Content uploaded by Martin Skote
Author content
All content in this area was uploaded by Martin Skote on Jul 30, 2014
Content may be subject to copyright.
Varicose instabilities in turbulent boundary layers
M. Skote
Department of Mechanics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden
J. H. Haritonidis
Department of Aerospace Engineering and Aviation, Ohio State University, Columbus, Ohio 43210
D. S. Henningson
Department of Mechanics, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden
共Received 7 November 2001; accepted 10 April 2002; published 31 May 2002兲
An investigation of a model of turbulence generation in the wall region of a turbulent boundary
layer is made through direct numerical simulations. The model is based on the varicose instability
of a streak. First, a laminar boundary layer disturbed by a continuous blowing through a slot is
simulated in order to reproduce and further investigate the results reported from the experiments of
Acarlar and Smith 关J. Fluid Mech. 175,43共1987兲兴. An isolated streak with an inflectional profile is
generated that becomes unstable, resulting in a train of horseshoe vortices. The frequency of the
vortex generation is equal to the experimental results. Comparison of the instability characteristics
to those predicted through an Orr–Sommerfeld analysis are in good agreement. Second, a direct
numerical simulation of a turbulent boundary layer is performed to point out the similarities between
the horseshoe vortices in a turbulent and a laminar boundary layer. The characteristics of streaks and
the vortical structures surrounding them in a turbulent boundary layer compare well with the model
streak. The results of the present study show that one mechanism for the generation of horseshoe
vortices in turbulent boundary layers is related to a normal inflectional instability of the
streaks. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1482377兴
I. INTRODUCTION
A. Detection of coherent structures
The occurrence of coherent vortices in wall-bounded tur-
bulent flows has been observed in a large number of inves-
tigations by different means. The experimental observations
have relied on dye injections or hydrogen bubbles introduced
in the flow. Lately, low Reynolds number flows have been
investigated numerically through direct numerical simula-
tions 共DNS兲. The flow field variables are all available at the
same time and thus more sophisticated detection methods
have been developed. Robinson1used the pressure success-
fully for revealing horseshoe vortices in a data base from a
DNS of a turbulent boundary layer. Singer and Joslin2also
used the pressure in a numerical simulation for visualizing a
horseshoe vortex generated by blowing through a slot.
Chong et al.3used the discriminant of the velocity gradient
tensor for identifying flow structures in turbulent boundary
layers. They found structures that to a great extent consist of
attached vortex loops. Zhou et al.4used the imaginary part of
the complex eigenvalue of the velocity gradient tensor to
identify hairpin structures in channel flow. The structures
originated from a vortical structure imposed in the flow. By
plotting the imaginary part a clear picture of the structure
was obtained and the shape was not sensitive to the level
chosen for visualization. Jeong and Hussain5and Schoppa
and Hussain6used an eigenvalue based on the Hessian of the
pressure for identification of vortices in a turbulent channel
flow, and used conditional sampling to extract the precise
form of the coherent structure.
B. Streamwise versus horseshoe vortex structures
Jeong and Hussain5did not detect any horseshoe vortices
in the channel flow simulation by Kim et al.7Instead they
extracted a coherent structure consisting of quasi-streamwise
vortices by conditional sampling. Jimenez and Moin8and
Hamilton et al.9observed, by shrinking the computational
box, that the self-sustained turbulence is linked to the quasi-
streamwise vortices, and does not depend on the outer part of
the flow. This scenario is consistent with the model of
Waleffe10 which states that the vortex is fed by energy from
the break up of the streak. Jimenez and Pinelli11 used a
method of reducing the influence of the outer flow in a nu-
merical simulation to show that the regeneration cycle is
independent on the outer flow. Thus, according to these find-
ings, there is little interaction between the inner and outer
flow. Consequently, it is possible to model the regeneration
of turbulence via a self-sustaining process involving low-
speed streak and quasi-streamwise vortex, independent on
the outer flow.
On the other hand, horseshoe vortices observed in
boundary layer flows reach into the outer flow. Experimental
evidence include the work of Acarlar and Smith12 and Haid-
ari and Smith13 in which vortices, generated by blowing fluid
through a slot in the wall, were studied in a laminar bound-
PHYSICS OF FLUIDS VOLUME 14, NUMBER 7 JULY 2002
23091070-6631/2002/14(7)/2309/15/$19.00 © 2002 American Institute of Physics
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
ary layer. The blowing was continuous in the experiment by
Acarlar and Smith while a pulsed injection was used by Hai-
dari and Smith. Recently, Adrian et al.14 have visualized
hairpin vortices in a turbulent boundary layer using particle
image velocimetry 共PIV兲. They show that hairpin packets
共groups of horseshoe vortices兲build up the turbulent bound-
ary layer. The number of vortices that constitute a packet is
lower in a low Reynolds number flow than in high Reynolds
number flows. Further experiments with conditional sam-
pling by Christensen and Adrian15 revealed that the outer
structure of the turbulent boundary layer includes spanwise
vortices at a spacing and angle which are Reynolds number
independent.
The experimental findings of Haidari and Smith was
confirmed by the DNS of Singer and Joslin.2They observed
different kinds of subsidiary vortices 共such as necklace or
U-shaped vortices兲and the initial vortex generated by the
blowing finally develops into a turbulent spot.
The size of the horseshoe vortices seems to vary within
the flow and also vary with Reynolds number. A turbulence
model for Reynolds average Navier–Stokes 共RANS兲calcu-
lations of turbulent flows has been developed by Perry
et al.16 based on size and strength of the horseshoe struc-
tures. This technique was recently used by Marusic17 to show
that packets of horseshoe vortices are statistically significant
structures.
C. Streak instability and turbulence regeneration
The vortex structures present in turbulent boundary lay-
ers seem to be related to streak instabilities. However, sev-
eral types of instabilities seem to occur. Robinson18 proposed
that a normal inflectional instability of the instantaneous ve-
locity profile may produce horseshoe vortices. Singer19
showed that a normal inflectional instability of the velocity
profile may be responsible for the generation of secondary
horseshoe vortices. Kim et al.20 were the first to show that a
normal inflectional instability of the instantaneous velocity
profile is of importance in the turbulence regeneration cycle.
They observed the inflectional velocity profiles in connection
with the rapid lift up of the low-speed streaks in the later part
of the process of the streak break up. Several models of the
turbulence regeneration cycle have been proposed by
Landahl,21,22 where inflectional instability of the local mean
velocity profile is a main ingredient.
On the other hand, in the model of Waleffe10 the basic
state is two-dimensional and consists of the turbulent mean
flow with a simple construction of the streak imposed. He
found that the dominating instability is sinuous and that it is
correlated with the spanwise inflection of the basic state.
Kawahara et al.23 and Schoppa and Hussain6also used such
a model and showed that the varicose mode is stable.
Schoppa and Hussain6argued that this is consistent with the
absence of horseshoe vortices in their examination of the
DNS data base generated by Kim et al.7
The references cited above form only a small part of the
work that has been put into the detection and analysis of
coherent structures. A recent article by Schoppa and
Hussain,24 where a detailed discussion of the state of the art
is included, categorizes the mechanisms into parent-offspring
and instability based scenarios. The former is characterized
as the generation of vortices by direct action of existing vor-
tices, whereas the second involves local instability of quasi-
steady flow. Within the instability based scenarios the sinu-
ous and varicose streak instabilities are the two main ones.
The view of Schoppa and Hussain24 seems to be that there is
a contradiction between these explanations. Our view is that
turbulence is such a rich phenomenon that there are most
likely several mechanisms at work regarding coherent struc-
ture generation, of which we here have chosen to study one.
D. Present study
In this work we will pursue the horseshoe vortex dynam-
ics, within the streak stability based scenario of varicose
type. In the experiments by Acarlar and Smith,12 hereafter
denoted AS, an artificial low-speed streak was generated in a
laminar boundary layer by blowing fluid through a slot in the
wall. The streak became unstable and horseshoe vortices
were formed and were followed downstream. In the present
study we reproduce the flow studied by AS through DNS.
Moreover, the hypothesis indicated by AS regarding the in-
stability causing the vortices is here further investigated. One
of the objectives in the AS experiment was to give insight to
the mechanisms and structures in a turbulent boundary layer.
In the present work, a stronger link to turbulence is made
through comparison with a simulation of a zero pressure gra-
dient turbulent boundary layer.
The experiments by Haidari and Smith13 could also have
been reproduced with our numerical code by turning off the
blowing after the injection was completed. However, the nu-
merical study by Singer and Joslin2was already performed
as a numerical compliment to the pulsed injection experi-
ments by Haidari and Smith. Note also that although the
pulsed injection is important for the study of a single vortex
developing downstream, the turbulent boundary layer seems
mostly to consist of packages of vortices, see, e.g., Adrian
et al.14
After a presentation of the numerical method and param-
eters in Sec. II, we present the results in Sec. III. The em-
phasis is on the results from the laminar simulation, which is
compared with the experimental results from AS. Further
investigations of the instability mechanism are made. Also
comparison with the turbulent simulation is done, from
which strong similarities between the two cases are pre-
sented.
II. NUMERICAL METHODOLOGY
A. Direct numerical simulations
The code used for the simulation is developed at KTH
and FFA.25 The program uses spectral methods with Fourier
discretization in the horizontal directions and Chebyshev dis-
cretization in the normal direction. Since the boundary layer
is developing in the down-stream direction, it is necessary to
use nonperiodic boundary conditions in the streamwise di-
rection. This is possible while retaining the Fourier discreti-
zation if a fringe region is added downstream of the physical
2310 Phys. Fluids, Vol. 14, No. 7, July 2002 Skote, Haritonidis, and Henningson
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
domain. In the fringe region the flow is forced from the
outflow of the physical domain to the inflow. In this way the
physical domain and the fringe region together satisfy peri-
odic boundary conditions. The fringe region is implemented
by the addition of a volume force whose form is designed to
minimize the upstream influence. The upstream influence of
the fringe region is of the same order as the height of the
computational box. Since our box is very long and thin, the
influence is marginal. All the results are taken from regions
well upstream of any fringe influence. For an analysis of the
fringe region technique, the reader is referred to the investi-
gation by Nordstro
¨met al.26
Time integration is performed using a third order
Runge–Kutta scheme for the advective and forcing terms
and Crank–Nicholson for the viscous terms.
All quantities are nondimensionalized by the freestream
velocity (U⬁) and the displacement thickness (
␦
*) at the
starting position of the simulation (x⫽0) where the flow is
laminar. At that position Re
␦
*⫽U⬁
␦
*/
⫽450 for all simu-
lations, except for some simulations performed at Re
␦
*
⫽290 for the comparison of frequency characteristics. The
length 共including the fringe兲, height and width of the com-
putation box were 260⫻7⫻14 in these units. The number of
modes was 432⫻65⫻72. The size and resolution were
checked to be sufficient for all cases.
The simulations were performed with an initial objective
of reproducing some of the results obtained in the experi-
ments of AS. In their experiments the slot was 63.5 mm in
length and 1 mm in width. The simulations were performed
with a slot with the same length but twice the width, i.e., 2
mm. This change in geometry results in a large decrease in
computational cost. The slot in simulation coordinates (
␦
*)
is approximately 30 long and 1 wide. The flow through the
slot is set by a velocity profile resembling a channel flow
parabola in the spanwise direction and is increasing from
zero to the maximum value during the first 10% of the slot
length at the upstream end, and is likewise terminated at the
downstream end. The blowing through the slot was contin-
ued without interruption through all of the simulations. To
avoid large transients in the beginning of the simulation we
ramped up the blowing from zero to the maximum value
during an initial time of 10(
␦
*/U⬁). The time step was con-
siderably decreased when the blowing through the wall is
applied. The strength of the blowing was varied from 6.5%
to 20% of the freestream velocity.
A low-speed streak is formed immediately above the slot
due to the lift-up of low-speed fluid to the flow further out in
the boundary layer.A disturbance on this streak was detected
and the frequency was observed during a long period of time,
and was then locked by letting a small 共1% of the original
blowing兲additional time-periodic blowing be superimposed
on the blowing forming the streak. The frequency of the
initial disturbance on the streak was locked to be able to
calculate the growth rate of the disturbance through a Fourier
transform in time.
A simulation of a turbulent boundary layer was per-
formed to investigate how the streak instabilities observed in
the isolated streak in the laminar boundary layer could be
applicable to a turbulent flow. The same code was used, but
the laminar boundary layer was disturbed at the beginning of
the computational box by a random volume force near the
wall. The length 共including the fringe兲, height and width of
the computation box were 600⫻30⫻34. The number of
modes was 640⫻201⫻128. The simulations were performed
at Re
␦
*⫽450 for the laminar inflow before the tripping,
which gives a turbulent Re⌰:343–636. The resolution in
plus units was ⌬X⫹⫽19, ⌬Z⫹⫽5.5, and ranging from
⌬Y⫹⫽0.04 close to the wall to ⌬Y⫹⫽5.6 at the coarsest
part of the grid.
B. The linear stability analysis
One of the main conclusions of this work will concern
the instability mechanism of a low-speed streak leading to
horseshoe-shaped vortices. Linear stability theory will be
used to describe the early stages of this instability. The dis-
turbance occurring due to the instability of the streak will be
denoted secondary disturbance, since the primary distur-
bance is the streak itself. The velocity profiles close to where
the secondary disturbance start to appear, below denoted U
⫽U(y), were analyzed by solving the Orr–Sommerfeld
共OS兲equation. The results from the OS equation are only
relevant as long as the disturbance is small enough and varia-
tions of the base flow 共streak兲in the horizontal directions and
time is much smaller than the length scale of the instability
waves. The OS equation is the linearized Navier–Stokes
equations for the disturbance,
⫺2
␣
2
⬙⫹
␣
4
⫽i
␣
R关共U⫺c兲共
⬙⫺
␣
2
兲⫺U⬙
兴.共1兲
The two-dimensional disturbance is written as a stream func-
tion
⫽
共y兲exp关i
␣
共x⫺ct兲兴⫽
共y兲exp关i共
␣
x⫺
t兲兴.共2兲
Because the secondary disturbance is characterized by
its frequency and its growth in space in the simulations, spa-
tial analysis of the OS equation will be used. In the case of
spatial analysis the eigenvalue problem 共1兲is solved for a
given Rand
, which is real. The solution is
(y)共eigen-
function兲and
␣
⫽
␣
r⫹i
␣
i共eigenvalue兲. The value of ⫺
␣
iis
the growth rate, and
␣
ris the streamwise wave number.
The results from the analysis of the OS equation are
compared with the actual behavior of the flow in the DNS.
The eigenvalue ⫺
␣
iis compared with the growth rate of the
disturbance. Furthermore, the eigenvalue
␣
ris compared
with the streamwise wave number of the disturbance. The
analysis of the time signal from DNS is done through a Fou-
rier transform in time of the velocity fields. For a given fre-
quency, we take the maximum over the spanwise and normal
directions. Thus, the results from DNS are contained in a
function uˆ(x). The growth rate of the disturbance is
⫽⫺Re
再
1
uˆ
d
dxuˆ
冎
,共3兲
and the streamwise wave number of the disturbance is
2311Phys. Fluids, Vol. 14, No. 7, July 2002 Varicose instabilities in turbulent boundary layers
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
␣
˜
⫽Im
再
1
uˆ
d
dxuˆ
冎
.共4兲
The use of the Orr–Sommerfeld equation proposed here
does not take the spanwise variation of the streak profile into
account. Although this could in principle be done, it would
defeat our purpose which is to show that an instability due to
the normal shear is the dominating growth mechanism re-
sponsible for the initiation of the horseshoe vortex. We are
interested in obtaining a simple model which we can apply in
the more complicated turbulent case, in the same manner as
Waleffe10 models the sinuous secondary instability of a tur-
bulent streak with only a spanwise varying shear. In addition,
a 2D stability analysis would also be superfluous since we
extract a result equivalent to such an analysis from the DNS
calculations. In fact, the DNS results can be considered as a
secondary instability calculation, where not only the span-
wise shear is taken into account, but also the streamwise
nonparallel effects, as long as the amplitude of the distur-
bance is small.
III. RESULTS
A. Initial observations
1. Comparison with AS experiment
The development of the streak downstream of the slot is
shown in Fig. 1. Only the part immediately after the slot is
shown. The light gray iso-surface represents the low-speed
streak, and the dark gray represents the low pressure. The
slot ends at x⫽60 and the first low-pressure structure is ob-
served at that point. The subsequent pressure structures de-
velop downstream and become stronger. Additional streaks
on either side are being induced by the pressure structure at
x⫽70. This will be further discussed in Sec. III B. Around
x⫽94 the last structure in the train of vortices is observed,
and the streak has been lifted upward. The low-pressure
structure vanishes, but the streak and the additional, induced
streaks persist downstream, as seen from Fig. 2, where the
region downstream of the breakup is also shown. The three
FIG. 1. The flow field downstream of the slot. The light gray structures
represent the low-speed streaks and the darker ones represent regions with
low pressure. Contour levels are ⫺0.08 for the streamwise velocity fluctua-
tions and ⫺0.01 for the pressure.
FIG. 2. The flow field far downstream of the slot. The light gray structures
represent the low-speed streaks and the darker ones represent regions with
low pressure. Contour levels are ⫺0.11 for the streamwise velocity fluctua-
tions and ⫺0.005 for the pressure.
FIG. 3. urms at 共a兲x⫽60, 共b兲x
⫽160.
2312 Phys. Fluids, Vol. 14, No. 7, July 2002 Skote, Haritonidis, and Henningson
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
streaks continue far downstream until more complicated low-
pressure structures occur at x⫽145, marked with an arrow in
Fig. 2. Here the flow has more of a turbulent nature, which is
also seen from the rms-values shown in Fig. 3. The urms
profile from a position at the end of the slot is shown in Fig.
3共a兲. This profile has a shape which is a result of an inflec-
tional instability, which will be further discussed in Sec.
IIID. The urms from a position far downstream (x⫽160) is
shown in Fig. 3共b兲. This profile resembles a profile from a
turbulent boundary layer. Thus, the more turbulent like flow
at the far downstream region is revealed both in the struc-
tures themselves and in the statistical profiles. The streak
spacing is actually 100 in viscous units in this region, further
indicating attributes of a turbulent boundary layer.
In AS no spreading of the structures were observed and
they argue that this is due to the sub-critical laminar bound-
ary layer in their experiment. They do however observe a
more turbulent like profile downstream and also three elon-
gated low-speed streaks, originating from secondary stream-
wise vortices. Our simulation continue further downstream
than the experiment by AS, and the persistent low-speed
streaks were observed downstream until the more compli-
cated vortices appeared at x⫽145, see Fig. 2.
It was shown, in the experiments by Haidari and Smith13
of a single vortex developing downstream, that the growth of
the structure is caused by both the interaction of the primary
vortex with surface fluid and inviscid deformation of the
vortex lines. The latter mechanism was however not found in
the numerical investigation of a single vortex by Singer and
Joslin.2Furthermore, the numerical simulation showed that
the secondary vortices formed beneath the hairpin vortex
legs were closely related to the initial injection and not gen-
erated by the primary vortex. Thus, it seems difficult to ob-
tain and observe the dynamics of a single vortex; it is
strongly linked to the generation process.
The low-pressure structures seen in Fig. 1 are vortex
loops, consisting of swirling flow. To illustrate that the low-
pressure regions consist of rotational flow, the imaginary part
of the complex eigenvalue of the velocity gradient tensor can
be used.27 Because the vorticity indicates both shear and ro-
tation, showing vorticity can be misleading when seeking
parts of the flow where rotating structures are of interest. The
imaginary part of the eigenvalue on the other hand, indicates
where swirling occurs.
In Fig. 4 the low-pressure structures in Fig. 1 are shown
without the low-speed streak to get a clearer picture of the
structures themselves. The structures in Fig. 5 consist of iso-
surfaces of the imaginary part of the eigenvalue. The strong
correlation indicate that the structures in Fig. 4 are due to
rapid rotation of the flow in the regions of low pressure.
Observe that the ⍀-shape of the last structure in Fig. 5 is
reminiscent of the structure observed by Zhou et al.4Also,
the kink of the legs about one-third of the length from the
upstream end are present in the last structure. Note that the
background flow in the present simulation is laminar
whereas it consisted of a turbulent mean flow in the study of
Zhou et al.4The kinked legs and the curled back head of the
last structure in Fig. 5 was also observed by AS at the same
downstream position.
A secondary vortex is observed above the primary horse-
shoe vortex in the two structures before the last one in Fig. 5.
The secondary vortex is also visible in one of the corre-
sponding pressure structures as marked in Fig. 1. The sec-
ondary vortex is visible at approximately the same position
as in AS. Zhou et al.4found not only secondary horseshoe
vortices developing upstream of the primary vortex, but also
downstream, which was not observed in the present simula-
tion. In the experiments of AS, a secondary vortex appears to
originate from the position above the legs of primary vortex.
It either grows to be an independent vortex, or agglomerates
with the upstream or downstream vortex. The same behavior
is observed in the present simulations.
Thus, the secondary vortices appearing upstream, above
the legs, of the primary one are in common with many of the
experimental and numerical investigations, while the genera-
tion of downstream secondary vortices depends on the
strength and duration of blowing.
FIG. 4. Iso-surface of low pressure just downstream of the slot. Same part
of the flow field as in Fig. 1. FIG. 5. Iso-surface of the imaginary part of the eigenvalue of velocity
gradient tensor. The figure shows the eigenvalue calculated from the same
velocity field as in Fig. 4. Contour level at 0.32.
2313Phys. Fluids, Vol. 14, No. 7, July 2002 Varicose instabilities in turbulent boundary layers
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
2. Near-wall turbulence
The presence of streaky structures in a near-wall turbu-
lent flow has been observed in many experiments and simu-
lations. These structures are low speed regions, where the
streamwise velocity is lower than the mean velocity, the
mean taken in the spanwise (z) direction for each x- and
y-position. They are narrow in the spanwise direction and
elongated in the streamwise direction with a spanwise spac-
ing of about 100 in wall units. Streaks lying at different
positions in zbreak down at different positions in x. Also, a
new streak seems to be born where the old one breaks down.
In a number of investigations, events referred to as burst
have been observed, and are generally considered to be part
of the streak break up.
An instantaneous flow field from the simulation of a tur-
bulent boundary layer is shown in Fig. 6. Only a part of the
computational box at approximately Re⌰⫽450 is shown.
The spanwise width is about 300 in wall units and the height
is 200. The light gray regions represent the low-speed
streaks. Also shown in the figure, in the dark gray color, are
regions of low pressure. The presence of horseshoe or hair-
pin vortices is well illustrated by this picture. The most
clearly visible ones are marked with arrows in Fig. 6. It is
observed that the vortices are strongly connected to the
streaks, since the vortices are positioned with their head
above a streak and their leg or legs on either side of the
streak. This feature is common to both the laminar and tur-
bulent streaks, cf. Figs. 1 and 6.
B. Horseshoe vortex formation
The mechanisms behind the formation of vortices from
the streak is here studied in detail in the laminar flow with an
artificial streak introduced. The proposed mechanism is that
the low-speed streak makes the streamwise velocity profile
highly inflectional. The instability is very strong 共with a large
growth rate兲. The disturbance grows downstream and higher
harmonics occur. The stability analysis is presented in Sec.
IIID.
The results in this section are taken from the simulation
at a Reynolds number Re
␦
*⫽U⬁
␦
*/
⫽450 at x⫽0, which
corresponds to a Reynolds number Re
␦
*⫽490 at the begin-
ning of the slot. The normal velocity blowing out of the slot
was Vw⫽0.0657, resulting in a slot Reynolds number of
ReVw⫽28.3. The blowing was introduced between x⫽30
and x⫽59 in the streamwise direction, and between z
⫽⫺0.48 and z⫽0.48 in the spanwise direction.
1. Vortex formation above the slot
One velocity field is studied, using plots in two dimen-
sions of different planes. In Figs. 7共a兲–7共f兲the planes are
from different positions in x, showing what happens with the
flow above the slot. The lines in the horizontal direction,
from blue to green, are the iso-lines of streamwise velocity,
while the arrows represents the normal and spanwise velocity
components. The first 关Fig. 7共a兲兴 figure shows the undis-
turbed laminar boundary layer at the point where the slot
starts. The next one 关Fig. 7共b兲兴 shows a plane further down-
stream. Here the injection is visible as the strong flow out
from the wall. The lines representing constant streamwise
velocity are bent outward and thus forming a low-speed
streak. The low-speed streak is formed because of the injec-
tion velocity that lifts up low-speed fluid from the near-wall
region higher up in the laminar boundary layer. In Fig. 7共c兲a
swirling flow is observed at either side of the low-speed
streak. As the vortical motion becomes stronger it deforms
the streak as seen in Fig. 7共d兲, where also the vortex is strong
enough to be represented with low pressure regions at the
center of the vortex. Iso-lines of constant low pressure are
shown as red lines. These low pressure regions that evolve
from the center of the vortex at either side of the streak are
the legs of the first low pressure structure seen in Fig. 1. The
plane in Fig. 7共d兲is located at the end of the slot, thus no
more injection velocity can be observed. In Fig. 7共e兲the low
pressure region is above the streak and the motion in the
region is a flow upward. The plane in Fig. 7共e兲is located a
short distance downstream of the plane in Fig. 7共d兲. Thus,
immediately after the legs have appeared an upward motion
is seen in Fig. 7共e兲in the low pressure region now located
above the streak, and hence forms the head of the first struc-
ture. At the other side 共downstream side兲of the low pressure
region the motion is a downward flow, as seen in Fig. 7共f兲.
This downward velocity at the downstream side of the head
indicates that the low pressure structure is a vortex loop.
Since the head is observed right after the legs, the structure is
very short, which was also observed in Fig. 1.
2. Vortex formation downstream of the slot
Now that the flow above the slot and around the first
structure has been studied, the flow further downstream will
be investigated. The same technique is used to get an idea
FIG. 6. Turbulent boundary layer. Only a part of the computational box is
shown. The light gray structures represent the low-speed streaks and the
darker ones represent regions with low pressure. Contour levels are ⫺0.07
for the streamwise velocity fluctuations and ⫺0.003 for the pressure. The
arrows point to some typical horseshoe vortices.
2314 Phys. Fluids, Vol. 14, No. 7, July 2002 Skote, Haritonidis, and Henningson
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
about what happens with the flow around the well developed
structure indicated as number three in Fig. 1. The structure in
the laminar simulation is compared with a typical structure
found in the turbulent field.
In Fig. 8 vertical xy-planes are shown. In Fig. 8共a兲the
plane is located at the centerline (z⫽0) in the laminar field.
The blue line is an iso-line of constant streamwise distur-
bance velocity and thus represent the low-speed streak, while
the red lines are iso-lines of low pressure. The arrows indi-
cate the normal velocity and the streamwise disturbance ve-
locity. The streamwise disturbance velocity is calculated by
subtracting the mean velocity 共the mean taken in the span-
wise direction兲at each point. The flow is from left to right
and arrows pointing to the left merely indicate low speed
compared to the mean. What is seen in Fig. 8共a兲is thus the
head of the pressure structure. The swirling flow around the
head is the relative motion when the mean streamwise veloc-
ity is subtracted. Contour levels are ⫺0.08 for the stream-
wise velocity fluctuations and from ⫺0.05 to ⫺0.01 for the
pressure.
In Fig. 8共b兲a structure from the turbulent simulation is
shown. The horseshoe vortex was identified with a pressure
plot as in Fig. 6. The structure is representative for a turbu-
lent structure since many can be identified in the same in-
stantaneous pressure field. The specific structure shown in
Figs. 8共b兲and 9共b兲is located approximately in the middle of
the computational domain 共x⫽200, z⫽1兲, and is similar to
the one in the upper right corner in Fig. 6. Then a horizontal
plane is cut through the center 共in the spanwise direction兲of
the structure and its head is seen as the low pressure region
in Fig. 8共b兲. Contour levels are ⫺0.04 for the streamwise
velocity fluctuations and from ⫺0.02 to ⫺0.01 for the pres-
sure.
The similarities between Figs. 8共a兲and 8共b兲are remark-
able. In both figures the center of rotation 共relative to the
local mean flow兲is displaced from the center of low pres-
sure. An additional, but weaker low pressure region is found
below the head of both structures. The head of the turbulent
structure in Fig. 8共b兲is located at y⫹⫽135.
In Fig. 9, vertical cross-stream (yz⫺) planes are shown.
The red contours represent low pressure and blue to yellow
lines are the iso-levels of streamwise velocity. The arrows
consist of normal and spanwise velocity components. In Fig.
9共a兲the legs of the structure in the laminar field are clearly
FIG. 7. 共Color兲Vertical planes in the spanwise (z) and normal (y) directions. Arrows represent the spanwise and normal velocity. Blue through green lines
represent constant streamwise velocity from 0 to 0.5. Red lines represent constant pressure. 共a兲x⫽30, 共b兲x⫽38, 共c兲x⫽53, 共d兲x⫽59, 共e兲x⫽60,
共f兲x⫽62.
2315Phys. Fluids, Vol. 14, No. 7, July 2002 Varicose instabilities in turbulent boundary layers
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
FIG. 8. 共Color兲Vertical planes in the
streamwise (x) and normal (y) direc-
tions. Arrows represent the streamwise
disturbance velocity and normal veloc-
ity components. The blue lines repre-
sent constant streamwise disturbance
velocity 共low-speed streak兲. The red
color represents constant pressure 共low
pressure兲.共a兲From the laminar simu-
lation. 共b兲From the turbulent simula-
tion at z⫽1.
FIG. 9. 共Color兲Vertical planes in the
spanwise (z) and normal (y) direc-
tions. Arrows represent the spanwise
and normal velocity. Blue through
green lines represent constant stream-
wise velocity. Red lines represent con-
stant pressure. 共a兲From the laminar
simulation. 共b兲From the turbulent
simulation at x⫽196.
2316 Phys. Fluids, Vol. 14, No. 7, July 2002 Skote, Haritonidis, and Henningson
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
visible as the two low pressure regions, and the flow is cir-
cling around the low pressure. Also seen are the induced
vortices further out from the centerline. These induced vor-
tices were also observed in the experiments by AS. The plane
in Fig. 9共a兲is located at x⫽70, thus showing the legs be-
longing to the structure whose head was shown in Fig. 8共a兲.
The blue to green contour lines represent streamwise veloc-
ity from zero to 0.5.
In Fig. 9共b兲, a cross-stream plane from the turbulent
simulation is shown. The plane is located at x⫽196 共refer-
ring to the coordinates in Fig. 8共b兲, which corresponds to a
distance of x⫹⫽184 共wall units兲upstream of the head lo-
cated at x⫽204 in Fig. 8共b兲. The legs belonging to the horse-
shoe vortex whose head was observed in Fig. 8共b兲are the
two low pressure regions located furthest from the wall, lo-
cated at z⫽5 and z⫽⫺3. The normal position of the legs is
y⫹⫽70, and they are separated with a distance z⫹⫽190.
The other low pressure regions close to the wall belong to
streamwise vortices. The blue to yellow contour lines repre-
sent streamwise velocity from zero to 0.7. In Figs. 8共b兲and
9共b兲every second point in all directions is omitted for clarity.
The positions of the head and legs of the horseshoe vor-
tex in the laminar simulation are in agreement with the ex-
perimental findings in AS. The strength of the transverse and
longitudinal vortices corresponding to the head and legs
were calculated in AS by assuming constant vorticity within
the vortex core. However, in the present DNS we find that
the vorticity varies through the core. For the vortical struc-
tures visualized by low-pressure in Figs. 8共a兲and 9共a兲, the
vorticity lines 共spanwise and streamwise, respectively兲
formed the same pattern as the corresponding pressure con-
tours. The vorticity ranged from ⫺1to⫺0.5 in the trans-
verse vortex and from ⫾1.5 to 0 in the longitudal vortices.
C. Frequency characteristics
In the experiments by AS the frequency of the roll up
was measured. Their observations led to the conclusion that
the frequency increased when the injection velocity or the
freestream velocity was increased. They present the results
as a nondimensionalized frequency (f
␦
*/U⬁) as a function
of slot Reynolds number (ReVw⬅wVw/
) and boundary
layer Reynolds number (Re
␦
*) at the beginning of the slot.
Here wis the width of the slot. The simulations were per-
formed at two Re
␦
*, each with three different ReVw, for
comparison with experimental results from AS. The Re
␦
*at
the beginning of the computational box were 450 and 290,
corresponding to 490 and 330 at the point where the slot
starts.
In the present simulations the frequency was calculated
using the time-signal of the velocity from various locations
in the flow. The frequency of the disturbance was observed
over the full extent of the slot at a number of positions in the
normal direction. When either of the two Reynolds numbers
FIG. 10. Nondimensional frequency f*of the disturbance versus Re
␦
*.
Symbols correspond to different injection velocities. ReVw⫽wVw/
.ReVw
⫽28.3 䊐;ReVw⫽33.6 䊊;ReVw⫽38.7 〫. Bold symbols represent experi-
mental data from AS.
FIG. 11. Nondimensional frequency f*of the disturbance versus Vw/U⬁.
Symbols correspond to different Reynolds number. Re
␦
*⫽490 䊐;Re
␦
*
⫽330 䊊. Bold symbols represent experimental data from AS.
FIG. 12. Time signal of the normal velocity component at x⫽60 and
x⫽70.
FIG. 13. Velocity profile at x⫽45.
2317Phys. Fluids, Vol. 14, No. 7, July 2002 Varicose instabilities in turbulent boundary layers
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
were changed, the frequency also changed. The frequencies
for three different ReVwat two Re
␦
*are plotted in Fig. 10,
together with the results from AS 共thick symbols兲. When the
frequency is plotted as a function of the two Reynolds num-
bers as was done in AS, it is observed that the frequency for
ReVw⫽28.3 is half of that observed by AS 共Fig. 10兲. Also,
reducing ReVwfurther in the simulation caused the vortex
generation to cease. In the experiment by AS, ReVw⫽28.3
was the largest slot Reynolds number for which an ordered
vortex generation was observed, while as low values as
ReVw⫽11.3 were shown to generate vortices.
Thus, the ReVwfor which vortex generation was ob-
served in the simulations was larger than the corresponding
ReVwin the experiments. For the value of ReVw⫽28.3, com-
mon to both simulation and experiment, the frequency ob-
served in the simulation was half of that observed in the
experiment. These discrepancies might be explained by the
value of the blowing velocity, which is half the value in the
simulation as compared to the experiment by AS. However,
the slot has double width in the simulation, making the slot
Reynolds number equal to the experimental value. If the
blowing velocity itself, normalized by the freestream veloc-
ity, is used as the parameter in the comparison, the frequency
for various blowing velocities compare well, as seen from
Fig. 11. Thus, the initial guess that the slot Reynolds number
in the simulation should be equal to the experimental value
to obtain the same frequency is not supported by Fig. 10.
Instead, it is the ratio of blowing velocity to freestream ve-
locity that apparently is the crucial parameter in this respect,
as indicated in Fig. 11. This was also suggested by AS, al-
though they present their frequency data as in Fig. 10.
From simulation data it was observed that the frequency
was doubled when going from a point above the slot to a
position further downstream, as shown in Fig. 12, where the
time signal of the normal component (v) of the velocity at
the two downstream locations at y⫽0.5 are shown. As was
shown in Sec. IIIB, the roll up of the structures starts right at
the downstream end of the slot (x⫽60), and the frequency
of the primary structures is thus the one measured at x⫽60
and not the frequency of double value which occurs further
downstream at x⫽70. The doubling of the frequency is con-
sistent with the growth of a second harmonic of the distur-
bance further investigated in the next section.
D. Stability analysis
In this section the laminar and turbulent simulations are
treated separately.
1. The laminar case
From the observations of their experiment, AS speculate
that a normal inflectional instability causes the oscillations
on the low-speed streak leading to vortex roll up. Also in the
experiments by Haidari and Smith13 an unstable normal ve-
locity profile was observed shortly before the vortex head
was developed.
As described in Sec. IIB, the spatial stability analysis is
performed with the OS equation. The input is the Reynolds
number, frequency of the disturbance, and the velocity pro-
file. The three inputs are well defined and taken from the
FIG. 14. Streamwise component. 共a兲
x⫽45. 共b兲x⫽55. rms-value;
- - - eigenfunction.
FIG. 15. Normal component. 共a兲x
⫽45. 共b兲x⫽55. rms-value;
- - - eigenfunction.
2318 Phys. Fluids, Vol. 14, No. 7, July 2002 Skote, Haritonidis, and Henningson
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
DNS. The output is the eigenfunction, which contains infor-
mation of the disturbance shape, and the eigenvalue, which
gives the growth rate and the streamwise wave number.
Throughout this section the laminar simulation with
Re
␦
*⫽490 at the beginning of the slot and a slot Reynolds
number of ReVw⫽28.3 will be considered. In Fig. 13 the
velocity profile at x⫽45 and z⫽0, corresponding to the cen-
ter of the slot, is shown. The profile is highly inflectional and
the OS analysis will give a large value of the growth rate.
Figures 14 and 15 show the eigenfunction from the OS
together with the rms-value of the velocity from DNS at
positions x⫽45 and x⫽55. The eigenfunctions are calcu-
lated using the instantaneous velocity profile at the two
x-positions as basic states. The rms-value from DNS is cal-
culated over one period of the disturbance, which was T
⫽14.8 共in units of
␦
*/U⬁兲and the corresponding frequency
was ⍀⫽0.425.
The eigenfunction in the streamwise direction is shown
in Figs. 14共a兲and 14共b兲, together with the corresponding urms
from the DNS. The solid line is DNS data and the dashed
line is from the OS analysis. The wall normal coordinate is
scaled with the boundary layer thickness. The sharp peak in
the profile is due to the shear layer instability. At both x
positions the shape is well predicted. The double inner peak
observed in the urms profile is slightly overpredicted by the
OS analysis at x⫽45 and is lacking at x⫽55.
The eigenfunction in the normal direction is shown in
Figs. 15共a兲and 15共b兲, together with the corresponding vrms
from the DNS. The profiles are well predicted by the linear
OS analysis. However, the second, outer peak is also slightly
overpredicted by the linear OS analysis. Observe that vrms is
not zero at the wall due to the injection through the slot.
The results shown in Figs. 14 and 15 are based on an
instantaneous two-dimensional approximation of the basic
state. The agreement between the calculated eigenfunctions
and the rms-profiles found in the fully three-dimensional
DNS is remarkable, indicating that the instability mechanism
is determined mainly by the local flow conditions.
The growth rate from DNS data is calculated from the
Fourier transform in time of velocity fields as a function of
x. When comparing the growth rate and streamwise wave
number from the OS analysis with the corresponding values
from DNS data, the DNS data has to be smoothed since
taking derivatives directly will give spurious oscillations.
The growth rate from the DNS data, denoted by
,is
calculated from the development in time of the maximum
value of the velocity in the downstream direction. The maxi-
mum value is extracted for different frequencies from the
Fourier transform in time. The transformed velocity is
uˆ共x,y,z,
兲⫽
冕
⫺⬁
⬁u共x,y,z,t兲e⫺i
tdt.共5兲
By taking the maximum over yand zand specifying which
frequency of interest, only the x-dependency is left, uˆ
⫽uˆ(x).
In Fig. 16 the maximum of uˆ in the first and second
harmonics are shown. The maximum occurs at the centerline.
By showing the logarithm of the maximum as in Fig. 16, a
curve fit is possible, shown as the dash–dotted line. Also in
Fig. 16 the linear approximations to both the first and second
harmonics are shown as the dotted lines. The slope for the
second harmonic is twice the slope for the first.
FIG. 16. Maximum of u. First harmonic;---second harmonic;
¯linear fit; • • • curve fit of first harmonic. FIG. 17. 䊊The growth rate from the OS analysis. Smoothed DNS
data 共curve fit兲.¯Linear approximation to DNS data.
FIG. 18. 䊊The streamwise wave number from OS the analysis.
Smoothed DNS data 共curve fit兲.¯Linear approximation to DNS data.
2319Phys. Fluids, Vol. 14, No. 7, July 2002 Varicose instabilities in turbulent boundary layers
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Now, the growth rate is calculated from Eq. 共3兲, which
can be written as
⫽1
兩
uˆ
兩
d
dx
兩
uˆ
兩
⫽d
dx共ln
兩
uˆ
兩
兲.共6兲
The linear approximation to the maximum of uˆ in the
first harmonic 共shown in Fig. 16兲is used for calculating the
growth rate, which becomes a constant and is shown as the
dotted line in Fig. 17. By using the curve fit of uˆ instead of
the linear approximation, the growth rate becomes as the
solid line shown in Fig. 17. The circles are the corresponding
growth rates calculated from OS using the instantaneous ve-
locity profiles.
The real part of the eigenvalue (
␣
r) from the OS analy-
sis, is shown in Fig. 18 as circles. To calculate the corre-
sponding
␣
rfrom DNS, which is denoted
␣
˜
, Eq. 共4兲is used.
This equation also involves derivatives in the downstream
direction which cause spurious oscillations. To equivalently
smooth the
␣
˜
, Eq. 共4兲is rewritten, by noting that uˆ ⫽ei⌰,in
the form
␣
˜
⫽Re
再
d⌰
dx
冎
.共7兲
Thus, it is a matter of smoothing ⌰, which is defined by
⌰⫽⫺ilnuˆ .共8兲
The resulting smoothed
␣
˜
is shown in Fig. 18. The linear
approximation becomes a constant and is also shown in the
figure.
Until the second harmonic has reached a substantial am-
plitude 共see Fig. 16兲the extracted results from the DNS cal-
culation represents the true varicose secondary instability,
including effects of both streamwise and spanwise variation
of the base flow. Thus, the agreement between the instability
analysis and DNS data presented in this section strengthen
the arguments for a normal inflectional instability being the
main contributor to the horseshoe vortex formation. This was
also found by Park and Huerre28 who studied streaks in
curved boundary layers and showed theoretically that the
sinuous mode is primarily induced by the spanwise shear,
while the varicose mode is triggered by the wall-normal
shear. In addition, Asai et al.29 studied the response to a
single low speed streak in a boundary layer excited by a
time-periodic signal of either sinuous or varicose type. The
growth of the sinuous mode evolved into a train of stream-
wise vortices and the varicose mode into horseshoe vortices.
To further show that the normal instability is of greatest
interest in our case, we show the maximum normal and span-
wise shear in Fig. 19. The maximum of the normal shear in
Fig. 19共a兲is located above the streak, and has a value of 1.3.
The maximum of the spanwise shear in Fig. 19共b兲is located
on either side of the streak, and has a value of 0.5.
In the secondary instability calculation of Andersson
et al.30 the spanwise shear in a typical streak was about the
same as the maximum found here. The calculated growth
rate, however, was about one order of magnitude lower than
the one we found from the DNS data and the presented sta-
bility calculations.
Andersson et al.30 found no varicose instability since
they did not have normal velocity profiles with strong normal
shear. In a parallel study 共Andersson et al.31兲, streak profiles
with strong normal shear were analyzed. In those cases the
varicose mode dominated over the sinuous mode.
2. The turbulent case
So far, the detailed analysis of the low-speed streak in an
otherwise laminar boundary layer has confirmed some of the
results from the experiment of AS. Furthermore, a thorough
analysis of the origin of the instability of the streak was
made with linear stability analysis. The simulations also
showed the development of more complicated structures fur-
ther downstream, where the statistics resembled turbulence.
These results, together with the striking resemblance of
the streak development between Figs. 6 and 1 lead to the
hypothesis that, at least to some degree, the break up of
streaks in a turbulent field is governed by the same mecha-
nisms as for the isolated streak in the laminar boundary layer.
To qualitatively show that streak instabilities exist in a
turbulent boundary layer that are of the same normal inflec-
tional type as in the laminar case, the OS analysis was per-
formed with velocity profiles from the turbulent velocity
field.
When a horseshoe structure in the turbulent field has
been identified, it can be followed backward in time, if ve-
FIG. 19. Contours of maximum 共over y兲shear with spacing 0.1. Contours
representing the largest values are indicated with thick lines. 共a兲Normal
shear (maxy
兩
u/
y
兩
), maximum⫽1.3. 共b兲Spanwise shear (maxy
兩
u/
z
兩
),
maximum⫽0.5.
2320 Phys. Fluids, Vol. 14, No. 7, July 2002 Skote, Haritonidis, and Henningson
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
locity fields from earlier times are available. Since the life
cycle of a structure is long 共over T⫽150
␦
*/U⬁兲, the re-
quirement for data storage is demanding. As the structure is
followed backward in time, it is found further upstream and
is weaker. At some point in time and space the structure
vanishes. Thus, at this point the birth of the structure can be
investigated. By examining the time signal of the velocity
from points just upstream of the first appearance of the struc-
ture, the frequency of the disturbance leading to the vortex
formation can be determined. One example of a time signal
of the streamwise velocity is shown in Fig. 20. The instabil-
ity wave appears at time 15588.
The point 共x⫽170, y⫽0.4, z⫽⫺6.5兲where the velocity
signal was examined is located just upstream of the first ap-
pearance of a structure. The newly born structure is shown in
Fig. 21. The figure shows the low pressure signature of the
structure at the time 15596 共referring to Fig. 20兲.
A velocity profile was extracted from the turbulent field
at a point where the disturbance was small compared to fur-
ther downstream, i.e., before roll up of the vortex. In this
particular case the point was located at 共x⫽170, z⫽⫺6.5兲at
the time 15584. This profile was used together with the ob-
served frequency in the OS equation.
To compare the DNS data with the eigenfunctions from
the OS analysis, the rms-profiles were extracted by collecting
statistics during a simulation over one period of the distur-
bance. The rms-profiles were taken from the same position as
where the frequency of the disturbance was observed for the
longest period of time. This position is 共x⫽170, z⫽⫺6.5兲in
the example discussed above. The time interval over which
the rms-profiles were taken was 15584–15596.
This whole procedure was performed for three indepen-
dent structures, each separated in time over 2000 (
␦
*/U⬁).
All three of the structures could be traced back to their point
of roll up, and the analysis of the velocity profiles gave simi-
lar results.
Furthermore, the OS analysis showed that the resulting
eigenfuctions are not sensitive to changes in Re
␦
*and fre-
quency 共
兲. The independence of Reynolds number is ex-
plained by the inviscid nature of the inflectional instability.
The insensitivity on
shows that the time scale of the dis-
turbance is not important for the instability mechanism. This
points towards an instability of a Kelvin–Helmholz charac-
ter. One example of the velocity profile just before roll up is
shown in Fig. 22. The frequency in this case was
⫽0.78
and the OS analysis gave a growth rate of ⫺
␣
i⫽0.024. The
eigenfunctions from the OS analysis were then compared to
rms-values taken over one period of the disturbance. The
results from this analysis are shown in Figs. 23共a兲and 23共b兲.
In the streamwise component 关Fig. 23共a兲兴, the double peak is
predicted by the linear analysis, even though the outer peak
is located further out in the urms profile. In the normal com-
ponent 关Fig. 23共b兲兴, the inner peak is located slightly closer
to the wall in the predicted profile. Also a tendency to a
second peak is seen, though the vrms profile has a much
stronger peak.
Although we have only investigated three randomly
picked events, the results are promising and a larger investi-
gation with an objective method for detecting structures, fol-
lowed by tracing them back in time to their point of origin
and the associated inflectional velocity profile, could provide
statistical evidence of the horseshoe vortex formation. This
is, however, beyond the scope of the present investigation.
FIG. 20. Time signal of the streamwise velocity component at x⫽170,
y⫽0.4, z⫽⫺6.5.
FIG. 21. Iso-surface of pressure at time 15 596. Contour level at ⫺0.004.
The height of the box shown is 4.5, corresponding to 80 in wall units.
FIG. 22. Velocity profile from a turbulent boundary layer.
2321Phys. Fluids, Vol. 14, No. 7, July 2002 Varicose instabilities in turbulent boundary layers
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
IV. DISCUSSION AND CONCLUSIONS
A DNS of a laminar boundary layer disturbed by a con-
tinuous blowing through a slot in the wall has been per-
formed. The objectives were to reproduce and further inves-
tigate the results reported from the experiments of Acarlar
and Smith.12 The blowing of fluid from the slot creates a
low-speed streak which exhibits a disturbance wave growing
downstream. It was argued that this secondary disturbance
originated from a normal inflectional instability in the
streamwise velocity profile. An analysis using the Orr–
Sommerfeld equations gave qualitative agreement in the
growth rate and streamwise wave number with the corre-
sponding values extracted from the DNS velocity fields. The
nonlinear effects gave rise to higher harmonics at the end of
the slot where the first low-pressure structure was found. The
structure consist of a vortex loop that evolves downstream to
form a horseshoe vortex. After the horseshoe vortex breaks
down the low-speed streak persist together with additional
streaks formed by the horseshoe vortex. Further downstream
more complicated structures appear and the streak spacing is
100 in wall units.
The frequency of the vortex generation was shown to
scale with the ratio between the blowing velocity and
freestream velocity. Good agreement with the experimental
data was obtained.
Also a DNS of a zero pressure gradient turbulent bound-
ary layer was performed, and horseshoe vortices were ob-
served using low-pressure identification. The similarities be-
tween structures in the turbulent field and the ones
originating from the low-speed streak in the laminar simula-
tion were presented. The origins of the horseshoe vortices in
the turbulent boundary layer were investigated by tracing
their evolution backwards in time, and the results suggested
that it was related to an inflectional instability of the streaks.
This is similar to the investigations of Johansson et al.32
which traced the evolution of typical structures associated
with the VISA events in low Reynolds number channel flow.
Their investigation did not contain any stability calculations
and can therefore not be directly compared to ours.
In addition, the results from our turbulent boundary layer
simulation is in general agreement with the investigation of
Adrian et al.,14 which shows that the low-speed streak in
turbulence may be spawned by horseshoe vortices. However,
in the numerical experiments of turbulence where the outer
part, and hence the hairpins, has been removed, the streak
prevails. See, e.g., the work by Jimenez and Moin8and
Hamilton et al.9The turbulence cycle is then dominated by
the sinuous instability, which is not surprising since the vari-
cose mode is related to hairpins which are suppressed by the
geometrical constrains.
Adrian et al.14 also propose a cause for the formation of
the vortices themselves. They write that ‘‘关hairpin兴packets
originate at the wall from a disturbance whose character is
not specified except that it creates a pool of low momentum
at the wall, i.e., a Q2 event from another hairpin, a bump, a
puff of low momentum through the wall, a random pressure
fluctuation, or a culmination of flow induced by surrounding
events such as wall-tangent flows that converge to a stagna-
tion point and thence erupt upwards.’’ Our findings support
this idea in the sense that the streak consists of low momen-
tum and the varicose instability follows as a consequence.
The inflectional instability considered in the present
work is of a different type from those investigated in
Waleffe,10 Kawahara et al.23 and Schoppa and Hussain,6who
model the turbulent velocity profile as a mean flow with the
streaky structure deforming the profile, rather than the in-
stantaneous profile considered here. They showed that it is
the sinuous mode which is unstable. Furthermore, it has been
shown in the secondary instability calculations by Andersson
et al.,30 that the growth rate of the sinuous mode scales with
the spanwise derivative of the mean flow, just as in the model
of Waleffe. Thus it is reasonable to assume that the sinuous
instability depends primarily on the appearance of the span-
wise inflection. Reddy et al.33 further showed that the sinu-
ous instability is inhibited by the appearance of normal shear.
We show in this work, as it has been implied in others
共e.g., Robinson,18 Asai et al.29兲, that the appearance of an
unstable normal velocity profile 共in many cases associated
with a normal inflection point兲is a precursor to the appear-
ance of horseshoe vortices. In terms of a streak instability,
Bottaro and Klingmann34 among others, have shown that this
is related to the varicose mode. Thus the sinuous streak in-
stability is correlated with a basic state with a spanwise in-
flection and the varicose mode with a basic state with a nor-
mal inflection, as has also been shown by Park and Huerre.28
It is reasonable to assume that both types of streak insta-
bilities are of importance in a turbulent boundary layer; the
sinuous type for the regeneration of near-wall turbulence, as
FIG. 23. 共a兲Streamwise component.
共b兲Normal component. rms-
value of velocity.---Eigenfunction
from the OS equation.
2322 Phys. Fluids, Vol. 14, No. 7, July 2002 Skote, Haritonidis, and Henningson
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
shown by Jimenez and Moin8and Hamilton et al.,9and the
varicose type for the production of horseshoe vortices popu-
lating the region away from the wall 共see, e.g., Robinson,18
Acarlar and Smith,12 Haidari and Smith,13 and Adrian
et al.14兲.
1S. K. Robinson, ‘‘Coherent motions in the turbulent boundary layer,’’
Annu. Rev. Fluid Mech. 23, 601 共1991兲.
2B. A. Singer and R. D. Joslin, ‘‘Metamorphosis of a hairpin vortex into a
young turbulent spot,’’ Phys. Fluids 6, 3724 共1994兲.
3M. S. Chong, J. Soria, A. E. Perry, J. Chacin, B. J. Cantwell, and Y. Na,
‘‘Turbulence structures of wall-bounded shear flows found using dns
data,’’ J. Fluid Mech. 357,225共1998兲.
4J. Zhou, R. J. Adrian, S. Balachandar, and T. M. Kendall, ‘‘Mechanisms
for generating coherent packets of hairpin vortices in channel flow,’’ J.
Fluid Mech. 387, 353 共1999兲.
5J. Jeong and F. Hussain, ‘‘Coherent structures near the wall in a turbulent
channel flow,’’ J. Fluid Mech. 332,185共1997兲.
6W. Schoppa and F. Hussain, ‘‘Genesis and dynamics of coherent structures
in near-wall turbulence: Anew look,’’Self-Sustaining Mechanisms of Wall
Turbulence, edited by R. L. Panton 共Computational Mechanics, Southamp-
ton, 1997兲.
7J. Kim, P. Moin, and R. Moser, ‘‘Turbulence statistics in fully developed
channel flow,’’ J. Fluid Mech. 177,133共1987兲.
8J. Jimenez and P. Moin, ‘‘The minimal flow unit in near-wall turbulence,’’
J. Fluid Mech. 225, 213 共1991兲.
9J. H. Hamilton, J. Kim, and F. Waleffe, ‘‘Regeneration of near-wall turbu-
lence structures,’’ J. Fluid Mech. 287,317共1995兲.
10F. Waleffe, ‘‘On a self-sustaining process in shear flows,’’Phys. Fluids 9,
883 共1997兲.
11J. Jimenez and A. Pinelli, ‘‘The autonomous cycle of near-wall turbu-
lence,’’ J. Fluid Mech. 389,335共1999兲.
12M. S. Acarlar and C. R. Smith, ‘‘A study of hairpin vortices in a laminar
boundary layer. Part 2. Hairpin vortices generated by fluid injection,’’ J.
Fluid Mech. 175,43共1987兲.
13A. H. Haidari and C. R. Smith, ‘‘The generation and regeneration of single
hairpin vortices,’’ J. Fluid Mech. 277, 135 共1994兲.
14R. J. Adrian, C. D. Meinhart, and C. D. Tomkins, ‘‘Vortex organization in
the outer region of the turbulent boundary layer,’’ J. Fluid Mech. 422,1
共2000兲.
15K. T. Christensen and R. J. Adrian, ‘‘Statistical evidence of hairpin vortex
packets in wall turbulence,’’ J. Fluid Mech. 431, 433 共2001兲.
16A. E. Perry, I. Marus
ˇic
´, and J. D. Li, ‘‘Wall turbulence closure based on
classical similarity laws and the attached eddy hypothesis,’’Phys. Fluids 6,
1024 共1994兲.
17I. Marus
ˇic
´, ‘‘The role of large-scale structures in wall turbulence,’’ Phys.
Fluids 13,735共2001兲.
18S. K. Robinson, ‘‘The kinematics of turbulent boundary layer structure,’’
NASA TM 103859, 1991.
19B. A. Singer, ‘‘The formation and growth of a hairpin vortex,’’ in Insta-
bility, Transition and Turbulence, edited by M.Y. Hussaini, A. Kumar, and
C. L. Street 共Springer, New York, 1992兲, pp. 367–376.
20H. T. Kim, S. J. Kline, and W. C. Reynolds, ‘‘The production of turbulence
near a smooth wall in a turbulent boundary layer,’’ J. Fluid Mech. 50,133
共1971兲.
21M. T. Landahl, ‘‘Wave mechanics of breakdown,’’J. Fluid Mech. 56,775
共1972兲.
22M. T. Landahl, ‘‘On sublayer streaks,’’J. Fluid Mech. 212,593共1990兲.
23G. Kawahara, J. Jimenez, M. Uhlmann, and A. Pinelli, ‘‘The instability of
streaks in near-wall turbulence,’’Center for Turbulence Research, Annual
Research Briefs 1998, pp. 155–170, 1998.
24W. Schoppa and F. Hussain, ‘‘Coherent structure generation in near-wall
turbulence,’’ J. Fluid Mech. 453,57共2002兲.
25A. Lundbladh, S. Berlin, M. Skote, C. Hildings, J. Choi, J. Kim, and D. S.
Henningson, ‘‘An efficient spectral method for simulation of incompress-
ible flow over a flat plate,’’Technical Report TRITA-MEK 1999:11, Royal
Institute of Technology, Stockholm, 1999.
26J. Nordstro
¨m, N. Nordin, and D. S. Henningson, ‘‘The fringe region tech-
nique and the fourier method used in the direct numerical simulation of
spatially evolving viscous flows,’’ SIAM J. Sci. Comput. 共USA兲20共4兲,
1365 共1999兲.
27M. S. Chong, A. E. Perry, and B. J. Cantwell, ‘‘A general classification of
three-dimensional flow fields,’’ Phys. Fluids A 2, 765 共1990兲.
28D. S. Park and P. Huerre, ‘‘Primary and secondary instabilities of the
asymptotic suction boundary layer on a curved plate,’’J. Fluid Mech. 283,
249 共1995兲.
29M. Asai, M. Minagawa, and M. Nishioka, ‘‘The instability and breakdown
of a near-wall low-speed streak,’’J. Fluid Mech. 455, 289 共2002兲.
30P. Andersson, L. Brandt, A. Bottaro, and D. S. Henningson, ‘‘On the
breakdown of boundary layer streaks,’’ J. Fluid Mech. 428,29共2001兲.
31P. Andersson, A. Bottaro, D. S. Henningson, and P. Luchini, ‘‘Secondary
instability of boundary layer streaks based on the shape assumption,’’
Technical Report TRITA-MEK 1999:13, Royal Institute of Technology,
Stockholm, 1999.
32A. V. Johansson, P. H. Alfredsson, and J. Kim, ‘‘Evolution and dynamics
of shear-layer structures in near-wall turbulence,’’ J. Fluid Mech. 224,579
共1991兲.
33S. C. Reddy, P. J. Schmid, P. J. Bagget, and D. S. Henningson, ‘‘On
stability of streamwise streaks and transition thresholds in plane channel
flows,’’ J. Fluid Mech. 365, 269 共1998兲.
34A. Bottaro and B. G. B. Klingmann, ‘‘On the linear breakdown of Go
¨rtler
vortices,’’ Eur. J. Mech. B/Fluids 15, 301 共1996兲.
2323Phys. Fluids, Vol. 14, No. 7, July 2002 Varicose instabilities in turbulent boundary layers
Downloaded 04 Jun 2002 to 130.237.222.14. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
