ)
E,
C(β,γ)=maxα
u(2)(α, β , γ)
E.

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38 L. Brandt, D.S. Henningson & D. Ponziani
0 10 20 30 40 50
−1.5
0
1.5
0 10 20 30 40 50
−800
−400
0
400
800
0 5 10 15 20
0
6000
12000
18000
0 5 10 15 20
−20
−15
−10
−5
0
5
yy
yy
iβU ˆv(2)
ˆu(2)
(c) (d)
ˆv(2),ˆw(2)
(b)
(a)
ˆ
TL
0
Figure 7. Second order solution corresponding to the non-
linear interactions of a couple of oblique Orr-Sommerfeld
modes associated to the wave numbers (0.5,±0.2,0.628),at
Re = 500: (a) forcing to the Orr-Sommerfeld equation (—–)
and to the Squire equation (- - -); (b) normal to wall (—–) and
spanwise component of velocity (- - -); (c) streamwise compo-
nent of velocity ; (d) forcing to the Squire equation associated
to the coupling term −iβUv(2).
The results show that the maximum amplification is attained for α≈2and
β≈0.15 (thus implying that the streaks are associated to spanwise wave
number β≈0.3) independently of the type of forcing modes. The results also
indicate that the maximum response is associated to low values of γ(γ≈0.25),
i.e. structures of large wall normal extent. We observe that in the case the
forcing is given by the Orr–Sommerfeld modes we find a lower maximum for
α≈0.1, β≈0.2andγ≈1.25.
One should also note that these figures tend to bias high wavenumbers,
since they are in practice more damped than low ones. Recall in fact that for
simplicity we have put the damping rate of the continuous spectrum modes to
zero. This implies, for example, that the increasing amplification for higher
value s of αin Figs. 8 a), b) and 9 a), b) would be damped for sufficiently
high values of the streamwise wavenumber. However this does not present a

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Boundary layer receptivity to free-stream disturbances 39
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a)
β
α
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
b)
γ
α
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c)
γ
β
Figure 8. Boundary layer response to forcing induced by
non-linear interactions of Orr-Sommerfeld modes. Contour
levels of the maximum amplification of the streamwise com-
ponent of velocity at Re = 300, y∞= 50: a) A(α, β)for
0.25 <γ<20.9; b) B(α, γ)for0.05 <β<1; c) C(β, γ)for
0.1<α<2. The maximum is 10145 and occurs at α=2.,
β=0.15, γ=0.25. Maximum contour level is 9500 and con-
tour spacing 1000.

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40 L. Brandt, D.S. Henningson & D. Ponziani
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a)
β
α
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
b)
γ
α
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c)
γ
β
Figure 9. Boundary layer response to forcing induced by
non-linear interactions of Squire modes. Contour levels of the
maximum amplification of the streamwise component of veloc-
ity at Re = 300, y∞= 50: a) A(α, β)for0.25 <γ <20.9; b)
B(α, γ)for0.05 <β<1; c) C(β, γ)for0.1<α<2. The
maximum is 10586 and occurs at α=2.,β=0.15, γ=0.25.
Maximum contour level is 10000 and contour spacing 1000.
problem in applying the results to a real free-stream turbulence case since re-
alistic free-stream turbulence spectra have little energy content in these higher
wavenumbers. This will be discussed in the next session.

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Boundary layer receptivity to free-stream disturbances 41
4.4. Filtering with turbulent energy spectrum and streak spacing
In the results presented so far we assumed unit energy in each Fourier compo-
nent of the free-stream disturbance. In order to predict which length scales may
be important in a real transition initiated by free-stream turbulence, we asso-
ciate each mode with a coefficient proportional to the energy spectrum of typical
homogeneous and isotropic turbulence. We use here the von K´arm´an spectrum,
which is proportional to κ4for large scales and matches the Kolmogorov-(5/3)-
law for small scales. It has the form
E(κ)=2
3
1.606(κL)4
(1.35 + (κL)2)17/6Lq (41)
where κ=α2+β2+γ2,Lis an integral length scale and qis the total tur-
bulent kinetic energy, defined as the integral over all κ’s of the spectrum. The
maximum of the energy is at κ=1.8/L.Wenotethatthisspectrum,givenin
Tennekes & Lumley (1972) is a good approximation to homogeneous turbu-
lence. The filtered results are reported in Figs. 10 and 11 for L=5,q=1;the
figures show that the filtering moves the more effective α’s and γ’s to smaller
values, while the β’s are less affected. The maximum amplification is attained
for α≈0.4 for the case the forcing is given by the Orr-Sommerfeld modes and
α≈0.3 for the Squire case; the corresponding values of βand γare respec-
tively 0.15 and 0.25 independently of the type of the forcing modes. We note
also that in the case the forcing is given by the Orr–Sommerfeld modes, a lower
maximum is still present at α≈0.1, β≈0.15 and γ≈0.4. Similar results
were obtained for different choices of the integral length scale L.
The objective of the present work is to find the wavenumbers associated to
free-steam disturbances which are most effective in the generations of stream-
wise vortices. Matsubara & Alfredsson (2001) in their experimental work ob-
served that the spanwise distance to the first minimum of two point velocity
correlations, which closely corresponds to half the streak spacing, stays almost
constant in the downstream direction. This suggests that the boundary layer
growth does not affect the streak development. When scaling these results
with the local displacement thickness the characteristic length scale close to
the leading edge is approximatively 20δ∗±10δ∗, i.e. centered around a span-
wise wavenumber β≈0.3. This is close to the β’s for which our simplified
temporal model predicts the largest response. In the experiments of Matsub-
ara & Alfredsson (2001) the growth of the boundary layer implies a variation
of the spanwise scale with the respect to the local displacement thickness. In
our model, we do not account for the growth of the boundary layer, but we are
still able to predict the first step of the receptivity process, i.e. the formation
of streamwise vortices.
5. Discussion and conclusion
In the present work we have investigated how free-stream disturbances affect
a laminar boundary layer. In particular, we have analyzed the receptivity to

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42 L. Brandt, D.S. Henningson & D. Ponziani
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a)
β
α
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
b)
γ
α
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c)
γ
β
Figure 10. Boundary layer response to forcing induced by
non-linear interactions of Orr-Sommerfeld modes filtered by
turbulent kinetic energy spectrum. Contour levels of the max-
imum amplification of the streamwise component of velocity
at Re = 300, y∞= 50: a) A(α, β)for0.25 <γ<20.9; b)
B(α, γ)for0.05 <β<1; c) C(β, γ)for0.1<α<2. The
maximum is 4554 and occurs at α=0.4, β=0.15, γ=0.25.
Maximum contour level is 4500 and contour spacing 500.

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Boundary layer receptivity to free-stream disturbances 43
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a)
β
α
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
b)
γ
α
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c)
γ
β
Figure 11. Boundary layer response to forcing induced by
non-linear interactions of Squire modes filtered by turbulent ki-
netic energy spectrum. Contour levels of the maximum ampli-
fication of the streamwise component of velocity at Re = 300,
y∞= 50: a) A(α, β)for0.25 <γ<20.9; b) B(α, γ)for
0.05 <β<1; c) C(β,γ)for0.1<α<2. The maximum is
6273 and occurs at α=0.3, β=0.15, γ=0.25. Maximum
contour level is 6000 and contour spacing 500.

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44 L. Brandt, D.S. Henningson & D. Ponziani
oblique waves in the free-stream and to continuous spectrum modes. In both
cases, we observe that the formation of streaks is the dominant feature. The
underlying mechanism can be reduced to a two-step process, first the generation
of streamwise vorticity and then the formation of streaks.
Previous investigators (Bertolotti 1997; Andersson et al. 1999; Luchini
2000) have considered the influence of streamwise vortices present in the free-
stream and have shown that the subsequent formation of streaks in the bound-
ary layer can be explained in terms of linear theory by the lift-up mechanism.
The most important feature of the process we have investigated is that stream-
wise vortices are non-linearly generated starting from wave-like disturbances
in the free-stream. This non-linear mechanism has already been observed in
the numerical experiments of Berlin & Henningson (1999). They isolated the
different order interactions to show that the streamwise independent modes are
the most excited. Here, we have used a perturbation model which has been
shown to provide an efficient theoretical tool to isolate the two-step process,
see also Ponziani et al. (2000).
The model has been validated by comparisons with DNS data for the case
the forcing is given by a couple of oblique waves. In order to apply the model
to study the boundary layer receptivity to free-stream turbulence, we have
exploited the fact that continuous spectrum modes can be used to represent
the free-stream turbulence spectrum (Jacobs & Durbin 2000). This assumption
has allowed us to further simplify the study accounting only for the response
of the boundary layer to couples of continuous spectrum modes. An extensive
parametric study has been carried out to isolate the most effective modes by
var ying the wave numb ers (α, β, γ ).
We have concluded that the formation of streaks is due to the second
order correction induced by the coupling term in the Orr-Sommerfeld, Squire
system and the receptivity is independent of the type of forcing modes. This
indicates that disturbances containing normal velocity in the free-stream, are
not more likely to force streamwise vorticity in the boundary layer, compared
to disturbances not containing v. On the other hand, Berlin & Henningson
(1999) draw the conclusion that the normal velocity in the free-stream is more
effective than other components. However, their conclusion was based on a
type of disturbance which grew in size in the normal direction as the normal
velocity increased. Thus their results are also consistent with the present ones
which show that this increase in amplification is rather a result of increase in
normal scale (or decrease in γ).
The large eddy simulations of Yang & Voke (1993) also indicated the
key influence of the wall normal component of the free-stream turbulence in-
tensity in provoking transition. Their results show in fact that the transition
process begins with the production of the Reynolds stresses due to the over-
lapping of regions of non zero fluctuating velocity vandmeanshear∂U/∂y.
From our analysis we find that the components with streamwise wave number
approximately zero are the ones crucial in generating disturbances inside the

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Boundary layer receptivity to free-stream disturbances 45
boundary layer, and that can be induced from non-linear interactions of either
Orr-Sommerfeld or Squire modes. The capability of modes with frequency and
streamwise wavenumber approximately zero to penetrate the shear layer has
been demonstrated by different authors (see Westin et al. 1994; Jacobs &
Durbin 1998; Matsubara & Alfredsson 2001; Hultgren & Gustavsson 1981),
and we may thus conclude that v–components active in generating Reynolds
stresses are the ones associated with nearly zero frequency and that they are
the ones associated with the second order solution in our model.
The results also show that the second order forcing does not depend on
the Reynolds number, thus recovering the O(Re2) scaling of the forced re-
sponse described by streamwise independent disturbances governed by the Orr-
Sommerfeld, Squire system. We may speculate on the implication of this scal-
ing on the Reynolds number dependence of the forced response in the spatial
problem. In a number of experiments it has been seen that the growth of
the streak amplitude in boundary layers subjected to free-stream turbulence is
proportional to Re, or equivalently that the energy growth is proportional to
Rex≈Re2or downstream distance. If we assume that the Reynolds number
dependence in the spatial case would be the same as in the temporal case in-
vestigated here, downstream growth of the streak amplitude predicted would
be proportional to Re2, i.e. over predicted by a factor of Re. However, the
result found here assumes a continuous deterministic forcing. Real turbulence
would better be described by a stochastic forcing in a number of wavenum-
bers. Bamieh & Dahleh (1999) have shown that a stochastic forcing reduces
the scaling of the maximum response of the temporal problem from Re2to
Re3/2. This is still a factor Re1/2too large. However, the growth in a realistic
free-stream turbulence case would probably further be reduced by the fact that
free-stream turbulence decays with downstream distance. In our model this
would correspond to a forcing which decreases with Re, thus further reducing
the growth of the streak amplitude.
Acknowledgments
This research was supported by the G¨oran Gustavsson Foundation, TFR (Tek-
nikvetenskapliga forskningsr˚adet), the C.M. Lerici Foundation and the research
funding obtained by Prof. R. Piva, University of Rome ”La Sapienza”.
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Fluid Mech.,250, 169-207 .
Hultgren,L.S.&Gustavsson, L. H. 1981. Algebraic growth of disturbances in a
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Jacobs, R. G. &Durbin,P.A.1998. Shear sheltering and continuous spectrum of
the Orr–Sommerfeld equation. Phys. Fluids.10 (8), 2006–2011.
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On the breakdown of boundary layer streaks
By Paul Andersson†‡, Luca Brandt†, Alessandro Bottaroand
Dan S. Henningson†‡
†Department of Mechanics, Royal Institute of Technology (KTH), S-100 44
Stockholm, Sweden
‡FFA, the Aeronautical Research Institte of Sweden,
P.O. Box 11021, S–161 11 Bromma, SWEDEN
Institut de M´ecanique des Fluides de Toulouse (IMFT), Universit´e Paul Sabatier,
118 route de Narbonne, 31062 Toulouse Cedex 4, France
A scenario of transition to turbulence likely to occur during the develop-
ment of natural disturbances in a flat-plate boundary layer is studied. The
perturbations at the leading edge of the flat plate that show the highest poten-
tial for transient energy amplification consist of streamwise aligned vortices.
Due to the lift-up mechanism these optimal disturbances lead to elongated
streamwise streaks downstream, with significant spanwise modulation. Direct
numerical simulations are used to follow the nonlinear evolution of these streaks
and to verify secondary instability calculations. The theory is based on a lin-
ear Floquet expansion and focuses on the temporal, inviscid instability of these
flow structures. The procedure requires integration in the complex plane, in
the coordinate direction normal to the wall, to properly identify neutral modes
belonging to the discrete spectrum. The streak critical amplitude, beyond
which streamwise travelling waves are excited, is about 26% of the free-stream
velocity. The sinuous instability mode (either the fundamental or the sub-
harmonic, depending on the streak amplitude) represents the most dangerous
disturbance. Varicose waves are more stable, and are characterized by a critical
amplitude of about 37%. Stability calculations of streamwise streaks employing
the shape assumption, carried out in a parallel investigation, are compared to
the results obtained here using the nonlinearly modified mean fields; the need
to consider a base flow which includes mean flow modification and harmonics
of the fundamental streak is clearly demonstrated.
1. Introduction
1.1. “Lift-up” effect and transient growth
For quite a long time the fluid mechanics community has recognized transition
to turbulence as a fundamental problem and has directed intense research ef-
forts toward its understanding. Even so, our current picture of the physical
51

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52 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
processes involved is far from complete. The classical starting point for theo-
retical investigations of transition is linear stability theory. Here, exponentially
growing solutions—in time or space—to the linearised Navier–Stokes equations
are sought. If such solutions are not found, the flow is predicted by the the-
ory to be stable. However, experiments show that the route to turbulence is
highly dependent on the initial conditions and on the continuous forcing that
background noise can provide (see for example Morkovin & Reshotko 1990;
Reshotko 1994, for reviews).
Experiments reveal that many flows, including for example Poiseuille and
boundary layer flows, may undergo transition to turbulence for Reynolds num-
bers well below the critical ones from the linear stability theory. For the case of
plane Couette flow the theory predicts stability at all Reynolds numbers (Ro-
manov 1973) while numerical and laboratory experiments point to a finite
transitional value (Lundbladh & Johansson 1991; Tillmark & Alfredsson 1992;
Dauchot & Daviaud 1995).
The reason for this discrepancy between the theory and the experiments has
been sought in the nonlinear terms of the Navier–Stokes equations. Examples
of nonlinear theories are given by Orszag & Patera (1983), Bayly, Orszag
& Herbert (1988) and Herbert (1988). However, examining the Reynolds–
Orr equation (Drazin & Reid 1981) a remarkably strong statement can be
made on the nonlinear effects: the nonlinear terms redistribute energy among
disturbance frequencies but have no net effect on the instantaneous growth rate
of the energy. This implies that there must exist a linear growth mechanism
for the energy of a disturbance of any amplitude to increase (Henningson &
Reddy 1994; Henningson 1996). The apparent need for an alternative growth
mechanism based on the linearized equations has recently led to intense re-
examination of the classical linear stability theory.
The first convincing alternative was proposed by Ellingsen & Palm (1975).
By introducing an infinitesimal disturbance without streamwise variation in a
shear layer, they showed that the streamwise velocity component can increase
linearly with time, within the inviscid approximation, producing alternating
low– and high–velocity streaks in the streamwise velocity component. Lan-
dahl (1975,1980) extended this result to the linear evolution of localized dist-
urbances and supplied the physical insight to the linear growth mechanism
with what he denoted the lift-up effect. He argued that vortices aligned in the
streamwise direction advect the mean velocity gradient towards and away from
the wall, generating spanwise inhomogeneities.
It is now clear that since the linearized Navier–Stokes operator is non-
normal for many flow cases (especially in shear flows) a significant transient
growth of a given perturbation might occur, before the subsequent exponential
behaviour. Such an algebraic growth involves non-modal perturbations and
can exist for subcritical values of the governing parameters.

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On the breakdown of boundary layer streaks 53
Indeed, early investigators of the lift-up and transient growth mechanisms
found considerable linear energy amplification before the viscous decay (Hult-
gren & Gustavsson 1981; Boberg & Brosa 1988; Gustavsson 1991; Butler &
Farrell 1992; Reddy & Henningson 1993; Henningson, Lundbladh & Johans-
son 1993). An overview of recent work can be found in the review articles
by Trefethen et al. (1993) and Henningson (1995). The initial disturbance
that yields the maximum spatial transient growth in a non-parallel flat plate
boundary layer flow was determined independently by Andersson, Berggren &
Henningson (1999a) and Luchini (2000) to consist of vortices aligned in the
streamwise direction. These vortices leave an almost permanent scar in the
boundary layer in the form of long-lived, elongated streaks of alternating low
and high streamwise speed.
1.2. ”Secondary” instability of streamwise streaks
If the amplitude of the streaks grows to a sufficiently large value, instabilities
can develop which may provoke early breakdown and transition, despite the
theoretically predicted modal decay. In the remainder of the paper we will re-
fer to the instability of the streak as a ”secondary” instability, to differentiate
it from the ”primary” growth mechanism responsible for the formation of these
flow structures. A (secondary) instability can be induced by the presence of
inflection points in the base flow velocity profile, a mechanism which does not
rely on the presence of viscosity. Controlled experiments on the breakdown of
periodically arranged (along the span) streaks produced by an array of rough-
ness elements have been conducted by Bakchinov et al. (1995). It was shown
that the instability of the streaks causes transition in a similar manner as do
the G¨ortler and cross-flow cases, i.e. via amplification of the secondary wave
up to a stage where higher harmonics are generated, and on to a destruction of
the spanwise coherence of the boundary layer. Alfredsson & Matsubara (1996)
considered the case of transition induced by streaks formed by the passage of
the fluid through the screens of the wind-tunnel settling chamber. They report
on the presence of a high frequency ”wiggle” of the streak with a subsequent
breakdown into a turbulent spot.
Today, the description of the establishment of steady streaky structures is
well captured by the theory. The work presented here aims at understanding
the instability of these streaks on the path to boundary layer turbulence. Par-
enthetically, we note also that streamwise vortices and streaks are an essential
ingredient of the near–wall turbulent boundary layer and that the instability of
streaky structures is one crucial feature of the near-wall cycle which is thought
to lie at the heart of the genesis and dynamics of turbulent coherent structures
(Jimenez & Pinelli 1999; Schoppa & Hussain 1997, 1998).
Some work has recently appeared in the literature on the instability of
streaks in channel flows (Waleffe 1995, 1997; Reddy et al. 1998) and, among
the findings reported, it is interesting to note that slip and no-slip boundaries
do not display significant differences in the instability scenario (Waleffe 1997).

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54 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
The present study focuses on the linear, inviscid breakdown of boundary layer
streaks. It is believed that the inviscid approximation captures the essential
features of the breakdown. This is supported primarily by the controlled exper-
iments of Bakchinov et al. (1995), who demonstrate unambiguously the role
of the critical layer in the development of the instability. The measurements
conducted by Boiko et al. (1997) on the instability of a vortex in a bound-
ary layer and the very carefully controlled experiments on the breakdown of
streaks in channel flow conducted by Elofsson, Kawakami & Alfredsson (1999)
further attest to the inflectional nature of the breakdown. The latter authors
generated elongated streamwise streaky structures by applying wall suction,
and triggered a secondary instability by the use of earphones. The growth rate
of the secondary instability modes was unaffected by a change of the Reynolds
number of their flow, over a subcritical range, and the regions of (sinuous-type)
oscillations of the streaks in cross-stream planes were reasonably well correlated
to the spanwise shear of the main flow. The numerical/theoretical comparative
viscous-inviscid investigations on the linear breakdown of longitudinal vortices
in a curved channel (Randriarifara 1998) and the numerous studies on the
secondary instability of G¨ortler vortices (Hall & Horseman 1991; Yu & Liu
1991; Bottaro & Klingmann 1996), show that the inviscid approach captures
correctly the dominant features of the instability with viscosity playing mainly
a damping role. These secondary instability studies bear a close resemblance
to the present one.
1.3. Mean field with optimal streaks and linear stability analysis
The equations governing the streak evolution are obtained by applying the
boundary layer approximations to the three-dimensional steady incompressible
Navier-Stokes equations and linearizing around the Blasius base flow. After
defining the disturbance energy density as the integral, in the wall-normal di-
rection, of the square of the disturbance velocity components, techniques com-
monly employed when solving optimal control problems are used to determine
the optimal disturbance (streamwise oriented vortices) and its downstream re-
sponse (streamwise streak). The output streak predicted by the theory of
Andersson et al. (1999a) and Luchini (2000) is remarkably similar to that
measured in the laboratory (see figure 1). The measurements were performed
in a pre-transitional flat plate boundary layer, where the largest amplitude of
the streamwise velocity was eleven percent of the free-stream velocity. The
streak is, in fact, a ”pseudo-mode” triggered in a flat-plate boundary layer
subject to significant outside disturbances.
The instability of these optimal streaks is studied here with different levels
of approximation. Two different representations are used for the mean field:
the simpler shape assumption, where the shape of the streak obtained from the
linearized equations is considered unmodified even at large amplitudes, and the
complete nonlinear development of the streak.

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On the breakdown of boundary layer streaks 55
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
1
2
3
4
5
6
7
y
U/Umax
Figure 1. Comparison between the streamwise velocity com-
ponent of the downstream response to an optimal perturba-
tion, and the u-r.m.s. data in a flat-plate boundary layer
subject to free-stream turbulence (—–, Reynolds-number-
independent theory). The symbols represent experiments from
Westi n et al. (1994) (◦,Reδ=203; +, Reδ=233; ×,Reδ=305;
,Reδ=416; ,Reδ=517). Here yhas been made non-
dimensional—and the Reynolds number is defined—using the
Blasius length scale δ=(Lν/U∞)1/2.
In both formulations the linear secondary stability calculation are carried
out on the basis of the boundary layer approximation, i.e. the mean field
to leading order will consist only of the streamwise velocity component (here
denoted U), consistent with the scaling hypothesis which led to the definition
of the streak. Such a mean field varies on a slow streamwise scale, whereas the
secondary instability varies rapidly in the streamwise direction x,asobserved
in the visualisations by Alfredsson & Matsubara (1996). Hence, our leading
order stability problem is the parallel flow problem, with perturbation mode
shapes dependent only on the cross-stream coordinates y(wall-normal) and z
(spanwise). The same approximation was made previously for the case of the
G¨ortler flow (Hall & Horseman 1991; Yu & Liu 1991; Bottaro & Klingmann
1996).
Due to the spanwise periodicity of the base flow—consisting of streamwise
aligned streaks superimposed on a flat-plate Blasius flow—a temporal Floquet
analysis is employed with the objective of determining which disturbance pat-
tern shows the highest potential for temporal growth. In particular we are
interested in determining if the maximum disturbance growth occurs for a sin-
uous or a varicose disturbance, and whether it is of fundamental or subharmonic

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56 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
type. In addition, the critical threshold amplitude of the streak for the onset
of the secondary instability is determined.
In §2 the two-dimensional eigenvalue problem arising from the governing
partial differential equation is formulated and the numerical methods adopted
are described. In §3 a scaling property of the mean field calculated by nonlinear
simulations is introduced; this property allows a reduced number of simulations
to cover a wide range of spanwise scales of the disturbance. Numerical experi-
ments on streak instability are also carried out using DNS and the results are
compared with the linear stability calculations. In §4 a parametric study of the
sinuous modes is presented and some comparisons with the shape assumption
calculations are discussed. The main conclusions of the work are summarized
in §5.
2. Governing Equations and Numerical Methods
2.1. Inviscid stability equations
The dimensionless, incompressible Euler equations linearized around the mean
field (U(y, z), 0, 0) are
ux+vy+wz=0,(1)
ut+Uux+Uyv+Uzw=−px,(2)
vt+Uvx=−py,(3)
wt+Uwx=−pz,(4)
and the system is closed by slip boundary conditions at the solid wall and by
decaying disturbances in the free stream;
(u, v, w)=(u(x, y, z , t),v(x, y, z, t),w(x, y, z , t))
are the perturbation velocities in the streamwise, wall-normal and spanwise
directions, respectively, tis time and p=p(x, y, z , t) is the disturbance pressure.
All velocities have been scaled with the free-stream speed U∞and the pressure
with ρU2
∞,whereρis the fluid density. The length scale is δ=(Lν/U∞)1/2,
with νkinematic viscosity and Ldistance from the leading edge. For later use
we define two Reynolds numbers using the two different length scales, Reδ=
U∞δ/ν and Re =U∞L/ν,whichrelateasRe =Re2
δ.
The presence of both wall-normal and spanwise gradients in the mean field
makes it impossible to obtain an uncoupled equation for either of the velocity
components. It is, however, possible to find an uncoupled equation for the
pressure by taking the divergence of the momentum equations, introducing
continuity and then applying equations (3) and (4) (Henningson 1987; Hall &
Horseman 1991). These manipulations yield
(∂
∂t +U∂
∂x)∆p−2Uypxy −2Uzpxz =0.(5)

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On the breakdown of boundary layer streaks 57
We consider perturbation quantities consisting of a single wave component in
the streamwise direction, i.e.
p(x, y, z , t)=Real{˜p(y, z)eiα(x−ct)},
where αis the (real) streamwise wavenumber and c=cr+iciis the phase
speed. The equation governing the pressure reduces to
(U−c)( ∂2
∂y2+∂2
∂z2−α2)˜p−2Uy˜py−2Uz˜pz=0; (6)
this constitutes a generalized eigenproblem with cin the role of eigenvalue and
needs to be solved for given mean field and streamwise wavenumber. Once the
pressure eigenfunctions are computed, the velocity components can be obtained
from the explicit expressions
iα(U−c)˜v=−˜py,(7)
iα(U−c)˜w=−˜pz,(8)
iα(U−c)˜u+Uy˜v+Uz˜w=−iα˜p. (9)
The pressure component ˜pis expanded in an infinite sum of Fourier modes
˜p(y, z)=
∞

k=−∞
ˆpk(y)ei(k+γ)βz,(10)
where βis the spanwise wavenumber of the primary disturbance field and
γis the (real) Floquet exponent. We note two symmetries: first, to within
renumbering of the Fourier coefficients γand γ±nyield identical modes for
any integer n, and second, equation (6) is even under the reflection z→−z.
These symmetries make it sufficient to study values of γbetween zero and
one half, with γ= 0 corresponding to a fundamental instability mode, and
γ=0.5 corresponding to a subharmonic mode (see Herbert 1988 for a thorough
discussion on fundamental and detuned instability modes). The mean field is
also expanded as a sum of Fourier modes
U(y, z)=
∞

k=−∞
Uk(y)eikβz (11)
and these expansions are introduced into equation (6) to yield an equation that
holds for each integer k:
∞

j=−∞
Uk−j∂2
∂y2−β2(j+γ)2−α2−2∂Uk−j
∂y
∂
∂y +2β2(k−j)(j+γ)Uk−jˆpj
=c∂2
∂y2−β2(k+γ)2−α2ˆpk.(12)
The appropriate boundary conditions are:
∂ˆpk
∂y =0 at y=0 and ∂ˆpk
∂y →0wheny→∞.(13)

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58 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
This problem consists of an infinite number of coupled ordinary differential
equations which must be truncated in order to find a numerical solution. The
complete system must be solved numerically, when the solution is sought for
an arbitrary value of the detuning parameter γ. If, however, even and odd
solutions in zare sought, the system of equations can be simplified for the
fundamental and subharmonic modes. In this case the numerical effort is de-
creased considerably because the dimension of the matrices arising from the
discretisation is halved.
In the fundamental (γ= 0) case, even (odd) modes are obtained by im-
posing the condition ˆp−k=ˆpk(ˆp−k=−ˆpk). This is equivalent to introducing
either a cosine or a sine expansion
˜p(y, z)=
∞

k=0
ˆpk(y)cos(kβz),(14)
˜p(y, z)=
∞

k=1
ˆpk(y)sin(kβz),(15)
into equation (6), to yield two different systems of ODE’s.
In the case of subharmonic disturbances (γ=0.5) the spanwise periodicity
of the fluctuations is twice that of the base flow. The subharmonic mode also
contains a symmetry which renders the decoupling into even and odd modes
possible. In this case the cosine and sine expansions are:
˜p(y, z)=
∞

k=0
ˆpk(y)cos(2k+1
2βz),(16)
˜p(y, z)=
∞

k=0
ˆpk(y)sin(2k+1
2βz).(17)
These two expansions produce two new systems of ODE’s which clearly yield
two different classes of solutions, as in the case of the fundamental modes.
Notice, however, that the sinuous fluctuations of the low-speed streaks, rep-
resented in the fundamental case by the sine expansion (15) are given, in the
subharmonic case, by the cosine expansion (16). This is because the subhar-
monic sinuous case treats two streaks, which oscillate out of phase (cf Le Cunff
& Bottaro 1993). Likewise, varicose oscillations of the low-speed streaks are
represented by the cosine series (14) for the fundamental mode, and by the sine
expansion (17) in the case of subharmonic perturbations.
For the sake of clarity, in the remainder of the paper only the definitions
of sinuous or varicose modes of instability will be employed, with reference to
the visual appearance of the motion of the low-speed streaks.Asketchofthe
different fundamental and subharmonic modes is provided in figure 2: it clearly
illustrates how in the subharmonic cases sinuous (varicose) fluctuations of the
low-speed streaks are always associated with staggered (in x) varicose (sinuous)
oscillations of the high-speed streaks.

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On the breakdown of boundary layer streaks 59
x
z
✲
✻
Fundamental sinuous Fundamental varicose
Subharmonic sinuous Subharmonic varicose
Figure 2. Sketch of streak instability modes in the (x−z)-
plane over four streamwise and two spanwise periods, by con-
tours of the streamwise velocity. The low-speed streaks are
drawn with solid lines while dashed lines are used for the high-
speed streaks.
2.2. Chebyshev polynomials in real space
The temporal eigenvalue system derived in section 2 is solved numerically using
a spectral collocation method based on Chebyshev polynomials. Consider the
truncated Chebyshev expansion
φ(η)=
N

n=0
¯
φnTn(η),
where
Tn(η)=cos(ncos−1(η)) (18)

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60 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
is the Chebyshev polynomial of degree ndefined in the interval −1≤η≤1.
We use
ηj=cosπj
N,j=0,1,... ,N,
as collocation points, that is, the extrema of the Nth-order Chebyshev polyno-
mial TNplus the endpoints of the interval.
The calculations are performed using 121 (N=120) Chebyshev collocation
points in y, and with the Fourier series in ztruncated after fifteen modes. The
wall-normal domain varies in the range (0,y
max), with ymax well outside the
boundary layer (typically ymax is taken equal to 50). The Chebyshev interval
−1≤η≤1 is transformed to the computational domain 0 ≤y≤ymax by the
use of the conformal mapping
y=a1+η
b−η,(19)
where
a=yiymax
ymax −2yi
and b=1+ 2a
ymax
.
This mapping puts half the grid points in the region 0 ≤y≤yi,withyichosen
to be equal to 8.
The unknown functions ˆpk=ˆpk(y) may now be approximated by
ˆpN
k(y)=
N

n=0
¯pn
k˜
Tn(y),
where ˜
Tn(y)=Tn(η)withη→ ybeing the mapping (19). The Chebyshev
coefficients ¯pn
k,n=0, ... ,Nare determined by requiring equation (12) to
hold for ˆpN
kat the collocation points yj,j=1,... ,N−1. The boundary
conditions (13) are enforced by adding the equations
N

n=0
¯pn
k˜
Tn,y(0) =
N

n=0
¯pn
k˜
Tn,y(ymax )=0,
where subscript n, y denotes the y-derivative of the n-th Chebyshev polynomial.
2.3. Chebyshev polynomials in complex space
The discretization leads to a generalized eigenproblem with the two matrices
containing only real elements; hence, the solutions will consist of either real
eigenvalues or complex conjugate pairs. No strictly damped solutions can be
found using these equations together with an integration path running along the
real y-axis from 0 to ymax, since the neglect of viscosity introduces a continuous
spectrum of singular neutral modes. Lin (1944) performed an asymptotic
analysis on the Orr–Sommerfeld equation, requiring the inviscid eigenvalue
problem (the Rayleigh equation) to be a limit of the viscous one when the
Reynolds number approaches infinity. For this to apply, he found that the
integration path in the inviscid case could be taken on the real axis if ci>0

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On the breakdown of boundary layer streaks 61
and that it should be taken in the complex plane for ci≤0, in such a way that
the singularities lie on the same side of the integration path as in the ci>0
case.
Information on the singularities is contained in the system’s determinant.
We are satisfied here with the approximate location of the singularities and to
simplify the analysis the mean field with the shape assumption is considered,
i.e.
U(y, z)=UB(y)+ As
2us(y;β)eiβz +As
2us(y;β)e−iβz ,(20)
with UBthe Blasius solution, usthe streak’s mode shape provided by the
analysis of Andersson et al. (1999a) and scaled so that max[us(y;β)] = 1, As
the amplitude of the streak and βthe spanwise wavenumber. Figure 1 show the
streak’s mode shape us. Introducing expansion (20) into equation (12) yields
an equation that holds for each integer k:
As
2us(∂2
∂y2−β2(k−1+γ)(k−3+γ)−α2)ˆpk−1−Asus,y
∂
∂y ˆpk−1+
(UB−c)( ∂2
∂y2−β2(k+γ)2−α2)ˆpk−2UB,y
∂
∂y ˆpk+
As
2us(∂2
∂y2−β2(k+1+γ)(k+3+γ)−α2)ˆpk+1 −Asus,y
∂
∂y ˆpk+1 =0,(21)
plus boundary conditions (13).
By rewriting equations (21) and (13) as a system of first-order equations
and studying the system’s determinant, the singularities can be identified an-
alytically as the roots of the equation
K

k=1{UB−c+Asuscos kπ
K+1}=0,
where Kis the number of Fourier modes. For small values of cithe approximate
location of each singularity in the complex y-plane can be identified with a
Taylor expansion around y=yr, i.e.
UB(yr)+Asus(yr;β)coskπ
K+1=cr+h.o.t.
and to first order these locations are:
ys=yr+ici
UB,y(yr)+Asus,y(yr)cos(kπ/(K+1)).(22)
These are, as might have been expected, the values of yfor which the base
flow velocity becomes equal to cat the discrete values of the zcoordinate
imposed by the truncated Fourier expansion in z. Clearly, yscrosses the real
yaxis when cichanges sign, so that the integration path has to go out into
the complex y-plane in order for the singularities to lie on the same side of the

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62 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
0.15 0.2 0.25 0.3 0.35 0.4
−0.01
−0.005
0
0.005
0.01
0.015
0.02
−0.2 −0.1 0 0.1 0.2 0.3
0
0.5
1
1.5
2
2.5
3
(a) (b)
ωi
As
yr
yi
Figure 3. (a) Temporal growth rate of the fundamental vari-
cose mode versus the streak’s amplitude for α=0.2 and β=0.45
(◦realintegrationpath,---complexintegrationpathwith
B=0.01, —– complex integration path with B=0.03). (b)
The three curved lines denote different integration paths: -
--B=0.01, -·-·-B=0.02, —– B=0.03. The thick segments of
stars denote the singular segments for three different streak’s
amplitudes. From the left: As= 0.15 (amplified modes do not
exist), 0.215 (the least stable discrete mode is neutral) and
0.35 (at least one amplified mode exists). Only for the latter
case the real integration path is suitable.
path. Integration in the complex plane is necessary when neutral curves are
sought. The mapping
yc=y−iB(ymax y−y2)1/2(23)
allows the computation of damped (and neutral) modes. It is introduced
into (19), that is yc→ ηc, to deliver a curve in the complex plane with end-
points in ηc=−1 and 1. Complex Chebyshev polynomials Tc
n(ηc) are defined
by using (18), and the unknown functions are approximated using this new ba-
sis. This can be done since the analytic continuation of a polynomial is given
by the same polynomial but with a complex argument.
As can be inferred from equation (22) the singularities corresponding to a
given set of problem parameters are confined to a finite segment in the complex
y-plane. For amplified modes this segment is found in the right half-plane in
figure 3(b); for damped modes it is displaced to the left half-plane, whereas for
neutral modes the singular segment is a subset of the real axis. In figure 3(a) the
results of three calculations of the temporal growth rate are plotted versus the
streak amplitude, employing the shape assumption for given streamwise, α=0.2,
and spanwise, β=0.45, wavenumbers. The circles are obtained by integrating
over the real y-axis; as the amplitude Asof the streak decreases, so does the
largest growth rate of the instability ωi,untilthevalueAS≈0.215 below which
only quasi-neutral modes are found. For such modes the real integration path

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On the breakdown of boundary layer streaks 63
is located on the wrong side of the singularities. If the complex integration
path denoted by a dashed line in both figures (corresponding to B=0.01 in
the mapping 23) is employed, the integration correctly follows the damped
mode down to an amplitude of about 18%. Clearly one can proceed to even
smaller amplitudes simply by increasing B, i.e. by displacing the integration
contour further into the negative yiregion. For example, the dotted-dashed
contour in figure 3(b) can be used, or the continuous line path (corresponding
to B=0.03). The latter integration path has been used and the resulting full
spectrum is shown in figure 4 for a streak amplitude of 0.18. The continuous
spectrum of singular neutral modes is displaced downward and an isolated,
damped mode can be identified at a phase speed close to 0.4. Provided that the
singular segment lies on the correct side of the integration path, changes in the
path do not affect this eigenvalue; the continuous spectrum is, instead, further
moved towards lower values of ωifor increasing B.ValuesofBbetween 0.01
and 0.03 have been used in most of the calculations identifying neutral modes
in the present paper.
Clearly also other integration paths are possible; in fact, any complex
detour that leaves the singularities to its right side in the complex y-plane
will yield the correct physical eigenvalues. Since the physical solutions vanish
rapidly at infinity there is no need for the integration path to return to the real
axis at y=ymax in order to enforce the boundary conditions at the free stream
(Peter Schmid, private communication). In some calculations, the mapping
yc=y−iD(2ymax y−y2)1/2(24)
corresponding to a quarter of an ellipse has also been used successfully (Dwas
taken equal to 0.006 in our calculations).
The growing and decaying solutions obtained by our procedure are the as-
ymptotic limits of amplified and damped modes of the viscous stability equa-
tions as the Reynolds number approaches infinity (Lin 1955). We re-emphasise
here that it is only by the use of this procedure that neutral (and damped)
modes can be defined without ambiguities.
2.4. DNS method
2.4.1. Numerical scheme
The simulation code (see Lundbladh et al. 1999) employed for the present com-
putations uses spectral methods to solve the three-dimensional, time-dependent,
incompressible Navier–Stokes equations. The algorithm is similar to that of Kim,
Moin & Moser (1987), i.e. Fourier representation in the streamwise and span-
wise directions and Chebyshev polynomials in the wall-normal direction, to-
gether with a pseudo-spectral treatment of the nonlinear terms. The time
advancement used is a four-step low storage third-order Runge–Kutta method
for the nonlinear terms and a second-order Crank–Nicolson method for the
linear terms. Aliasing errors from the evaluation of the nonlinear terms are
removed by the 3
2-rule when the FFTs are calculated in the wall-parallel plane.

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64 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−0.025
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
wi
cr
Figure 4. Eigenvalue spectrum of the fundamental varicose
mode for α=0.2, β=0.45 and As=0.18, displayed as temporal
growth rate versus phase speed. It is obtained using a complex
integration path with B=0.03.
In wall-normal direction it has been found more efficient to increase resolution
rather than using dealiasing.
To correctly account for the downstream boundary layer growth a spatial
technique is necessary. This requirement is combined with the periodic bound-
ary condition in the streamwise direction by the implementation of a “fringe
region”, similar to that described by Bertolotti, Herbert & Spalart (1992). In
this region, at the downstream end of the computational box, the function λ(x)
in equation (25) is smoothly raised from zero and the flow is forced to a desired
solution vin the following manner,
∂u
∂t =NS(u)+λ(x)(v−u)+g,(25)
∇·u=0,(26)
where uis the solution vector and NS(u) the right hand side of the (unforced)
momentum equations. Both g, which is a disturbance forcing, and vmay
depend on the three spatial coordinates and time. The forcing vector vis
smoothly changed from the laminar boundary layer profile at the beginning of
the fringe region to the prescribed inflow velocity vector. This is normally a
boundary layer profile, but can also contain a disturbance. A convenient form
of the fringe function is as follows:
λ(x)=λmax[S(x−xstart
∆rise
)−S(x−xend
∆fall
+1)],(27)
where λmax is the maximum strength of the damping, xstart to xend the spatial
extent of the region where the damping function is nonzero and ∆rise and ∆fall

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On the breakdown of boundary layer streaks 65
xl ×yl ×zl nx ×ny ×nz Reδ0
δ0(resolution)
Box1 1940×34.4×22.06 576 ×65 ×32 272.2
Box2 1940×34.4×22.06 576 ×65 ×32 332.1
Box3 1702×34.4×16.68 512 ×81 ×8264.6
Box4 3096×34.4×16.68 1024 ×81 ×8591.6
Box5 3404×34.4×14.71 1024 ×81 ×8948.7
Tabl e 1 . Resolution and box dimensions for the simulations
presented. The box dimensions includes the fringe region, and
are made dimensionless with respect to δ0, the Blasius length
scale at the beginning of the computational box. The param-
eters zl and nz represent the full span and the total number
of Fourier modes, respectively. Note that zl corresponds in all
cases to one spanwise wavelength of the primary disturbance.
the “rise” and “fall” distance of the damping function. S(a) is a smooth step
function rising from zero for negative ato one for a≥1. We have used the
following form for S, which has the advantage of having continuous derivatives
of all orders:
S(a)=


0a≤0
1/[1 + exp( 1
a−1+1
a)] 0 <a<1
1a≥1
(28)
This method damps disturbances flowing out of the physical region and smoothly
transforms the flow to the desired inflow state, with a minimal upstream influ-
ence.
In order to set the free-stream boundary condition closer to the wall, a gen-
eralization of the boundary condition used by Malik, Zang & Hussaini (1985)
is implemented. Since it is applied in Fourier space with different coefficients
for each wavenumber, it is non-local in physical space and takes the following
from,
∂ˆu
∂y +|k|ˆu =∂ˆv0
∂y +|k|ˆv0,(29)
where kis the absolute value of the horizontal wavenumber vector and ˆu is
the Fourier transforms of u.Herev0denotes the local solution of the Blasius
equation and ˆv0its Fourier transform.
2.4.2. Disturbance generation and parameter settings
The presented numerical implementation provides several possibilities for dist-
urbances generation. The complete velocity vector field from the linear results

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66 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
by Andersson et al. (1999a) is used for the primary disturbance. These opti-
mally growing streaks, here denoted vd, are introduced in the fringe region by
adding them to the Blasius solution to yield the forcing vector v=v0+vd.
In order to trigger a secondary instability of the streaks a harmonic lo-
calized wall-normal volume force is implemented. The harmonic forcing, g=
(0,F,0), is constructed as an exponentially (in space) decaying function centred
at y=y0and x=xloc:
F=Cexp [−((x−xloc)/xscale)2]exp [−((y−y0)/yscale)2]g(z)f(t),(30)
where the constant Cdetermines the strength of the forcing and the parameters
xscale and yscale its spatial extent. The time dependence is provided by the
function
f(t)=S(t/tscale)cos(ωt),(31)
where ωis the angular frequency and the function Shas been used again in
f(t) to ensure a smooth turn on of the forcing (of duration tscale )inorderto
avoid problems with transients that may grow and cause transition in the flow.
It is also possible to choose the spanwise symmetry of the forcing, to separately
excite two classes of secondary disturbances; in
g(z)=cos(βz +φ),(32)
we choose φ=0or π
2for varicose or sinuous symmetries, respectively.
The box sizes and resolutions used for the simulations presented in this
paper are displayed in table 1. The dimensions of the boxes are reported in
δ0, which here denotes the boundary layer thickness at the beginning of the
computational box. Box1 is used to produce the non-linear streaks and study
their secondary instability, while Box2 is employed to verify the scaling property
introduced in section 3.2; Box3, Box4 and Box5 are, instead, used to test the
DNS against the linear results (see figure 7(a)). The Reynolds numbers based
on δ0are also reported in the table. For the calculations presented on the
secondary instability induced by harmonic forcing, we use xloc = 300δ0from
the beginning of the computational box, and y0=3δ0with xscale and yscale of
35δ0and 3δ0, respectively.
3. DNS Results
3.1. Nonlinear development of the streaks
Nonlinear mean fields are computed solving the full Navier–Stokes equations in
a spatially evolving boundary layer, using the optimal streaks as initial condi-
tions. The complete velocity vector field from the linear results by Andersson et
al. (1999a) is used as input close to the leading edge and the downstream non-
linear development is monitored for different initial amplitudes of the pertur-
bation. This is shown in figure 5(a), where all energies are normalized by their
initial values. The dashed line corresponds to an initial energy small enough for
the disturbance to obey the linearixed equations. For this case the maximum
of the energy is obtained at x=2.7; note that this location of maximum energy

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On the breakdown of boundary layer streaks 67
0.5 1 1.5 2 2.5 3
0.05
0.15
0.25
0.35
0.45
0.5 1 1.5 2 2.5 3
0
5
10
15
20
25
(a) (b)
E
E0
x
❅❅❅❅❅❅❘
A
x
Figure 5. (a) The energy of the primary disturbance, E,nor-
malized with its initial value, E0, versus the streamwise coor-
dinate, x,forβ=0.45 and Reδ=430. Here xhas been made
non-dimensional using the distance Lto the leading edge. The
arrow points in the direction of increasing initial energies, E0=
2.92 ×10−2,3.97×10−2,5.18×10−2,7.30 ×10−2,9.78×10−2,
1.36×10−1,1.81 ×10−1,2.33×10−1,2.91×10−1(E0is com-
puted at x=0.3). The dashed line represents the optimal linear
growth. (b) The downstream amplitude development for the
same initial conditions as in (a). The amplitude Ais defined
by equation (34). (The two lines have been circled for future
reference).
is weakly dependent on the initial amplitude, even for quite large values of the
initial energy.
A contour plot in the (z,y)-plane of the nonlinear mean field corresponding
to the circled line in figure 5(a) at x= 2 is shown in figure 6(b). This velocity
field may be expanded in the sum of cosines
U(y, z)=
∞

k=0
Uk(y)cos(kβz) (33)
where U0differs from the Blasius solution UBby the mean flow distortion term.
To be able to quantify the size of the primary disturbance field an amplitude
Ais defined as
A=1
2max
y,z (U−UB)−min
y,z (U−UB).(34)
When the shape assumption is adopted, Acoincides with As. Figure 5(b) dis-
plays the downstream amplitude development for the same initial conditions as
figure 5(a). One can note that the amplitude reaches its maximum value up-
stream of the position where the energy attains its peak, and starts to decrease
at a position where the energy is still increasing. This is due to the thickening
of the boundary layer.

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68 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
0 2 4 6 8
0
1
2
3
4
5
6
7
0 2 4 6 8
0
1
2
3
4
5
6
7
(a) (b)
y
z
y
z
Figure 6. (a) Contour plot in a (z, y)-plane of the primary
disturbance streamwise velocity using the shape assumption.
Thespanwisewavenumberisβ=0.45, the streamwise position
x=2 and the amplitude As=0.36. (b) Contour plot in a z-y
plane of the nonlinear mean field corresponding to the circled
line in figure 5(a) at x=2 (where A=0.36). Here Reδ=430. In
both figures the coordinates yand zhave been made non-
dimensional using the local Blasius length scale δ,atthe
streamwise position x=2. In fact, for all y, z plots hereafter
the cross-stream coordinates have been scaled using the local
Blasius length scale.
The effect of the nonlinear interactions on the base flow are shown by
the contour plots in figures 6. Figure 6(a) displays the primary disturbance
obtained using the shape assumption with As=0.36, while 6(b) shows a fully
nonlinear mean field, characterized by the same disturbance amplitude. In the
latter case, the low-speed region is narrower, therefore associated with higher
spanwise gradients, and displaced further away from the wall.
Abaseflowliketheonepresentedinfigure6(b)isrepresentativeofflat-
plate boundary layer flows dominated by streamwise streaks as encountered in
experiments (Bakchinov et al. 1995; Westin et al. 1994; Kendall 1985, 1990)
and simulations ( ).
3.2. Scaling of the mean field
In Andersson et al. (1999a) a scaling property of the optimal streamwise streaks
in the flat-plate boundary layer was found to exist. In a linearized setting,
they considered an upstream velocity perturbation at the leading edge of the
flat plate, uin(0), and its downstream response, uout(x), a distance xfrom the
leading edge, and maximized the output energy E(uout (x)) = E(x, β, E0,Re)
over all initial disturbances with fixed energy E(uin(0)) = E0.

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On the breakdown of boundary layer streaks 69
The disturbance energy can be written
E(u(x)) = 2π/β
0∞
0
(u2+v2
Re +w2
Re)dy dz, (35)
where v=v√Re and w=w√Re are the cross-stream velocities in boundary
layer scales. The optimal disturbances, which were calculated using the lin-
earized, steady boundary layer approximation, were found to consist of stream-
wise vortices developing into streamwise streaks. Since streamwise aligned vor-
tices contain no streamwise velocity component, the energy at the leading edge
E0can be written as
E0=E0
Re,where E0=2π/β
0∞
0
(v2+w2)dy dz, (36)
with E0is independent of the Reynolds number. The boundary layer equa-
tions governing (u, v, w, p), here p=pRe, contain no explicit dependence on the
Reynolds number; furthermore, all velocities are O(1) a distance sufficiently far
downstream of the plate leading edge. Hence, the streamwise velocity compo-
nent will dominate in the disturbance energy (35) and the output energy obeys
the scaling law
E(x, β, E0) = lim
Re→∞ E(x, β, E0,Re).(37)
This scaling property holds also when the solutions are obtained from the
Navier–Stokes equations, if u=0atx= 0. In figure 7 both linear and
nonlinear solutions obtained from the Navier–Stokes equations are presented
which verify (37).
Figure 7(a) displays E/E0versus xfor the spanwise wavenumber, β=0.45.
The solid line corresponds to a solution obtained using the linearized, steady
boundary layer approximations. The other three lines represent results ob-
tained from solving the Navier–Stokes equations, for three different Reynolds
numbers, with initial disturbance energies small enough to yield a linear evo-
lution. Figure 7(a) shows that the boundary layer approximation is valid and
yields solutions in agreement with those obtained from the Navier–Stokes sim-
ulations.
Figure 7(b) depicts two curves representing the spatial development of E
using an initial energy, E0, large enough to induce substantial nonlinear effects.
The two curves, which represent solutions to the Navier–Stokes equations for
the same initial energy and spanwise wavenumber, collapse onto one, although
they correspond to two different Reynolds numbers. From figure 7(b) we con-
clude that the scaling property (37) holds also when the velocity field of the
primary disturbance is fully nonlinear.
To clarify the implication of (37), consider the same dimensional problem
with the dimensional energy denoted E∗scaled with two different length scales,

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70 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
0.5 1 1.5 2 2.5 3
0
0.3
0.6
0.9
1.2 x 10−3
0 1 2 3
0
0.5
1
1.5
2
2.5
3
3.5 x 10−3
(a) (b)
E
E0
x
E
E0
x
Figure 7. The spatial energy growth versus the streamwise
coordinate, for the spanwise wavenumber, β=0.45. (a) With
three small-amplitude solutions to the Navier–Stokes equa-
tions for Reynolds numbers (- - - Re =1×106,-·-·-
Re =5×105,···Re =1×105), and (—–) one Reynolds num-
ber independent solution from the linearized, steady bound-
ary layer approximations. (b) With an initial amplitude large
enough to induce considerable nonlinear effects, A=0.30. Here
the two curves represent solutions to the Navier–Stokes equa-
tions for two different Reynolds numbers (—– Re =5×105,
and + Re =7.5×105).
Land L1.Wewrite
E∗(x, β, E0,Re)=E∗(x1,β1, E 1
0,Re
1),(38)
where the variables are scaled as
x∗=xL =x1L1,β
∗=βU∞
νL =β1U∞
νL1
and Re =U∞L
ν,Re
1=U∞L1
ν;
(39)
here x∗and β∗are the dimensional downstream position and spanwise wave
number, respectively. The disturbance energies scale as
E∗=EU2
∞δ2=E1U2
∞δ2
1and E0=E1
0.(40)
Introducing c2=L/L1=δ2/δ2
1and rewriting the right-hand expression in (38),
in the variables x, β, Re we obtain
c2E(x, β, E0,Re)=E(c2x, β/c, E0,Re/c
2).(41)
Now letting the Reynolds numbers tend to infinity and using (37) we get
c2E(x, β, E0)=E(c2x, β /c, E0),(42)
for each c>0.

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On the breakdown of boundary layer streaks 71
01234567
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
β
x
*
Figure 8. Loci of spanwise wavenumbers, β, and streamwise
positions, x, representing the known solutions from the linear
or nonlinear calculations at x=2 with β=0.45. The streamwise
position and the spanwise wavenumber have been made non-
dimensional using the distance from the leading edge Land
the Blasius length scale δ=(Lν/U∞)1/2, respectively.
An important physical implication of (37) can now be inferred from re-
lation (42). Since (42) is Reynolds number independent, a non-dimensional
solution E(x, β, E0) represent a continuous set of physical solutions in (x∗,β∗)-
space, for a fixed E0. We have seen that an initial array of streamwise aligned
vortices at the leading edge will result in an array of streamwise streaks down-
stream. Since the streamwise and spanwise length scales are coupled, increasing
the spanwise length scale at the leading edge will yield the same downstream
behaviour of the solutions but on a larger streamwise length scale.
In figure 8 the curve (c2x, β/c), with x=2 and β=0.45, is shown. From (42)
the results along this curve are known and correspond to a rescaling of the
solutions calculated at x=2 and β=0.45 (represented by the star in figure 8).
Note that (42) implies that Eincreases linearly as the streamwise coordi-
nate xincreases, and the spanwise wavenumber βdecreases (cf. figure 8). The
increase in the energy of the streak E∗is a result of the widening of the cross-
stream spatial extent of the disturbance. Since the shape of the streak velocity
profile is the same, this implies that the amplitude remains constant along the
curve xβ2=constant. In contrast, the energy of the corresponding initial vor-
tex E∗
0remains the same for this parameter combination. This implies that
the amplitude of the initial vortex, Av, increases linearly with the spanwise
wavenumber, i.e. Av∼β. Thus the amplitude of the initial vortex needed to
produce a fixed amplitude of the streak along the curve xβ2=constant decreases
in a manner inversely proportional to the spanwise wavelength.

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72 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
0 2 4 6 8
0
1
2
3
4
5
6
7
1 1.5 2 2.5 3
0
0.005
0.01
0.015
0.02
(a) (b)
y
z
−αi
x
Figure 9. (a) Isocontours of r.m.s. values of the streamwise
velocity component of the secondary disturbance (ωr=0.211)
for the fundamental sinuous mode at x=2, obtained from the
DNS. The dashed line represents the contour of the constant
value of the mean field corresponding to the phase speed of
the disturbance ( U=cr=0.80). (b) Spatial growth rates, αi
versus x; —–, DNS data with Reδ=430 and β=0.45; ∗, linear
temporal inviscid stability calculations using mean fields at
each corresponding x-position and streamwise wavenumber
αr=0.260.
3.3. Secondary instability results from DNS
In this section direct numerical simulations of the secondary instability of
streaks in a spatially growing flat-plate boundary layer are compared to the
results from the inviscid secondary instability theory. This is done to ensure
that the inviscid approximation is appropriate and can be used in further in-
vestigations and parametric studies of streak instabilities.
The spatial stability problem is defined by the use of a real frequency ω
and a complex wavenumber α=αr+iαi. Here the spatial growth rate −αi
is obtained from the maximum of the streamwise velocity component of the
secondary disturbance. The secondary disturbances are triggered using the
harmonic forcing introduced in section 2.4.2, allowing for the two symmetries
of fundamental type which can be excited separately. The amplitude of the
volume force is selected low enough to yield linear secondary disturbances,
avoiding the appearance of higher harmonics in the frequency spectra.
To choose the forcing frequency for the DNS, temporal linear secondary
stability calculations for the sinuous mode are performed using the nonlinear
mean field corresponding to the circled line in figure 5(a), at the local position
x=2. The selected mean field has amplitude A=0.36, close to the threshold
value for secondary instability in plane channel flow (Elofsson et al. 1999). The
maximum temporal growth is found for α=0.257, corresponding to a secondary
disturbance frequency of 0.211.

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On the breakdown of boundary layer streaks 73
The direct numerical simulations for the fundamental sinuous mode of the
secondary instability are carried out using this forcing frequency, ω=0.211,
and the velocity fields are Fourier transformed in time to obtain the amplitude
variation in the streamwise direction and the cross-stream distribution of the
disturbance velocity at the frequency of the forcing.
Figure 9(a) shows the urms distribution of the fundamental sinuous mode
at x=2. Note how the disturbance fluctuations follow quite closely the dashed
line representing the contour of constant value of the mean field velocity corre-
sponding to the phase speed of the secondary instability, U=cr=0.80. The
solid line in figure 9(b) represents the spatial growth rate of the sinuous mode
obtained from the direct numerical simulations. Here the secondary instability
is excited at the streamwise position x=0.85. However, since the local forcing
does not input pure eigenmodes the values of the growth rates are measured
from an x-position downstream of the forcing, where the onset of an ”eigen-
function” is identified.
Linear temporal stability calculations, using the real part of the streamwise
wavenumber obtained from the direct numerical simulations, αr=0.260, are also
performed, employing mean fields extracted at different streamwise positions
from the DNS. In order to compare the spatial results to the growth rates ob-
tained from the temporal inviscid stability problem (12), (13) a transformation
first proposed by Gaster (1962) is employed:
ωi=−αi
∂α/∂ω .(43)
From the temporal eigenvalues, Gaster’s transformation (43) provides estimates
of spatial growth rates (cf. the stars in figure 9(b)). The agreement between
the stability theory and the full simulation results can be regarded as good,
since the linear stability calculations are inviscid and performed under the
assumptions of parallel mean flow. Note that, as one could expect, the inviscid
approximation provides a slight overestimate of the amplification factors, and
that closer agreement is found as the Reynolds number increases; here Re=500
000 at the streamwise position x=2.7.
Using the same saturated mean field, direct numerical simulations are also
carried out for the fundamental varicose mode of the secondary instability.
Attempts to identify instabilities are made with different frequencies and for
different streamwise and wall normal positions of the forcing in the direct sim-
ulations. Also, linear stability calculations at x=2 and for a range of different
streamwise wavenumbers are performed. Both methods produce only stable
solutions for this symmetry of the disturbances.
However, linear calculations using as base flow the streaks obtained with
the largest initial energy tested (see figure 5), produce small positive temporal
growth rates for the fundamental varicose instability. We then proceed as for
the sinuous case: the largest growth rate, ωi, is identified to correspond, at x=2,
to a streamwise wavenumber α=0.250 and a frequency ωr=0.217. This value
is used in the DNS forcing and the spatial growth rates obtained are compared

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74 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
0 2 4 6 8
0
2
4
6
8
1 1.5 2 2.5 3
−0.01
−0.005
0
0.005
0.01
(a) (b)
y
z
−αi
x
Figure 10. (a) Isocontours of r.m.s. values of the streamwise
velocity component of the secondary disturbance (ωr=0.217)
for the fundamental varicose mode at x=2, obtained from the
DNS. The dashed line represents the contour of the constant
value of the mean field corresponding to the phase speed of the
disturbance ( U=cr=0.863). (b) Spatial growth rates αi
versus x; —–, DNS data with Reδ=430 and β=0.45; ∗, linear
temporal inviscid stability calculations using mean fields at
each corresponding x-position and streamwise wavenumber
αr=0.252.
to linear stability calculations performed at different streamwise positions for
αr=0.252. The results are shown in figure 10(b); the inviscid analysis gives
small positive growth rates, while in the DNS the perturbation growth rates
remain close to neutral as xexceeds 2.
Figure 10(a) shows the urms distribution of the fundamental varicose mode
together with the contour of constant value of the mean field velocity corre-
sponding to the phase speed of the secondary instability, i.e. U=cr=0.863.
Note also here the close correspondence between the critical layer (displayed in
the figure with a dashed line) and regions of intense u-fluctuations.
From the above calculations and comparisons we draw as first conclusion
that the secondary instability of streamwise streaks is initially of sinuous type,
and that the essential stability features can successfully be captured by an
inviscid approach. The above statements are further comfirmed by the results
described below.
4. Inviscid Secondary Instability Results
4.1. The shape assumption versus the nonlinearly developed mean field
As a preliminary investigation, the secondary instability of streaks approxi-
mated by the shape assumption was parametrically studied (Andersson et al.
1999b). Comparison of the results with those obtained from calculations where

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On the breakdown of boundary layer streaks 75
0 2 4 6 8
0
1
2
3
4
5
0 2 4 6 8
0
1
2
3
4
5
0 2 4 6 8
0
1
2
3
4
5
0 2 4 6 8
0
1
2
3
4
5
y
z
✲
✻
Fundamental sinuous Subharmonic sinuous
Fundamental varicose Subharmonic varicose
(a) (b)
(c) (d)
2π
β
2π
β
2π
β
2π
β
Figure 11. Contours of constant absolute values of the
streamwise velocity component of four different kinds of modes
obtained using the shape assumption. The dashed lines rep-
resent the contours of the constant value of the mean field
corresponding to the phase velocities of the disturbances.
The sinuous modes are calculated with parameters α=0.150,
β=0.45 and As=0.36 at x=2 (see figure 6a)(cr=0.627 and
ωi=0.00301 for the fundamental mode; cr=0.661 and
ωi=0.0104 for the subharmonic mode). The varicose modes
are calculated with parameters α=0.150, β=0.45 and As=0.38
at x=2 (cr=0.379 and ωi=0.00998 for the fundamen-
tal mode; cr=0.371 and ωi=0.00297 for the subharmonic
mode). Note that the real and imaginary parts of the subhar-
monic modes have a period of 4π/β. However, their absolute
value s are 2π/β-periodic.
the base flow is the nonlinearly developed streak demonstrate the inapplicability
of the shape assumption for this type of studies (except for sinuous symmetries
where a qualitative agreement could still be claimed).
In figures 11 and 12 the u-eigenfunctions, obtained with the shape assump-
tion approximation and the nonlinear mean field, respectively, are displayed for
the parameters indicated.

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76 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
Figure 11(a) shows the fundamental sinuous mode which is characterized
by out-of-phase oscillations on either side of the low speed-streak, whereas
the near-wall region is relatively quiescent. The subharmonic sinuous mode
(figure 11b) has real and imaginary parts of usymmetric around the z=0
and z=2π/β axes. This eigenfunction shows a striking resemblance to that
obtained by Ustinov (1998) who solved the linearized Navier–Stokes equations
in time; it is also the same high frequency mode triggered in the experiments
by Bakchinov et al. (1995). In figure 11(c) the fundamental varicose mode
is displayed. In-phase fluctuations are spread in a z-range around π/β and
halfway through the undisturbed boundary layer height. Some effect is also
noticeable around z=0and2π/β, close to the wall. This mode is also very
similar to that computed by Ustinov (1998). The subharmonic varicose mode
(figure 11d) is characterized by almost the same phase speed as that of its
fundamental counterpart, but here the real and the imaginary parts of this
u-eigenfunction are anti-symmetric around the axes z=0andz=2π/β.
The fundamental u-eigenfunction displayed in figure 12(a) was obtained
using the same mean field and streamwise wavenumber as the direct numerical
simulations shown in figure 9(a). The agreement between figures 9(a) and 12(a)
is very good and in fact, for this symmetry, there is also a fair agreement with
the u-eigenfunction displayed in figure 11(a), which was obtained using the
shape assumption. Also, the sinuous subharmonic u-eigenfunction 12(b) is in
fair agreement with the one obtained using the shape assumption displayed
in figure 11(b). In contrast, the fundamental and subharmonic varicose u-
eigenfunctions are in poor agreement with both the urms plot of figure 10(a),
obtained from direct numerical simulations, and the u-eigenfunctions of fig-
ures 11(c) and 11(d). As shown in figures 12(c) and 12(d) the u-eigenfunctions
are considerably less diffuse and strongly concentrated around the isoline U=
cr.
The growth rates of the varicose (fundamental and subharmonic) symme-
tries are highly over-predicted when the shape assumption is used. The positive
growth rates of the fundamental varicose case are found to be even larger than
those for the fundamental sinuous case, which contradicts experiments and
previous, comparable, calculations (Schoppa & Hussain 1997; 1998)
In comparing the mean fields obtained from the shape assumptions to
the fully nonlinear ones, we find that the inflection point in the wall-normal
direction is smoothed by the nonlinear modification, cf. figure 13(a). This
figure shows the streamwise velocity profiles centred on the low-speed streak
(i.e. z=π/β) for the two types of mean fields and for a large amplitude.
This-zlocation has been chosen since it is where a wall-normal inflection point
first appears when the streak’s amplitude is increased, and also where the
varicose eigenfunctions achieve their peak values. As reported in a number
of experimental and numerical studies (Swearingen & Blackwelder 1987; Yu
& Liu 1991; Bottaro & Klingmann 1996; Matsubara & Alfredsson 1998),

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On the breakdown of boundary layer streaks 77
02468
0
1
2
3
4
5
6
7
02468
0
2
4
6
8
02468
0
1
2
3
4
5
6
7
02468
0
2
4
6
8
y
z
✲
✻
Fundamental sinuous Subharmonic sinuous
Fundamental varicose Subharmonic varicose
(a) (b)
(c) (d)
2π
β
2π
β
2π
β
2π
β
Figure 12. Contours of constant absolute values of the
streamwise velocity component of four different kinds of modes
obtained using the nonlinear mean fields. The dashed lines
represent the contours of the constant value of the mean field
corresponding to the phase velocities of the disturbances. The
sinuous modes are calculated using the nonlinear mean field
corresponding to the circled line in figure 5(a), at streamwise
position x=2, where A=0.36, for a streamwise wavenumber
α=0.280 (cr=0.821 and ωi=0.0144 for the fundamen-
tal mode; cr=0.839 and ωi=0.0125 for the subharmonic
mode). The varicose modes are calculated using the mean
field with largest streak’s amplitude (see figure 5(b)) at posi-
tion x=2,whereA=0.378, for a streamwise wavenumber
α=0.275 (cr=0.866 and ωi=0.00218 for the fundamen-
tal mode; cr=0.876 and ωi=0.00243 for the subharmonic
mode). In all calculations Reδ=430 and β=0.45.
the wall-normal inflection point can be related to the varicose mode, whereas
sinuous instabilities correlate well to the spanwise mean shear.
The reason for the overpredicted varicose amplification factor when using
the shape assumption can be deduced from inspection of figure 13: an inflection
point appears for yclose to 3 in this case, but it disappears when the base flow

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78 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
−0.5 0 0.5 1
0
2
4
6
8
10
(a) (b)
y
U/U∞
y
Uk/Uk,max
✻
Figure 13. (a) The total streamwise velocity versus the wall-
normal coordinate at z=π/β for: - - - -, the nonlinear mean
field, and - ·-·-, the shape-assumption-approximated mean
field. Here both amplitudes are A=As=0.33, the streamwise
position is x=2 and the spanwise wavenumber is β=0.45; the
Blasius profile is drawn, for reference, with a solid line. (b) The
different Fourier modes Uk(y) representing the streamwise ve-
locity of the nonlinear mean field in (a) versus the wall-normal
coordinate. The dashed line corresponds to the mean flow dis-
tortion, (U0−UB), and the arrow points in the direction of
higher-order modes. Every mode is normalized with its max-
imum ((U0−UB)max=0.11, U1,max=0.26, U2,max=6.6×10−2,
U3,max=1.1×10−2,U4,max=9.2×10−4.) Here Reδ=430.
contains all harmonics of the streak. Both the direct numerical simulations
and the linear stability calculations using the nonlinearly distorted mean fields
produce stable varicose modes for A=0.33. In fact, only for the largest streak
studied in this paper, corresponding to an amplitude A=0.373, a slightly unsta-
ble varicose mode is found. The discrepancy in the secondary stability results
between the two cases can be traced to the mean flow distortion (U0−UB)
(see the dashed line in figure 13b). Calculations employing a “nonlinear” mean
field constructed without the mean flow distortion (where U0is replaced by
UB) result in varicose perturbations with positive growth rates for A=0.33.
Furthermore, the phase speed of the secondary instability is considerably
increased when using the nonlinear mean fields. This can be explained by
observing that nonlinearities “move” the primary instability outwards from the
wall. In figure 13(b) the individual Fourier modes from the cosine expansion of
the nonlinear mean field from figure 13(a) are shown, normalized to unit value.
Note that higher-order modes are displaced away from the wall. The phase
speed of the secondary instability equals the mean field velocity at the critical
layer, cf. figure 11. Therefore, as the critical layer is moved outwards, where

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On the breakdown of boundary layer streaks 79
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
ωi
β(x)
Figure 14. Results for the fundamental sinuous growth rates,
ωi, versus the spanwise wavenumbers along the curve in fig-
ure 8. The star gives the value of the amplification factor
obtained when A=0.36, α=0.30, β=0.45 and x=2, i.e.
ωi=0.014.
the mean flow velocity is higher, the phase speed of the secondary instability
is also increased.
4.2. Parametric study
In figure 5(b) the downstream amplitude development of streamwise streaks for
a spanwise wavenumber β=0.45 and for different initial amplitudes is shown.
Most of the linear stability calculations are performed using the velocity fields
with the amplitudes found at x=2. This position was chosen since it is close to
the point where the primary disturbance energy attains its maximum value.
According to the scaling property of the mean fields derived in section 3.2,
the results obtained for the parameters used in this section, x=2 and β=0.45,
can be rescaled to apply for all values of β(see figure 8). This implies that
a result from a secondary instability calculation obtained using a mean field
corresponding to a point on the curve in figure 8 can be rescaled to yield
the amplification value for all points on this curve. Since ωi/β=constant, the
value of the constant can be determined from a secondary instability result
for a specific parameter combination. The line representing the growth rates
of the fundamental sinuous symmetry is displayed in figure 14. In practice,
however, the scaling property relating ωiand βis limited to an intermediate
range of β. Since their distance of amplification is so short, large values of
βwill need a very large initial disturbance amplitude at the leading edge,
while disturbances represented by low values of βwill saturate far downstream,
where Tollmien–Schlichting instabilities may become important and change the

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80 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
0.1
0.15
0.2
0.25
0.3
0.35
0.4
α
x
Figure 15. Isocontours of the growth rate ωi,inthe(x, α)-
plane, for the sinuous fundamental mode, employing the mean
field corresponding to the circled line in figure 5. The maxi-
mum contour level is 0.014 and the spacing is 0.0014. The ∗
represents the maximum growth rate, ωi=0.0147, obtained at
position x=1.88 and for a streamwise wavenumber α=0.259.
transition scenario. For larger values of β, the corresponding x-position is closer
to the leading edge and viscous effects may have a damping influence on the
amplification of unstable waves.
Primary disturbances with βin the range [0.3,0.6] (here considered with
respect to a fixed streamwise position, x=1), have the largest transient amplifi-
cation (Andersson et al. 1999a; Luchini 2000). The spanwise distance selected
in the controlled experiments by Bakchinov et al. (1995) corresponds to a
value of βequal to 0.45 (at the location of their roughness elements), and
this is also the scale of boundary layer fluctuations in the presence of free-
stream turbulence (Westin et al. 1994). It could be speculated that this partic-
ular range of βis particularly appropriate when dealing with boundary layer
streaks, since it corresponds to a spanwise spacing of about 100 wall units once
viscous length scales are introduced. The spacing of 100 viscous wall units
is not only obeyed by quasi-regular streaks in turbulent but also in laminar
and transitional boundary layers (see Blackwelder 1983; Kendall 1985, 1990;
Westi n et al. 1994). It is also the typical transverse ’box’ dimension for turbu-
lence to survive in the minimal channel simulations of Jimenez & Moin (1991)
and Hamilton, Kim & Waleffe (1995). In the present case, if a simple measure
of the friction velocity is adopted by the use of the Blasius wall shear, it is
easy to see that streaks spaced 100 wall units apart are present in a subcritical
(with respect to TS waves) boundary layer if βis in the range (0.3, 0.63).

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On the breakdown of boundary layer streaks 81
0 0.2 0.4 0.6
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6
0
0.005
0.01
0.015
0.02 (a) (b)
✻
A

✒
A
ωi
α
cr
α
Figure 16. Temporal growth rate (a) and phase speed (b)
versus streamwise wavenumber for the fundamental sinuous
modes for different amplitudes of the primary disturbance (- -
-A=28.8,- ·-·-A=31.7,——A=34.5, —+— A=36.4, —∗—
A=37.3). The arrows point in the direction of increasing A.
Figure 15 shows the isocontours of the growth rate ωi, for the fundamental
sinuous instability in the (x, α)-plane for the mean field corresponding to the
circled line in figure 5. The growth rates do not vary significantly for the range
of xbetween 1.6<x<2.2, and in this interval, the maximum growth rate is
obtained for nearly the same value of the streamwise wavenumber (α=0.259).
Stability calculations are, therefore, performed on the mean field at x=2, where
the primary disturbance has saturated and, for the cases with lower initial
energy, the streak amplitude achieve its maximum value (cf. figures 5).
An extensive parametric study is carried out for the sinuous fundamental
(γ= 0), arbitrarily detuned (0 <γ<0.5) and subharmonic (γ=0.5) symme-
tries, which were the only ones found to be significantly unstable. At first the
Floquet parameter is set to zero, i.e. fundamental modes are focused upon. In
figure 16(a) the growth rate of the instability ωi=αciis plotted against the
streamwise wavenumber, for the different amplitudes of the streaks, obtained
with the DNS. One can note that on increasing the amplitude, not only do the
growth rates increase but their maxima are also shifted towards larger values
of the streamwise wavenumber α. As shown in figure 16(b) the phase speeds
of the fundamental sinuous modes are but weakly dispersive.
Next, we examine the effect of changes in the spanwise wavelength of the
secondary disturbance, i.e. we study the effect of the detuning parameter
γ. It is often assumed that the preferentially triggered secondary instability
modes have the same transverse periodicity as the base flow; this is not at all
evident here. The full system of equations (12)-(13) has been solved without
resorting to symmetry considerations to yield the results displayed in figures 17,
corresponding to a streamwise wavenumber αequal to 0.255 and for mean
fields with amplitudes large enough to lead to instabilities for the chosen α.A

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82 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
0 0.1 0.2 0.3 0.4 0.5
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5
0
0.005
0.01
0.015
0.02 (a) (b)
✻
A
✻A
ωi
α
cr
α
Figure 17. Temporal growth rate (a) and phase speed (b)
versus the Floquet parameter for sinuous modes, for four dif-
ferent amplitudes of the primary disturbance (symbols as in
figure 16).
monotonic behaviour is observed in the γrange of [0, 0.5] except for the case
of lower amplitude A.
The behaviour of the amplification factor of the subharmonic modes for
different streamwise wavenumbers and different amplitudes of the primary dis-
turbance is shown in figure 18(a). For amplitudes larger than about 0.30, the
subharmonic symmetry produces lower maximum growth rates than the fun-
damental symmetry. Note, however, that for lower amplitudes the sinuous
subharmonic symmetry represents the most unstable mode. The phase speed
for the subharmonic symmetry, displayed in figure 18(b), is larger than in the
fundamental case and the waves are slightly more dispersive.
4.3. The neutral conditions of streak’s breakdown
A study has been conducted to identify the marginal conditions of breakdown,
with each neutral point ωi= 0 calculated for a range of αby means of the
complex integration technique discussed in section 2.3. The steady base velocity
profiles obtained with the DNS at x=2 are used here. The results are displayed
in figures 19(a-b) for the two sinuous symmetries, together with contour levels
of constant growth rates.
It is immediately observed that a streaks amplitude of about 26% of the
free-stream speed is needed for breakdown to occur. Although this critical value
is achieved for small values of α(where the parallel flow assumption becomes
questionable) we are roughly around the values reported by P. H. Alfredsson
(private communication, 1998), who stated that ”amplitudes of at least 20%”
are needed for an instability of the streaks to emerge, and by Bakchinov et
al. (1995) who in their experiments produced streaks with A≈20% and
generated their controlled excitation with a vibrating ribbon. In the case of
plane Poiseuille flow, the experiments of Elofsson et al. (1999) show that the

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On the breakdown of boundary layer streaks 83
0 0.2 0.4 0.6
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6
0
0.005
0.01
0.015
0.02 (a) (b) ✻
A

✒
A
ωi
α
cr
α
Figure 18. Temporal growth rate (a) and phase speed (b)
versus streamwise wavenumber for the subharmonic sinuous
modes for different amplitudes of the primary disturbance (—
×—A=25.6,——A=27.2, - - - A=28.8, -·-·-A=31.7,——
A=34.5, —+— A=36.4, —∗—A=37.3).
0.27 0.29 0.31 0.33 0.35 0.37
0.1
0.2
0.3
0.4
0.5
0.6
0.25 0.27 0.29 0.31 0.33 0.35 0.37
0.1
0.2
0.3
0.4
0.5
0.6
(a) (b)
AA
αα
Figure 19. Neutral curves for streak instability in the (A, α)-
plane for (a) fundamental sinuous mode, (b) subharmonic sin-
uous mode (contour levels: ωi=0, 0.0046, 0.0092).
threshold amplitude for streaks’ breakdown is 35%, irrespective of the Reynolds
number. This Reynolds-number independence was also observed in direct nu-
merical simulations of Couette flow by Kreiss, Lundbladh & Henningson (1994)
who reported that “the disturbances in the calculations are found to reach an
amplitude of order one for all Reynolds numbers before the rapid secondary
instability sets in”.
One can notice that the subharmonic mode is unstable for lower amplitudes
than the fundamental mode and that the growth rates for larger amplitudes
are quite close for the two symmetries. The direct numerical simulations and
experiments of oblique transition in a boundary layer conducted by Berlin,

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84 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
Lundbladh & Henningson (1994) and Berlin, Wiegel & Henningson (1999)
show that a subharmonic breakdown of the streaks precedes transition to tur-
bulence.
We do not present any results for the varicose instabilities here. In fact,
both the fundamental and the subharmonic symmetries resulted in weak in-
stabilities for amplitudes larger than A=0.37 with growth rates smaller than
one fifth of the corresponding sinuous growth rates. Therefore a breakdown
scenario triggered by a varicose instability seems unlikely.
It appears then that there is not a dominating mode but rather that fun-
damental and detuned sinuous instabilities have the same probability of being
observed. Hence, the knowledge provided by these results must be combined
with that of the inflow disturbance spectrum, i.e. the prevailing receptivity
conditions. The present study furnishes possible scenarios which should be
confirmed by careful experiments, i.e. with controlled harmonic disturbances
to try and trigger specific modes.
5. Conclusions
We have investigated one of the mechanisms which is a possible precursor of
transition to turbulence in a boundary layer, namely the linear instability of
streaks produced by the non-modal streamwise evolution of free-stream distur-
bances. Such a breakdown has been observed in experiments carried out by the
Swedish (Westin et al. 1994; Alfredsson & Matsubara 1996) and the Russian
groups (Gulyaev et al. 1989; Bakchinov et al. 1995): they generated streaky
structures and visualized their development, breakdown and the formation of
turbulent spots, via smoke injection.
There is starting to be a good correspondence between experiments and
theory, and most of the segments of transition induced by the breakdown of
streaks are now elucitated (at least qualitatively). Our study aims at the mod-
elling of only one part of this process. More complete pictures are starting to
emerge, often based on simple model systems, particurlarly for the description
of the self–sustained process that makes near–wall turbulence viable ( for a
recent account refer to the book by Panton 1997). Although similarities exist
between the wall turbulence process and the breakdown of laminar streaks, it
is best not to draw definite parallels because of the widely different space and
time scales involved in the two cases.
Clearly, other steps can be envisioned to lead to early transition to turbu-
lence (i.e. strong nonlinearities, resonant interactions, etc.) and the present
work represents but one brick in the building of a comprehensive picture. One
has to further appreciate the fact that in an actual experiment irregular streaky
structures are often observed, i.e. with non-uniform spacing and with neigh-
bouring streaks in different stages of development, see Bottin, Dauchot & Davi-
aud (1998) for an example in plane Couette flow. Thus, these structures do not
necessarily become unstable together at a given x-position, but their breakdown
will likely occur in an irregular manner. These aspects are linked to the flow

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On the breakdown of boundary layer streaks 85
receptivity, the understanding of which is, hence, crucial. For recent progress
in this direction the reader is referred to Luchini & Bottaro (1998) and Airiau
& Bottaro (1998).
The local, large Reynolds number limit has been considered here, with
the implication that this simplified approach captures the essential features of
the instability. The inviscid assumption means that one has to be careful in
choosing the integration path for the eigenvalue calculation, and a simple proce-
dure for identifying the singularities in the complex y-plane and for integrating
around them has been outlined. With our approach, inspired by Lin (1944),
neutral and damped inviscid modes can be computed, and a quasi-linear be-
haviour of the growth rate of the instability with the streak’s amplitude is
found, in agreement with the careful channel flow measurements by Elofsson
et al. (1999).
We have shown here that both the linear and nonlinear spatial develop-
ment of optimal streamwise streaks are well described by the boundary layer
approximation and, as a consequence, Reynolds number independent for large
enough Reynolds numbers. This results in a boundary layer scaling property
that couples the streamwise and spanwise scales, implying that the same solu-
tion is valid for every combination of xand βsuch that the product xβ2stays
constant. The parameter study of streak instability is therefore representative
of a wide range of intermediate values of βfor which saturation occurs at a
reasonable x: large enough so that the boundary layer approximation may still
be valid and small enough so that Tollmien–Schlichting waves may not play a
significant role.
The different modes of instability have been catalogued and studied. At a
preliminary stage secondary instability calculations of the shape-assumption-
approximated mean fields were carried out; however, most results presented
here are performed employing the fully nonlinear mean fields. In comparing
the two levels of approximations we conclude that the shape assumption must
be abandoned in secondary instability studies of streamwise streaks in flat-plate
boundary layers. The secondary instability results are very sensitive to a slight
change in the shape of the mean field velocity profile and, even if the sinuous
modes are reasonably well captured by the shape assumption, the growth rates
of varicose modes are widely over-predicted.
When considering the nonlinear mean field we find that the sinuous modes
are by far the dominating instabilities. The varicose modes become unsta-
ble only for very large amplitudes (around 37% of the free-stream speed) and
should, therefore, be rarely observed in natural transition. This is in agreement
with DNS and experiments, where the sinuous modes of instability are most
often reported for the streak breakdown.
Noticeable is the fact that the sinuous, detuned instability waves can be
more amplified than the fundamental modes. The subharmonic modes are in
fact found to first become unstable with a critical streak amplitude of about
26% of the free-stream velocity. Plots of the eigenfunctions for fundamental

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86 P. Andersson, L. Brandt, A. Bottaro & D.S. Henningson
and subharmonic modes demonstrate clearly that the instability is concentrated
around the critical layer, and both types of sinuous modes of breakdown are
found to be almost non-dispersive. When the streak amplitude is large enough
(around 30% of the free-stream velocity) both the fundamental and detuned
modes have positive growth rates; thus, they might both be observed and
to decide on their downstream fate is a matter of environmental bias. It is
noteworthy that both experiments (Bakchinov et al. 1995) and DNS (Berlin et
al. 1994) did show subharmonic breakdown of the streaks, although neither
paper stated so explicitly.
Neutral curves have been obtained here in the amplitude–streamwise wave-
number plane; their identification should prove useful for controlling transition
and near-wall turbulence.
Future experiments under controlled conditions may attempt to trigger
some of the modes described here. Also, an interesting direction of future re-
search concerns the search for a possible absolute instability of the streaks.
Finding a self-sustaining instability mechanism could provide a firmer connec-
tion with the birth of turbulent spots.
Acknowledgments
We thank Theo Randriarifara for making his stability code available for some
of the validation tests, Paolo Luchini and Peter Schmid for interesting and
fruitful discussions, and the referees for valuable comments that helped us
produce a more physically relevant account of the breakdown of boundary
layer streaks. Part of the work was performed during the first author’s stay
in Toulouse, supported by “Internationaliseringsmedel f¨or Doktorander” dnr
930-766-95, dossier 71, at the Royal Institute of Technology.
References
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90 References
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Transition of streamwise streaks in zero
pressure gradient boundary layers
By Luca Brandt and Dan S. Henningson1
Department of Mechanics, Royal Institute of Technology (KTH), S-100 44
Stockholm, Sweden
1. Introduction
A scenario for bypass transition likely to occur in a flat plate boundary layer
flow under free–stream turbulence is studied. The disturbances at the lead-
ing edge of the flat plate that show the highest potential for transient energy
growth consist of streamwise aligned vortices. Due to the lift-up mechanism
these optimal disturbances are transformed downstream into elongated stream-
wise streaks with significant spanwise modulation. The initial disturbance that
yields the maximum spatial transient growth in a non-parallel flat plate bound-
ary layer flow was determined by Andersson, Berggren & Henningson (1999) by
applying the boundary layer approximations to the three-dimensional steady
incompressible Navier-Stokes equations and linearizing around the Blasius base
flow. If the disturbance energy of the streaks becomes sufficiently large, sec-
ondary instability can take place and provoke early breakdown and transition,
overruling the theoretically predicted modal decay. A possible secondary insta-
bility is caused by inflectional profiles of the base flow velocity, a mechanism
which does not rely on the presence of viscosity. Experiments with flow visu-
alizations by for example Matsubara & Alfredsson (2001) have considered the
case of transition induced by streaks formed by the passage of the fluid through
the screens of the wind-tunnel settling chamber. They report on the presence
of a high frequency ”wiggle” of the streak with a subsequent breakdown into a
turbulent spot.
In Andersson et al. (2001) Direct Numerical Simulations (DNS), using a
numerical code described in Lundbladh et al. (1999), are used to follow the
nonlinear saturation of the optimally growing streaks in a spatially evolving
boundary layer. The complete velocity vector field from the linear results by
Andersson et al. (1999) is used as input close to the leading edge and the down-
stream nonlinear development is monitored for different initial amplitudes of
the perturbation. Inviscid secondary instability calculations using Floquet the-
ory are performed on the obtained mean flows and it is found that the streak
critical amplitude, beyond which streamwise traveling waves are excited, is
about 26% of the free-stream velocity. The sinuous instability mode (either
1Also at FOI, The Swedish Defense Research Agency, Aeronautics Division, S-17290 Stock-
holm, Sweden
93

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94 L. Brandt & D.S. Henningson
the fundamental or the subharmonic, depending on the streak amplitude) rep-
resents the most dangerous disturbance. Varicose waves are more stable, and
are characterized by a critical amplitude of about 37%.
Here, also using DNS, we study the transition process resulting from the
sinuous secondary instability. A velocity vector field from the simulations pre-
sented in Andersson et al. (2001) is used as inflow condition. In those simula-
tions a spanwise antisymmetric harmonic volume force was added to the non
linear streaks to trigger their sinuous secondary instability in order to check the
linear stability calculations. Here the saturated streaks, vs, and the secondary
instability mode, vd, obtained filtering the velocity field at the frequency ωof
the forcing, are introduced as inflow condition by adding them to the Blasius
solution to give the forcing vector v=v0+vs+Avdeiωt. An amplifica-
tion factor Ais used for the secondary instability to give transition within the
computational box. The late stages of the process are investigated and flow
structures identified. They are different from the case of transition initiated
by Tollmien-Schlichting waves and their secondary instability (see Rist & Fasel
1995 as example) or by-pass transition initiated by oblique waves (Berlin et al.
1999). In these latter two scenarios Λ-vortices with strong shear layer on top,
streamwise vortices deforming the mean flow and inflectional velocity profiles
are observed. Berlin et al. (1999) speculated that the pattern of Λ-vortices
appearing is then independent on the presence of Tollmien-Schlichting waves,
but depends only on the streawise streaks and the oblique waves. These two are
key elements also in the present case, but a different spatial symmetry property
of the amplifying disturbance gives different flow structures. The present case
shows analogies with streak instability and breakdown found in the near wall
region of a turbulent boundary layer (see Schoppa & Hussain 1997 or Kawahara
et al. 1998).
2. Results
In this section we give an overview of the full transition of a streamwise streak
subjected to sinuous secondary instability. Time averaged statistics and Fourier
analysis of the results are presented while instantaneous flow structures are
discussed in the next section. Our simulation starts at Reδ∗
0= 875 (x=0)and
if not stated differently, in the results presented the coordinates are made non
dimensional using the inflow boundary layer thickness δ∗
0. The computational
box is 6.86 δ∗
0wide, corresponding to one spanwise wavelength of the streak, and
10.7 δ∗
0high. A simulation with the inlet moved further downstream (Reδ∗=
1044) is also performed to have some fully developed turbulence within the
computational box using the same number of modes. The length of the boxes
is 380 δ∗
0. 1440×97 ×72 spectral modes are used respectively in the streamwise,
wall-normal and spanwise directions.
To extract information on the frequency content of the flow, sixteen velocity
field are saved during one period of the secondary instability mode. We then
transform these velocity fields in time and in the spanwise direction to Fourier

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Transition of streamwise streaks 95
0 50 100 150 200 250 300
10−10
10−8
10−6
10−4
10−2
100
(0,β)
(1,β)
(2,β)
E
x
Figure 1. Energy in different Fourier modes (ω, β)versusthe
downstream position. Frequencies: zero (streak), one (sec-
ondary instability), two (higher harmonic). — β=0,---
β=1,···β=2.
space and use the notation (ω, β ), where ωand βare the frequency and spanwise
wavenumber, each normalized with the corresponding fundamental frequency
and wavenumber. The energy in some of the modes is displayed in figure 1,
where the zero frequency mode represents the streak. The secondary instability
mode (ω= 1) is present at the beginning of the computation, while the higher
harmonics are excited as the flow evolves downstream. The energy growth is
exponential for a long streamwise extension and the growth rate of the first
harmonic (ω= 2) is twice the one of the fundamental secondary instability,
and similarly for higher frequencies (not reported here), the growth rate is
proportional to the harmonic order.
It is interesting to note that the energy content is of the same order for
modes with different spanwise wavenumbers but with the same frequency. This
result is different from the one obtained when the same analysis is applied to
a case of transition initiated by two oblique waves (Berlin et al. 1999) or by
Tollmien–Schilichting waves (Laurien & Kleiser 1989; Rist & Fasel 1995). In
these cases nonlinear interactions are important to select the modes dominating
the transition process, namely the streamwise independent ones, while here
streaks are induced from the start and they develop to a highly nonlinear
stage before they become unstable to time dependent disturbances; thus the
harmonics in the spanwise direction are generated during the streak growth
and are responsible for the large spanwise shear of the flow. The instability of
such a flow is then characterized by modes strongly localized in the spanwise

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96 L. Brandt & D.S. Henningson
0 1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
U
y/δ∗
Figure 2. Average streamwise velocity in outer coordinates
at different streamwise positions;x= 126: —– (thin line), x=
185: - - - , x= 215: - ·-,x= 268: ··· ,x= 399:—– (thick
line).
direction so that a number of wavenumbers βis needed to correctly capture
them (see Andersson et al. 2001). The growth in the different harmonics starts
to saturate around position x= 200 and soon the energy becomes of the same
order for the different ω’s. From this point (x≈220) the Fourier transform
in time of the whole velocity fields is no longer accurate since not enough
frequencies are resolved. In fact higher and higher harmonics are excited until
the energy spectra fill out.
Mean velocity profiles at various locations in the transitional zone are dis-
played in figure 2, where the wall normal coordinate is made non dimensional
with the local boundary layer thickness δ∗. The evolution from the laminar
flow to a turbulent one can be seen. At position x= 215 a strong inflectional
mean profile is present exactly during the large growth of the skin friction co-
efficient, not reported here. In the outer part of the boundary layer one can
see an over–shoot of the velocity before approaching the final value. The same
behavior of the mean flow was observed by Wu et al. (1999) in their simulations
of transition induced by free–stream turbulence.
At the early stages of transition, the averaging of the streamwise velocity
provides information on the evolution of the streak during the process, since
the spanwise modulation dominates in the rms values. These are displayed
in figure 3 also for the other two velocity components. In the experiments of
Matsubara & Alfredsson (2001) of transition induced by upstream–generated
grid turbulence the urms value attained by the streaky structure before the
breakdown is about 11 −12%. In our case, instead, the streak amplitude at the
beginning of transition is about 19%, but we do not have a continuous forcing
by the free–stream turbulence which is able to locally nonlinearly trigger the

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Transition of streamwise streaks 97
0 1 2 3 4 5 6 7 8
0
0.05
0.1
0.15
0.2
0 1 2 3 4 5 6 7 8
0
0.02
0.04
0.06
0.08
0 1 2 3 4 5 6 7 8
0
0.05
0.1
urms
vrms
wrms
y/δ∗
Figure 3. Time averaged Reynolds stresses in outer coor-
dinates at different streamwise positions;x= 126: —– (thin
line), x= 185: - - - , x= 215: - ·-,x= 268: ··· ,x= 399:—–
(thick line).
inflectional instability of the flow. However, the same qualitative behavior of
the urms is observed compared to the experiments, i.e. the peak is sharpening,
moving closer to the wall and reaching values of approximatively 12 −13%.
As the flow develops downstream, the rms values of the wall normal and
spanwise velocity components increase especially in the outer part of the bound-
ary layer, around y≈3. This corresponds to the wall normal region where the
secondary instability is localized. One can also note that the spanwise velocity
fluctuations are larger than the wall normal ones, and a considerable value of
wrms ≈11% is attained at x= 215. It is also interesting to notice that at
x= 268 the mean velocity profile, figure 2, and the urms are very close to
the turbulent ones, especially close to the wall, but the vrms and wrms are
characterized by large values in the upper part of the boundary layer. These
oscillations represent periodic structures, formed in the transition region which
survive downstream.
Profiles of the time-averaged turbulence kinetic energy production normal-
ized with wall parameters are shown in figure 4 at the three different streamwise
locations within the turbulent region and compared with the DNS data of Skote
(2001). Spalart (1988) noticed that his DNS profiles of turbulent production
at three different momentum thickness Reynolds numbers are self-similar. He
explained it with the fact that the decrease of Reynolds stresses is compensated

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98 L. Brandt & D.S. Henningson
0 10 20 30 40 50 60 70 80 90 100
0
0.05
0.1
0.15
0.2
0.25
0.3
P+
y+
Figure 4. Time averaged non-dimensional turbulence kinetic
energy production, P+=−uv+∂U+
∂y+, near the wall. - - -
: present simulations at Reθ= 845 (x= 360), Reθ= 875
(x= 375) and Reθ= 910 (x= 400); —–: Skote’s simulations
at Reθ= 685.
by the increase of mean velocity gradient for these relatively low Reynolds num-
bers. This seems to be true also in the present case, since all the profiles show
a maximum of P+=0.25 at y+= 12. The agreement in the profile of the ki-
netic energy production let us believe that the dynamics of turbulence is active
at the downstream end of our computational box. Thus the flow observed is
still influenced by the deterministic inflow conditions and by the transitional
process only in the upper part of the boundary layer.
3. Instantaneous flow structures
A three-dimensional picture of the secondary instability mode is displayed in
figures 5 and 6. These are obtained from the Fourier transformed velocity fields
discussed in the previous section, filtering at the fundamental frequency. The
mode is characterized by a streamwise wavelength λx=11.9 and a frequency
ω=0.43; only one wavelength λxis shown in the figures. Isosurface of positive
and negative streamwise velocity are plotted in figure 5 to show the antisym-
metry of this kind of instability. The low speed region is located around z=0,
where the fluctuations are stronger. The result is a spanwise oscillation of the
low speed streak. The spanwise velocity, seen in figure 6, is in fact character-
ized by alternating positive and negative values, with a symmetric distribution
of the disturbance with respect to the streak.
As observed in a number of experiments and numerical studies, see Le
Cunff & Bottaro (1993) as example, the sinuous instability can be related to
the spanwise inflectional points of the mean flow. Andersson et al. (2001) have
shown that the secondary instability modes are concentrated around the critical

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Transition of streamwise streaks 99
y
z
x
Figure 5. Isosurface of positive (dark grey) and negative
(light grey) streamwise velocity component of the secondary
instability eigenmode. The coordinates are made non dimen-
sional using the local boundary layer thickness.
layer, i.e. the layer of costant value of the mean field velocity corresponding to
the phase speed of the disturbance which is u=0.81U∞in the present case,
thus confirming the inviscid nature of the considered instability.
In figure 7 the instantaneous streamwise velocity component of the pertur-
bation is shown in a longitudinal plane perpendicular to the wall for z=0,
corresponding to the center of the undisturbed low speed streak and in a plane
parallel to the wall, at y=0.47. The perturbation velocity field is defined as
the difference between the solution velocity field and the mean value in the
spanwise direction for each value of xand y. It can be clearly seen that the
sinuous instability consists of harmonic antisymmetric streamwise oscillations
of the low speed region.
In figure 7a) one can note that the perturbation is first seen in the outer part
of the boundary layer. The disturbance moves then towards the wall until the
wall-shear is considerably increased. At the end of the computational box some
periodicity can still be seen in the disturbance in the outer part of the boundary
layer, while close to the wall the flow is now turbulent. In figure 7b) two streaks
can be seen within the computational box at the end of the transition process,

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100 L. Brandt & D.S. Henningson
y
z
x
Figure 6. Isosurface of positive (dark grey) and negative
(light grey) spanwise velocity component of the secondary in-
stability eigenmode. The coordinates are made non dimensio-
nal using the local boundary layer thickness.
with a spacing of about 130z+. The spanwise dimension of the box is in fact
for x>350 less than 275 plus units, larger than the minimal channel studied
by Jim´enez & Moin (1991), in which a turbulent flow could be sustained. On
the other hand, the computational box is apparently too small to allow the
formation of some of the large structures present in the outer region.
The flow field from the laminar to the turbulent region is shown in fig-
ure 8. The light grey isosurface represents the low speed streaks, while the
dark grey represents regions of low pressure. These corresponds to strong rota-
tional fluid motions and are used to identify vortices. Also visualizations using
negative values of the second largest eigenvalue of the Hessian of the pressure
(see Jeong & Hussain 1997) are performed and no relevant differences are ob-
served. The main structures observed during the transition process consist
of elongated quasi-streamwise vortices located on the flanks of the low speed
streak. Vortices of alternating sign are overlapping in the streamwise direction
in a staggered pattern and they are symmetric counterparts, both inclined away
from the wall and tilted in the downstream direction towards the middle of the
undisturbed low speed region. The strength and extension of these vortices and
the spanwise motion of the low speed streak increase downstream before the

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Transition of streamwise streaks 101
x
a)b)
zy
Figure 7. Visualization of streaks breakdown using stream-
wise velocity component of the perturbation in a wall normal
x–yplane and wall parallel x–zplane. Grey scale from dark
to light corresponding to negative to positive values. The flow
is from bottom to top. x−y-plane at z=0,x−zplane at
y=0.47. 185 ≤x≤360 in the streamwise direction.

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102 L. Brandt & D.S. Henningson
x
z
y
Figure 8. The flow field from the laminar to the turbulent
region. The x-values correspond to the range 185 ≤x≤360.
The light grey structures are the low speed streaks and the
darker ones are regions with low pressure. Contours level are
-0.14 for the steamwise velocity fluctuations and -0.014 for the
pressure for x<268 and -0.0065 further downstream. The
streamwise scale is one third of the cross stream ones.
breakdown. Towards the end of the box the flow has a more turbulent nature
and more complicated low-pressure structures occur. It also seems that there
is no connection between the laminar and turbulent-region low speed streak,

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Transition of streamwise streaks 103
since the streak is disrupted at transition and those which appear downstream
are not a continuation from upstream.
In other studied transition scenarios (Rist & Fasel 1995, Berlin et al. 1999),
positive and negative streamwise vortices are also present on the side of the
low speed region but they are not staggered in the streamwise direction so that
the typical Λ-structures are seen. This is due to the different symmetry of the
streamwise vorticity of the fundamental secondary instability; in the present
case the vorticity disturbance is symmetric, while in the varicose case, observed
in the simulations of oblique transition by Berlin et al. (1999), the streamwise
vorticity is antisymmetric.
The vortex structures present in a turbulent boundary layer seem to be
related to streak instabilities. Waleffe (1997) found that the dominating insta-
bility is sinuous and it is correlated with the spanwise inflection of the basic
mean flow. Kawahara et al. (1998) and Schoppa & Hussain (1997) also used
a similar approach and showed that the varicose mode is stable. The identi-
fied structures show a close resemblance to the one detected in our transitional
boundary layer (Schoppa & Hussain 1997). On the other hand, Skote et al.
(2001) show that the appearance of an unstable wall-normal velocity profile is
a precursor to the appearance of horseshoe vortices, thus associated to varicose
instability of the turbulent streaks.
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