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

FREESTREAM-TURBULENCE INDEPENDENCE OF SECONDARY INSTABILITY OF
CROSS-FLOW VORTICES IN SWEPT-FLAT-PLATE BOUNDARY LAYER
Kosuke Nakagawa
Department of Mechanical Engineering
Tokyo University of Science
2641 Yamazaki, Noda-shi, Chiba, 278-8510
7522702@ed.tus.ac.jp

Takahiro Ishida
Aeronautical Technology Directorate
Japan Aerospace Exploration Agency (JAXA)
6-13-1 Osawa, Mitaka, Tokyo 181-0015
ishida.takahiro@jaxa.jp

Takahiro Tsukahara
Department of Mechanical Engineering
Tokyo University of Science
2641 Yamazaki, Noda-shi, Chiba, 278-8510
tsuka@rs.tus.ac.jp

ABSTRACT
Laminar-to-turbulent transition on a swept-flat plate iscaused by the breakdown of the cross-flow vortex via the highfrequency secondary instability. In order to determine common
transition mechanisms, this parametric study has analyzed
turbulent transitions under various environmental conditions
including wall-roughness elements and freestream turbulence.
We performed direct numerical simulations of the Falkner–
Skan–Cooke boundary layer with various conditions of isolatedroughness and/or different peak wavelengths of freestream
turbulence. In the roughened case, where the flow wasaccompanied by the stationary cross-flow vortex, the shortwavelength freestream turbulence promoted the secondary
instability with hairpin vortices, while the long-wavelength
disturbance delayed the cross-flow-vortex break down
compared to the short wavelength condition. The wavelengthof freestream disturbance played a key role to promote highfrequency secondary instability on the cross-flow vortex. The
main finding here is, with or without freestream turbulence, thehigh-frequency secondary instability is commonly the trigger
for the breakdown to turbulence.

flow vortex. Högberg & Henningson (1998) reported that thefrequency ω of secondary instability was the order of 100. The
frequency was non-dimensionalized by the external chordwise
velocity U0 and the displacement thickness δ*. The highfrequency secondary instability was classified into y- and ztype modes, based on the relative position against the crossflow vortex. These frequencies were found to be unstable forthe cross-flow vortices (Brynjell-Rakola et al., 2015; 2017).
Many studies reported that the FST modulated thestationary cross-flow vortex, and induced a traveling vortex.
Deynhel & Bippes (1996) performed a wind-tunnel test on theswept-flat-plate model under various FST intensity conditions.
They found that the threshold of FST intensity Tu, above which
the traveling vortex dominates the turbulent transition process.
The threshold was approximately 0.2%. Downs & White(2013) showed the effects of FST on the stationary mode andunstable modes of the cross-flow vortices by wind tunnel tests.
The unstable disturbances were intensified by the FST.
Although the turbulent transition process via the isolatedroughness elements and the intensity of the FST have beenstudied, there is less understanding of the interaction between
the stationary vortex (due to a wall roughness) and the FST.
There are few studies on the effects of the wavelength
distribution of the FST. In this study, we focused on the peakwavelength of the FST and compare the turbulent transition
processes depending on the characteristics of the FST alone
and the interaction with the stationary cross-flow vortex whichis induced by the cylindrical roughness. We aimed to obtain
critical conditions of the transition process related to the crossflow vortex that is exposed to FST by discussing the turbulent
transition process.

INTRODUCTION
A swept wing has a backward angle against the fuselage.
Near its leading edge, the cross-flow instability is dominant andpromotes the turbulent transition. The cross-flow instability
forms a cross-flow vortex. As the cross-flow vortex evolves
downstream, the turbulent transition is induced after anonlinear saturation. At the saturation state, a high-frequency
disturbance called secondary instability occurs. The highfrequency secondary instability is the main factor of the
turbulent transition of the cross-flow vortex. A cross-flow
vortex via the surface roughness forms a stationary vortex,
while a traveling vortex via environmental disturbance such as
freestream turbulence (FST), or both, may occur. The crossoverof stationary and traveling vortices is still unclear. Theinteraction of roughness and freestream disturbance should beclarified.
Brynjell-Rakola et al. (2017) simulated flows past an
isolated cylindrical roughness, which is a kind of surface
roughness, on the swept-flat plate by direct numericalsimulation. They investigated the turbulent transition process ofthe stationary cross-flow vortex. They also detected thelocation of high-frequency secondary instability on the cross-

NUMERICAL METHOD
The base flow was Falkner–Skan–Cooke (FSC) similarity
solution (Brynjell-Rakola et al., 2017). We performed DNS ofincompressible three-dimensional boundary-layer flows. The
governing equations are the continuity and the Navier–Stokes
equations. The computational domain was (Lx × Ly × Lz) = 200–800δ0* × 25.14δ0* × 27δ0* in the chordwise (x), spanwise (y),and wall-normal (z) directions, respectively. Here, δ0* is thedisplacement thickness at the inlet of the computational domain.
The number of grids was 1024–4096 × 128 × 128. A periodic
boundary condition was imposed in the y- direction, and thenon-slip condition was applied at the wall. In the x- direction,
the convective outflow condition was applied. The cylindrical1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
roughness with a height of kz and a diameter of 6.0δ0* wasimplemented by an immersed boundary method, and its center
was located at x/δ0* = 20.59. The spanwise roughness spacingof 25.14δ0* corresponds to the most unstable stationary crossflow mode. The Reynolds number Re (= U0δ0*/ν) was 337.9,
where U0 is the external chordwise velocity at the inlet. The
details of the numerical method were reported in our previous
paper (Ishida et al., 2022). Validation carried out is discussed
in the next section.
The method of simulating freestream turbulence (FST) was
referred to as Watanabe & Maekawa (2015), allowing us toselect arbitrarily a turbulent intensity (Tu) and a peakwavelength (λmax/δ0* (=Ly/δ0*/k)). The energy spectrum of FST
is given as
E(k) = k4 exp(−2(k/kmax)2).

inducing a cross-flow vortex earlier than in the shortwavelength condition. From the comparison between theroughness condition and without roughness condition, thestationary or traveling cross-flow vortex is necessary for the
early transition process via the FST-induced secondary
instability.

(1)

Figure 1 shows the energy spectrum for different peakwavelengths. The turbulence intensity of the disturbance isdefined as Tu = (ui2/3)1/2×100[%], where ui indicates thedisturbance velocity. In this study, we fixed Tu = 0.1 %. The
disturbance was damped in the near-wall region at the inletaccording to the shape of the chordwise FSC velocity profile tosatisfy the non-slip condition.

Figure 3 shows the visualizations of vortex structures forthe only cylindrical roughness case and the only FST case. The
vortex structure is visualized by the iso-surface of the second
invariant of the velocity gradient tensor Q and colored by
u’w’/U02. The vortex emerges at the increasing position of thefriction coefficient plotted in Fig. 2. These are the finger vortex
via the high-frequency secondary instability. The vortex
structures are the same as those reported by Brynjell-Rahkola
et al. (2017) and Wassermann & Kloker (2003) and are thetypical structure that can be observed in the turbulent transition

max / 0** = 25.14
max / 0 = 5.0

1E() / Emax

Figure 2. Friction coefficient distribution.

0.5

0 0
10

(a)

101

102

 / 0*Figure 1. Energy spectrum for freestream turbulence.
RESULTSFirst, we demonstrate diverse phenomena in the turbulent
transition processes depending on the cylindrical roughness andthe freestream turbulence (FST). Figure 2 shows the chordwise
distribution of the friction coefficient, Cf. Upstream, the friction
coefficients are close to the laminar value of the FSC flow.
Downstream, the friction coefficients drastically increase. The
increase is owing to the proceeding of the transition process.
Under the roughness condition, the turbulent transition is muchearlier at x/δ0* ≈ 100 in the short-wavelength condition of
λmax/δ0* = 5.0. Högberg & Henningson (1998) reported a highfrequency secondary instability on the shear layer after crossflow vortex saturation. As in the present result, the stationary
cross-flow vortex near the wake of cylindrical roughness isforced to have a high-frequency secondary instability by thehigh-frequency disturbance of the FST. Therefore, the turbulent
transition in the short-wavelength condition occurs earlier thanin the long-wavelength and no-FST conditions. On the other
hand, without roughness, the long-wavelength condition of
λmax/δ0* = 25.14 causes an early turbulent transition. In this case,the cross-flow vortex is absent for a long extent and, of course,
the secondary instability should not emerge in this range. In thelong-wavelength condition, the peak wavelength corresponds
to the most unstable stationary cross-flow mode (λ/δ0* = 25.14),(b)

Figure 3. Visualization for instantaneous vortex structurecolored by u’w’/U02 between −0.015 (blue) to 0.015 (red). Isosurfaces are u/U0 = 0.3 (blue) and Q = 0.005. Bottom figure isenlarged view of the finger vortex. (a) shows the condition ofkz/δ0* = 1.1 and Tu = 0.0 %. (b) shows the condition withoutroughness, but with long-wavelength freestream turbulence (Tu= 0.1 %,λmax/δ0* = 25.14).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)roughness condition, where we aim to extract the frequenciesof the hairpin and finger vortexes. The frequency of the highfrequency instability reported by Högberg & Henningson(1998) was ω ≈ 0.9. Brynjell-Rahkola et al. (2015) reported thefrequency of the unstable fluctuations at the cross-flow vortex,
which was ω ≳ 0.9. Our results shown in Fig. 5(a) indicate thatthe frequency of the secondary instability are independent of
the secondary vortex structure: both two cases exhibit a peak at
ω = 0.6–0.9, which correspond to that of the previous studies.
Figure 5(b) shows the case with long-wavelength FST, inwhich there only occurs the finger vortex. In this case, lowfrequency disturbances occur most strongly which is consistent
with a report by Högberg & Henningson (1998). This spectrum
implies strong growth for a traveling wave. Owing to the
spanwise advection characteristic of traveling waves, thefrequency of secondary instability becomes low. However,independently of the FST wavelength, the secondary vortex
provides the stationary vortex with an increase in the highfrequency disturbance.

(b)

(a)
102

PSDu'

101
100
10-1
-2

10

With roughness condition ( kz / 0 = 1.1)
Tu = 0.0 %Tu = 0.1 %, max / 0* = 5.0*

10-1

Figure 4. Visualization for instantaneous vortex structure offreestream turbulence condition (Tu = 0.1 %). Iso-surfaces andcolor are same as Fig. 3 but Q = 0.05. (a) shows the longwavelength condition (λmax/δ0* = 25.14) and kz/δ0* = 1.1. (b)shows the short-wavelength condition (λmax/δ0* = 5.0) and kz/δ0*= 1.1.

100



101

(b)

102

PSDu'

101

processes in this case. The finger vortex has u'w' < 0 regions. It
suggests the onset of Reynolds shear stress. Under the case of
the roughness and FST, there are different vortex structures.
Figure 4 shows the flow fields with the same iso-surfaces as
Fig. 3. Under the short-wavelength case, hairpin vortices occur
near the wake of roughness. Hairpin vortices provide the wakevortex with the transition to turbulence via high-frequency
secondary instability. The hairpin vortex has a sweep andejection region, where fluid motions of blowing down and up
around the vortex, respectively (Robinson; 1991), resulting inu’w’ < 0. The characteristics of fluid motions are the same as
the finger vortex. Under the long-wavelength condition, theformation of the finger vortex is the same as in Fig. 3. There
are also fluid motions of u’w’ < 0 around the finger vortex. In
addition, the correspondence with the friction coefficient
distribution (Fig. 2) indicates that vortices are the trigger of
turbulent transition.
We show the common characteristics between the hairpin
and finger vortexes, which are induced by the high-frequency
secondary instability in the swept-flat-plate boundary layer, bymeans of the frequency analysis of the disturbance velocity
components. In the frequency analysis, the sampling period Δt*
and averaging time T* equal 0.098 and 390, respectively.
Measurement positions are at the secondary vortices shown in
Figs. 3 and 4. Figure 5(a) shows the frequency analysis with

100
Without roughness condition(Tu = 0.1 %,max / 0* = 25.14)*
x / 0* = 350
x / 0* = 390
x / 0 = 430
-2

10-1
10

10-1

100



101

Figure 5. Frequency analysis of chordwise disturbance velocity(u’/U0). (a) shows independency on secondary vortex structurewith roughness condition (kz/δ0* = 1.1). Black: obtained at (x, y,z)/δ0* ≈ (370, 2.6, 2.4). Red: (x, y, z)/δ0* ≈ (50, 14.5, 1.0). (b)shows without roughness condition for λmax/δ0* = 25.14 at (y,z)/δ0* ≈ (10.0, 2.4).
From these results, we clarified that the turbulent transitionof the swept-flat-plate boundary layer proceeds through thehairpin vortex or finger vortex owing to the high-frequency
secondary instability at the shear layer. This is a featurecommon to all the transition processes, independent of thesurface roughness and FST conditions.
CLYRINDRICAL ROUGHNESS HEIGHT
Under conditions of cylinder roughness and FST, we foundthat the vortex structures and turbulent transition positions

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)
0.02

without FST

Cf

0.015

with FST (Tu = 0.1%, max / 0* = 25.14)kz / 0** = 1.1
kz / 0* = 1.9
kz / 0* = 1.3
kz / 0* = 2.0
kz / 0* = 1.5
kz / 0 = 1.7
FSC flow

0.01

0.005
100

200

300

x/0*

(b)
0.02

with FST (Tu= 0.1%, max / 0* *= 5.0)kz / 0** = 1.1
kz / 0* = 1.5
kz / 0 = 1.3
kz / 0 = 1.7

Figure 7. Top view for the short-wavelength condition of kz/δ0*= 1.5. Isosurfaces are same as Fig. 4.

FSC flow

0.015

Cf

The fluctuation component is determined as u’/U0 = u/U0 –ū/U0, where ū is the time-averaged component. Figure 8 shows
the contribution of each production term to the turbulent
transition. Under no-FST conditions, Pk2, Pk8, and Pk9 take adominant role in the production. Among these, Pk2 and Pk8 are
the production terms that provide u'w' and v'w' < 0. These
correspond to the finger vortex shown in Fig. 3. The highest
contribution of Pk9 is caused by the fluctuation induced by thefinger vortex and the spanwise velocity gradient. The growncross-flow vortex provides a minimum velocity gradient with aflow field. Malik et al. (1999) called this velocity gradient asmode z, which is one of the secondary instability modes. In
other words, Pk9 increases with the growth and saturation of
the cross-flow vortex. That is a characteristic trend of the
turbulent transition conditions caused by the finger vortex.
According to Vincentiis et al. (2022), for a swept-wingboundary layer, the production term of the spanwise velocity
gradient takes a dominant role in the turbulence transition. For
the two-dimensional flow (Loiseau et al., 2014), the dominant
production term was also owing to the spanwise velocity
gradient, which was created by the cylindrical roughness.
Figure 9 shows the contribution of the production term
under the FST condition. Comparing the no-FST condition (Fig.
8) and the short-wavelength FST condition (Fig. 9(a)), thecontributions of Pk2 and Pk8 are more significant in the short-

0.01

0.005
50

100

150

x/0*

Figure 6. Friction coefficient distribution. (a) shows longwavelength conditions and no freestream turbulence
conditions. (b) shows short-wavelength conditions.
differ depending on the wavelength of the FST. We investigate
the dependence of secondary vortex structures and transition
positions on the height of the cylinder roughness.
Figure 6 shows the friction coefficient distribution fordifferent cylinder roughness heights. In the no-FST condition,
the transition position shifts upstream as the cylinder roughness
height increases. The same is observed under conditions with
FST. With long-wavelength FST, the transition position shifts
upstream with increasing height are not remarkable in kz/δ0* ≥1.5 than in kz/δ0* < 1.5. In the high roughness height condition,
the influence of the roughness-induced disturbance dominates
the turbulent transition process. Figure 7 shows the flow field
for the roughness height kz/δ0* = 1.5 in the long-wavelength
condition. Hairpin vortices occur near the roughness. Note, thesecondary vortex is a finger vortex with kz/δ0* = 1.1 (see Figure4(a)). The secondary vortex inducing the turbulent transitionchanged to a hairpin vortex as the roughness height increased(see figure 7). The low roughness results (kz/δ0* = 1.1) showthat the switch of secondary vortex structure between hairpin
and finger vortices depends on the FST wavelength. However,
the roughness-height study clarified that both the cylindricalroughness height and the FST wavelength affect the switch of
secondary vortex structure. Although many factors affect thetransition process, they undergo the secondary instability.
Finally, we focus on the disturbance components occurringon the stationary cross-flow vortices and discuss them from theturbulent energy production. The production terms are thefollowing:
Pk1 = −u’2 ∂u/∂x, Pk2 = −u’w’ ∂u/∂z, Pk3 = −u’v’ ∂u/∂y,Pk4 = −u’w’ ∂w/∂x, Pk5 = −w’2 ∂w/∂z, Pk6 = −v’w’ ∂w/∂y,
Pk7 = −u’v’ ∂v/∂x, Pk8 = −v’w’ ∂v/∂z, Pk9 = −v’2 ∂v/∂y,
Pk = Pk1 + Pk2 + … + Pk9.

Figure 8. Contribution to the integral value of production terms
scaled by Pk of kz/δ0* = 1.1. The integral region is x∈[25, 400],y∈[0, 25.14], z∈[0, 10].

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(a)transition via the finger vortex, the fluctuation component is
low within x/δ0* < 150 (see figure 4(a)). Therefore, turbulent
production is also low. It is possible to classify the secondary
vortex structures by the turbulent production near the cylinder.
The classification by production term is the same as theobservations of the vortex structures in the flow field.

(b)

Figure 10. Roughness height and wavelength dependency ofintegral value of production term. The integral region is x∈[25, 150], y∈[0, 25.14], z∈[0, 10].

SUMMARY
We performed direct numerical simulations for turbulent
transitions in a swept-flat-plate boundary layer with isolated
cylinder roughness and the freestream turbulence (FST). We
investigated the similarities/differences between secondary
vortices via the high-frequency instability.
The stationary cross-flow vortex paths the growth,saturation, and breakdown. In the saturation term, highfrequency secondary instability induces the finger vortex on the
cross-flow vortex. The FST induces secondary instability
earlier than the no-FST condition. In short-wavelength FST
conditions, high-frequency secondary instabilities occurred
immediately after the roughness. It provided hairpin vortices.
In the long-wavelength FST condition, the finger vortexes
occurred earlier than in the no-FST condition. FST skipped thesaturation term and produced secondary instability. In the
without roughness condition, FST induces a traveling crossflow vortex. The traveling cross-flow vortex is also breakdown
via high-frequency secondary instability. As a unified result ofdifferent conditions, the turbulent transition process follows apath of velocity shear layer formation. At the late stage of thetransition process, the flow field becomes breakdown owing tosecondary instability. The increasing roughness height changed
only the secondary vortex structure. An analysis of the
production terms clearly shows that the dominant production
term differs depending on the secondary vortex structure. The
production term with spanwise velocity gradient changes
significantly concerning the growth of the cross-flow vortex.
These differences depend on the presence/absence of asaturation term of the cross-flow vortex. However, theproduction terms related to the secondary vortex are similar.

Figure 9. Contribution to the integral value of production terms
scaled by Pk of kz/δ0* = 1.1. (a) is short-wavelength condition.
The integral region is x∈[25, 150], y∈[0, 25.14], z∈[0, 10].
(b) is long-wavelength condition. The integral region is x∈[25,300], y∈[0, 25.14], z∈[0, 10].
wavelength condition. The sweep/ejection by the hairpin vortex
induces Pk2 and Pk8. The short-wavelength FST provides astationary vortex with a secondary vortex earlier. Near thecylinder roughness, the spanwise velocity gradient is low. The
Pk9 is also lower than the grown cross-flow vortex. With thelong-wavelength FST (Fig. 9(b)), the contributions of Pk2, Pk8,
and Pk9 are comparable. In this case, the finger vortex induces
turbulent transition. It is the same as no-FST conditions.
However, compared to the no-FST condition, the transition
occurs upstream (see Fig. 2). The contribution of Pk9 is lower
owing to the cross-flow vortex growth. The contributions of
Pk2 and Pk8 are common to all conditions, irrespective of the
FST. The generation of turbulence energy by Pk2 and Pk8 are acharacteristic feature for the system in the cross-flow instability
dominant. However, the contribution rate of Pk differs under
different conditions. It is due to the growth of the cross-flow
vortex.
Figure 10 shows the dependence of the integral value of the
production term (Pk) on roughness height and FST wavelength.
The black line in Fig.10 is based on the results of production
terms and the secondary vortex structures of the flow field. The
clear threshold between secondary vortices is not yet evident.
In the chordwise region x/δ0* < 150, the hairpin vortex provides
high value of Pk. However, under the conditions of turbulent

ACKNOWLEDGEMENT
This work was supported by JST SPRING, Grant Number
JPMJSP2151. We appreciate Professor D. Watanabe at
Toyama University for his valuable advice to simulate
freestream turbulence. Some of the present numericalsimulations utilized supercomputer resources at the512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Cybermedia Center, Osaka University and the Cybermedia
Center, Tohoku University.
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