12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

ON THE MULTI-MODAL GROWTH OF DISTURBANCES IN A LAMINARSEPARATION BUBBLE SUBJECTED TO FREESTREAM TURBULENCE

Tomek Jaroslawski*, Olivier Vermeersch and Maxime Forte
Dept. of Multi-Physics for Energetics
ONERA
Toulouse, France
thomas.jaroslawski@onera.fr*, olivier.vermeersch@onera.fr, maxime.forte@onera.fr

Erwin Gowree and Jean-Marc Moschetta
Dept. of Aerodynamics and Propulsion
ISAE-SUPAERO
Toulouse, France
erwin-ricky.gowree@isaesupaero.fr, jean-marc.moschetta@isaesupaero.fr

ABSTRACT

integral length scale (Λu ) on boundary layer transition in LSBs
have not been addressed to the same extent as for attachedboundary layers. Häggmark et al. (2000) provided some of
the first experimental results on the effects of grid generated
freestream turbulence (FST) on an LSB generated over a flat
plate subjected to an adverse pressure gradient. They found
low frequency streaky structures in the boundary layer upstream of the separation and in the separated shear layer fromsmoke visualisation and spectral analysis. More recently, Istvan & Yarusevych (2018) experimentally investigated the effects of FST on an LSB formed over a NACA0018 aerofoil
for a chord based Reynolds numbers of 80000 and 150000
using Particle Image Velocimetry (PIV). They found that increasing the level of FST leads to a decrease in the chordwise
length of the LSB due to a downstream shift of the separation
point and an upstream shift of the reattachment point. Hosseinverdi & Fasel (2019) used direct numerical simulations(DNS) to investigate the role of isotropic FST on the hydrodynamic instability mechanisms and laminar to turbulent transition in an LSB. They reported that the FST induced Klebanoffmodes (streaks) upstream of the separation location, proposingthat the boundary layer transition process was made up of two
mechanisms. The first consisted of low frequency Klebanoffmodes (streaks) induced by the FST and the second being a
Kevin-Helmholtz instability enhanced by the FST. Dependingon the level of FST either one or both of these mechanisms
would dominate the transition process.
Hence the common notion of the modal instability being ”bypassed” appears to be under question as evidence fromthe above authors suggests the coexistence of two competingtransition mechanisms in an LSB. However, there is a lackof knowledge on how freestream turbulence and the integral
length scale of turbulence affect the stability of an LSB subjected to FST. The present work investigates the effects offorcing an LSB with a large range of Tu and Λu on the flow
development, stability and transition of the bubble. The aim of
this study is to experimentally investigate the laminar separation bubble, focusing our attention on the coexistence of modaland non-modal instabilities, their interaction and effects on thetransition process.

Experiments were conducted to study the transition andflow development in a laminar separation bubble (LSB)
formed on an aerofoil. The effects of a wide range offreestream turbulence intensity (0.15% < Tu < 6.26%) andstreamwise integral length scale (4.6mm < Λu < 17.2mm) areconsidered. The coexistence of a modal instability due to the
LSB and a non-modal instability caused by streaks generated
by freestream turbulence is observed. The presence of streaks
in the boundary layer modifies the mean flow topology of the
bubble. These changes in the mean flow field result in the modification of the convective disturbance growth, where an increase in turbulence intensity is found to dampen the growth ofthe modal instability. For a relatively fixed level of Tu, the variation of Λu has modest effects, however a slight advancementof the non-linear growth of disturbances and eventual breakdown with the decrease in Λu is observed. The data shows thatthe streamwise growth of the disturbance energy is exponential for the lowest levels of freestream turbulence and gradually becomes algebraic as the level of freestream turbulence
increases. Once a critical turbulence intensity is reached, there
is enough energy in the boundary layer to suppress the LSB,
which in turn, results in the non-modal instability to take over
the transition process. Linear stability analysis is conducted in
the fore position of the LSB, and accurately models unstablefrequencies and eigenfunctions for configurations subjected tolevels of turbulence intensity levels up to 3%. Increasing theTu resulted in the Reynolds number dependence to increase,suggesting that a viscous, rather than an invicid formulationof the stability equations is appropriate for modeling modal
instabilities in the fore portion of the studied LSB.

INTRODUCTION

5

At low Reynolds numbers (Re < 5 × 10 ) viscous effects
are significant, such that the presence of a strong enough adverse pressure gradient can cause a laminar boundary layer
to separate from the wall. These types of flows are common in engineering applications such as low-pressure turbines(Volino, 1997) and micro-aerial vehicles (Jaroslawski et al.,2022). The effects of freestream turbulence intensity (Tu) and1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
EXPERIMENTAL SETUP

Config.

The experiments were conducted at atmospheric conditions in the ONERA Toulouse TRIN 2 subsonic wind tunnel.
The maximum freestream turbulence level (measured near theleading edge of the aerofoil, cf. Fig. 1a) in the test section
with the aerofoil mounted was found to be below 0.15 %. The
freestream velocity was fixed at U∞ ≅ 6m/s for all test configurations, corresponding to a chord based Reynolds number,
Rec = U∞ c/ν of 125000. The angle of attack, AoA, was fixed◦
to a value of 2.3 throughout all the experiments. Velocity
measurements are acquired using a Dantec Dynamics Streamline Pro system with a 90C10 module and a 55P15 boundary
layer probe mounted on a two-dimensional traverse, at a sampling frequency of 25 kHz. Freestream turbulence measurements were conducted using a 5 µm Dantec 55P51 X-Wireprobe. Freestream turbulence is generated in a controlled manner using a variety of regular and fractal grids (refer to Fig. 1b,c) set up in a way such that turbulence interacting with the
bubble would be approximately isotropic and homogeneous.
The evolution of the grid generated turbulence was characterised before the leading edge and above the aerofoil. The
tested configurations are presented in table 1. Infrared Thermography measurements were also conducted to validate the
spanwise flow homogeneity of the bubble (which are not presented in the present paper). The experimental setup is presented in Fig. 1a.

NG
C0
C1
C2
C3
C4
C5
C6
C7

2

Ω d ṽd U
2(U − α ) ( 2 − α ṽ) − 2 ṽdydy2Λu (mm)Λv (mm)

0.92
0.82
0.91
0.81
0.92
1.07
1.07
1.02
1.10

0.15
0.64
1.21
1.23
1.31
1.63
2.97
4.16
6.26

210
4.6
8.7
10.3
8.3
12.3
15.4
16.8
17.2

181
3.1
5.5
6.7
5.6
8.3
10.6
11.4
13.3

0.05 to xS /c = 0.700 ± 0.025 for the natural case. In the presence of freestream turbulence forcing the mean flow toplogyof the LSB changes, in particular a slight delay of boundary
layer separation is observed, the height decreases significantly
and the mean transition position advances upstream as can beobserved in Fig. 2b. At the highest level of Tu (Fig. 2c),no LSB is observed as there is enough energy injected fromthe freestream turbulence into the boundary layer to suppress
the laminar separation. The measurements, in accordance withprevious studies (Istvan & Yarusevych, 2018; Simoni et al.,2017; Hosseinverdi & Fasel, 2019), show that with the increaseof Tu, the streamwise extent of the separation bubble is reduced. This is a result of an earlier onset of pressure recovery,
caused by the shear layer transitioning in the aft position of the
LSB.
The power spectral density (PSD) of the streamwise velocity fluctuations was calculated for each configuration, withthe chordwise evolution presented in Fig. 4. In the cases
where an LSB was present, the PSD exhibits a characteristic frequency band which is amplified downstream (cf. Fig.
4a-c). When the LSB was subjected to FST the chordwise development and distribution of the spectra is significantly modified. First the unstable frequency band is broadened, which isa consequence of significant energy content within a broader
range of frequencies in the FST, resulting in measurable velocity fluctuations over a broader frequency range earlier upstream. Second, increasing the freestream turbulence level, results in the unstable frequency band to be slightly shifted to ahigher frequency range compared to the natural case. The frequency shift of the wave packet is attributed to the decreased
streamwise length and height of the LSB, and has been observed in Hosseinverdi & Fasel (2019).
The effect of increasing the level of Tu on the chordwise22
evolution of the disturbance energy growth (E = urms /Ue ) ispresented in Fig. 3a, where the trend of disturbance growth
gradually changes from exponential, at lower levels of Tu,
to algebraic for the more extreme Tu levels, where energy
saturation is observed earlier. Algebraic or transient growthis associated with a non-modal instability, commonly due tostreaks in boundary layer flows subjected to elevated levelsof freestream turbulence (Tu > 1% ) and has been well documented for zero-pressure gradient attached boundary layers(Matsubara & Alfredsson, 2001; Fransson et al., 2005). These
different energy growth behaviours suggest that different instability mechanisms are present in the flow, and their con-

Linear Stability Theory (LST) has been employed to
study the convective streamwise amplification of disturbancesin the LSB. The Orr-Sommmerfeld given by Eq. 1, can reliably predict the primary amplification of instability waves forparallel flows and in the fore position of a LSB (Yarusevych &Kotsonis, 2017).

4

Tu(%)

Table 1. Freestream turbulence test matrix. Turbulence
isotropy, turbulence intensity (Tu), streamwise and vertical integral length scale (Λu and Λv , respectively) at the leadingedge of the aerofoil (x/c = 0).

LOCAL LINEAR STABILITY ANALYSIS

2

vrms /urms(1)

iUe δ1 d ṽ2 d ṽ
4=−− 2α+ α ṽ)(αReδ1 dy4
dy2
where Reδ1 is the Reynolds number based on displacement thickness, ṽ is the wall-normal perturbation, Ω is the angular frequency and the complex wave number is defined as
α = αr + iαi , where i is the imaginary unit. When αi > 0 the
disturbance is attenuated and amplified when αi < 0.
Calculations were conducted using ONERA’s in housestability code, where a spatial formulation of the problem is
employed, such that Ω is defined and the eigenvalue problem
is solved for α, therefore modelling the convective amplification of single frequency disturbances. Equation 1 is solved numerically using Chebyshev polynomial base functions and thecompanion matrix technique to treat eigenvalue non-linearity.
The mean streamwise velocity profiles at discrete streamwise locations are used as input for the LST calculations.
In the stability analysis the higher order spatial gradients arehighly sensitive to noise, therefore hyperbolic tangent fits areused for the calculations. This method has been shown to provide accurate linear stability predictions on HWA velocity profiles of separated shear layers (Boutilier & Yarusevych, 2012).

RESULTS
The mean streamwise velocity and urms contours (which
are composed of 21 streamwise velocity profiles separated by
0.025c in the chordwise direction) are presented in Fig. 2a andshow the presence of an LSB that extends from xS /c = 0.375 ±212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
(7) X-Wire(1) Hotwire boundary layer probe(4) Pitot tubey(2) Locations of FST characterisation (5) 2D traverse(a)J. NedićS. Tavoularis
J. NedićandS.Turbulence
Tavoularis(3)
generating grid(6) IR measurement region
x and

6
t0 t0

t0

(5)

: Reference Tu measurement

t0

(3)

0.4

(7)

0.3

x x
(2)2 2x3 x3

L0 Lh0 h(1)

0.2

L0 hL0 h
0.1

50 mm

AoA

(6)

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FSG
(a) (a)FSG

Grid mesh, M(a) (a)

t0 t0

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Flow
M M

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x / M > 20

c

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6 6

tt0 tt0

0

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(b) (b)tt0(c)

J. NedićS. Tavoularis
J. NedićandandS. Tavoularis
0.8

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tt0

t0

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L0 hL0 h(c) (c)
RG80
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h

x2

h

MLf 0 Mh
Lf 0 h(a) grid:
FSG(a)Fractalgrid:
C7 FSG
Fractal
C7

(d)
RG18
(d)
RG18
Regulargrid:
C0-C6
Regular
grid:
C0-C6

Figure2. Scaleddrawingsof theturbulence
generating
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Note
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t0

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Tu = 1.21%, !u = 8.71 mm
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to removeedge
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tially. Once the turbulence forcing reaches a critical level, theas shown
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and
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in
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waswas
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Consequently,
the ratio between
subsequent
modal via the non-modal instability. Damping of the modal
observed that an increase1�(1−N ) in )Λu at the leading edge of the aerodeterminedas R= ttr= t1�(1−N= 0.355.
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withwith
previous
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20
Vassilicos
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eventual
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This the&length
due to the influence of freestream turbulence. In other words,& Lavoie
2014;
Valente& Vassilicos
ratio
betweenof s&
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theratio
betweenthelengthexternal freestream turbulence forcing reduces the size of theis in wasagreement
who suggested
that
the smaller
elements
set setto Breuer
0.5;
as (2018),
anexample,
L1 =L0.5L
three
other
grids
(RG(
0 . Theelements
was
to to
0.5;
as of
anthe
example,
other
grids
1 = 0.5L
0 . Theseparation bubble, such that the region of instability growthscaleswerecloserthat
shear
layer,
resultingin threetheandandRG18)
were“regular”
squaregrids,eachwithelements
closely
matching
RG18) weresquaregrids,
eachHosseinverdi
with elements
match
is brought closer to the wall, resulting in damping effects ofreceptivityof the“regular”
boundarylayeroftoFSG.
increase.
& closely
secondandandfourth-levelelements
RG18
alsoalso
matchedthe theeﬀectivesecondfourth-levelelements
of FSG.
RG18
matchedeﬀec
the disturbances in the shear layer. Previous experiments on
Fasel(2019)
brieflysuggestedthat
theintegrallengthscales(Hurst& Vassilicos
2007)
of FSG.
(Hurst& Vassilicos
2007)
of FSG.
forced bubbles, found a damping effect on the disturbance
ranging from 0.9δ1 to 3δ1 had little effect on the energy growth
growth. For example, Kurelek et al. (2018) found that both
relative to the Tu, which is observed in the experimental results
tonal and broadband acoustic forcing resulted in the damphere. Furthermore, a smaller integral length scale resulted in a3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)

0.05
0.05

0.04
0.04
0.03
0.03

(a)

(b)

20

10

20

10

10

10

10

0

100

10-10

10-10

10

0.15%
0.64%, 6.72 mm
1.21%, 8.71 mm
1.23%, 10.31 mm
1.31%, 8.31 mm
1.63%, 12.33 mm
2.97%, 15.42 mm
4.16%, 16.81 mm
6.26%, 17.18 mm

10

0.02
0.02

0.01
0.01

0
0.2
0
0.2

(b)

0.3
0.3

0.4

0.4
0.5

0.5
0.6

0.7

0.6

0.7

0.8

0.8

10-20

-20

101

100

102

103

10

101

(c)

10-5

102

103

(d)

1020

1020

1010

1010

100

100

10-10

10-10

x/c

-10

10

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(c)
0.05
0.04

10-20
10

1.23%, 10.31 mm
1.21%, 8.71 mm
1.31%, 8.31 mm

0.03
0.01
0.3

0.4

0.5

0.6

0.7

10

2

3

10

10-20

101

102

103

Figure 4. Chordwise evolution of the PSD. Frequency bands
correspond to the vertical dashed lines which indicate the mostamplified frequency band used in the stability analysis in thefollowing section. Red and blue curves denote xS and xR , respectively. NB: Spectra are separated by an order of magnitude
for clarity. a) Tu = 0.15%. Frequency band: [110-150 Hz]; b)Tu = 1.21%. Frequency band: [180-220 Hz]; c) Tu = 2.97%.
Frequency band: [255-295 Hz]; d) Tu = 6.28%;

!u

0.02
0
0.2

1

0.8

Figure 3. The disturbance energy growth for a) integrated
over the entire energy spectrum and b) integrated over the frequency range of the most amplified wave packet and c) configurations with are latively fixed Tu and varying Λu .

is indicative of a viscous modal instability (Veerasamy et al.,2021). Rist & Maucher (2002) showed that LSBs with smaller
wall normal-distances could exhibit a viscous modal instability. Therefore, even when the LSB is subjected to elevatedlevels of freestream turbulence, LST can predict the most amplified frequencies and eigenfunctions. This suggests that amodal instability is still present at elevated levels of freestream
turbulence when the bubble is present. Furthermore, the unfiltered (Fig 6d) distrubance profiles agree remarkably well withthe theoretical optimal perturbation profile (Luchini, 2000) for
configurations with Tu > 1%. Although not presented in thepresent paper, all configurations where Tu > 1% exhibit agreement with the optimal perturbation profile at multiple chordwise positions, demonstrating self similarity of the disturbance
profiles over most of the boundary layer. With the only exception being outside of the boundary layer, with urms /umax not
tending to zero since freestream turbulence is present, in contrast to theory, which has no freestream disturbances outside of
the boundary layer. In configurations where Tu < 1% (Fig 6c),
experimental disturbance profiles do not agree with theory, implying that only a modal instability was present. Furthermore,
all configurations subjected to freestream disturbance levels of
Tu < 3% show agreement with LST. Therefore, the results imply the coexistence of both modal and non-modal instability
mechanisms, confirming the observations made by Hosseinverdi & Fasel (2019). The damping of the disturbance growthhigher initial level of disturbance energy in the boundary layer
and has been also observed by Hosseinverdi (2014), howeverin their work, the saturation of the energy growth was foundto be independent of Λu . Based on the experimental observations here and past numerical simulations, an effect of integral
length scale could be present and further investigation is warranted. However, it is likely that the effect will be small compared to the Tu, in light of the results here and Hosseinverdi &Fasel (2019).
Results of the linear stability analysis are presented Fig.
5, where an overlaid plot between PSD from experiment andαi shows that LST is capable of predicting the most amplified
frequencies from experiment, even in the presence of elevatedlevels of Tu (NB. Only two configurations are presented forbrevity). Furthermore, the eigenfunctions of the most amplified frequency predicted by LST are presented in Fig. 6a,b, andare in acceptable agreement with the experiment for the filtered(for the most amplified frequency band) fluctuating streamwise
velocity profile in the wall-normal direction. The eigenfuction
exhibits two distinct peaks at approximately y/δ1 = 1, corresponding roughly to the inflection point and y/δ1 = 0.3, which412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
(b)

(a)
10-3

0.15

10-3

Tu = 0.15%
0.05

10-5
10
10

-7

5
Exp.
LST

0.05

10-5

Exp.
Theory

4

4

3

3

2

2

1

1

0
-6

-0.05

10-8

5

0.1

10-4

0
-6

Tu = 0.15%

0.15

Tu = 1.21%
0.1

10-4

(c)

(a)

100 200 300 400 500

10

-0.05

-7

-0.1

10

-0.15

10-8

-0.1
-0.15
100 200 300 400 500

Figure 5. Comparison between the amplified frequencies
predicted by LST to the experimental spectra for (a) Naturalcase (x = 0.400c); (b) Tu = 1.21%Λu = 8.7mm (x = 0.425c);
NB: Two different y−axes for αi and the power from the PSD,
therefore direct comparisons between the two are not be made.

0

0of the modal instabilities (cf. Fig. 3b) is attributed to thestreaks modifying the mean flow.

0

0.5

1

0

0.5

Filtered: [110-150] Hz.

Full Spectrum.

(b)

(d)

1

Tu = 1.21%
5Further insights can be obtained upon examination of thecontours of the non-dimensional frequency (Ω) and αi as afunction of the Reynolds number plotted in Fig. 7 for the baseline configuration and a forced one. For lower Tu, a larger
range of unstable frequencies is present, which is due to theinflectional point being further away from the wall. Upon inspection of the contours of the LST growth rates (Fig. 7a),a dependence on Reδ1 is observed, suggesting that viscosity
needs to be considered in the LST calculation in the fore position of the LSB. Moreover, increasing Tu results in the rangeof unstable frequencies to decrease and the Reynolds number
dependence to increase (Fig. 7b), due the inflection point shifting closer towards the wall, resulting in an increased effectof viscosity. For example, the increased dependence of αi onReδ1 with increased Tu, is clearly demonstrated in Figs. 7cand d where the unexcited bubble converges at Reδ1 ≈ 8000
compared to Reδ1 ≈ 13000 for the elevated levels of Tu. This
result implies that invicid LST calculations are unsuitable toaccurately model the stability in the fore position of the LSB(especially in the presence of elevated levels of FST) at the low
values of Reδ1 considered in the present work.

5

Exp.
LST

Exp.
Theory

4

4

3

3

2

2

1

1

0

0

0.5

1

Filtered: [180-220] Hz.

0

0

0.5

1

Full Spectrum.

Figure 6. Experimental filtered disturbance profiles in thewall-normal direction compared to the eigenfunction for themost amplified frequency from LST. Experimental streamwise
disturbance profiles are computed by applying a bandpass filter corresponding to the lost amplified frequency band from the
PSD (a,c). Unfiltered experimental disturbance profiles compared to optimal perturbation theory (b, d).

CONCLUSION
The present investigation examined the effects of varying
the freestream turbulence intensity and integral length scaleon the flow development and transition in a laminar separation
bubble. The current work provides experimental evidence on
the coexistence of modal and non-modal instabilities in a laminar separation bubble. It is shown, through experiment andtheory, that even at relatively high/moderate turbulence intensity levels the modal instability is still operational in an LSB
and the primary growth can be satisfactorily predicted with the
Orr-Sommerfeld formulation. The damping of the streamwise
growth of disturbances is due to the presence of streaks causedby the elevated levels of freestream turbulence, which modify
the mean flow topology of the bubble through the introductionof non-modal disturbances (streaks) into the boundary layer.
A dependence of αi on Reδ1 is observed, suggesting that viscosity effects need to be considered in LST calculations in thefore position of the LSB.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

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Re
0

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1

= 100

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Figure 7. Contours of LST predicted spatial growth rates (αi ) as a function0 of the dimensionless(Reδ1 ) for the natural case for a mean velocity profile at x = 0.400c (a) and an0.08
elevated freestream turbulence case for a mean velocity
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