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

INTERACTION BETWEEN WAVEPACKETS AND STREAKS IN
TURBULENT JETSPetrônio A. S. Nogueira
Aerospace Engineering
Monash University
3800, Clayton, Australia
petronio.nogueira@monash.edu

André V. G. Cavalieri
Engenharia Aeronáutica
Instituto Tecnológico de Aeronáutica
12228-900, São José
dos Campos, Brazil
andre@ita.br

Oliver Schmidt
Mechanical and Aerospace Engineering
University of California
CA 92093, La Jolla, USA
oschmidt@ucsd.edu

Vincent Jaunet
Fluides, Thermique, Combustion
Institut PPrime
86962, Poitiers, France
vincent.jaunet@ensma.fr

Peter Jordan
Daniel Edgington-Mitchell
Fluides, Thermique, Combustion
Aerospace Engineering
Institut PPrime
Monash University
86962, Poitiers, France
3800, Clayton, Australia
peter.jordan@univ-poitiers.fr
daniel.mitchell@monash.edu

ABSTRACT

no sound, as they are virtually invariant in the streamwise
direction. However, streaks may impact sound radiation indirectly by means of their effect on the development of theKH wavepacket, i.e., via non-linear interactions between these
two structures. A study of such interaction in a reduced-ordermodel was performed by Nogueira & Cavalieri (2021), whoused a confined shear-layer model to evaluate the temporal dynamics involving streaks and wavepackets. They showed thatthese structures interact via oblique waves, and that streamwise elongated rolls are also an important part of the dynamics. But these results are yet to be extended for real turbulent
jets.
This work aims to overcome this limitation. To achievethis, we propose a model to evaluate the interaction between these structures using PSE linearised around a streakcontaining mean flow, similar to the formulation proposed by
Sinha et al. (2016). The model allows us to track the spatial evolution of the originally axisymmetric wavepacket in
the presence of streaks and to evaluate how this interaction
may change its noise-producing characteristics. The paper isorganised as follows: first, a formulation for the Parabolised
Stability Equations for jets with streaks is presented. This isfollowed by a description of the methods for obtaining streaks,
which are based on experimental results and resolvent analysis. Finally, a parametric sweep of streak amplitude and azimuthal wavenumber is performed, where some trends relevant
for sound generation are highlighted. The paper is closed witha summary of the main conclusions.

We propose a formulation to study the effect of streaks inthe spatial development of wavepackets. To this end, a modified version of the parabolised stability equations (PSE) linearized around a streak-containing mean flow is used, which
considers a series of azimuthal wavenumbers in the solution.
In the present case, streaks are obtained from experiments using spectral proper orthogonal decomposition applied to particle image velocimetry data, and extrapolated in the radial direction using a Gaussian fit. Streaks and rolls predicted byresolvent analysis are also used in the analysis to evaluate theeffect of streamwise vortices in the development of the noisegenerating structures. Results show that streaks non-triviallymodify the spatial support of the Kelvin-Helmholtz wavepackets and their phase velocity, which may lead to changes in thesound generation efficiency of the jet. New structures acrossthe shear layer induced by the presence of streaks are also observed further downstream for high streak amplitudes.

Introduction
Recent works have identified the presence of three structures in turbulent jets: Kelvin-Helmholtz (KH) wavepackets(Cavalieri et al., 2013), Orr structures (Tissot et al., 2017) andstreaks (Nogueira et al., 2019a). As summarised by Pickering et al. (2020), these structures are a result of distinct
flow mechanisms and are dominant in different regions of
the frequency-azimuthal wavenumber spectrum. The KelvinHelmholtz mode is generated by a linear stability mechanism caused by the presence of an inflection point in the jetbase flow. These structures usually follow a spatial patternof growth-saturation-decay, and are present at mid-high frequencies and most azimuthal wavenumbers. These structures
are also responsible for most of the sound radiated at low polar angles in subsonic jets (Cavalieri et al., 2012). Streaks,on the other hand, are a result of the lift-up effect (Ellingsen& Palm, 1975) and are only active at very low frequencies(Strouhal number Stm → 0) and non-zero azimuthal wavenumber. Despite their high amplitude, these structures generate

Parabolized Stability Equations
This study is based on the parabolised stability equations(PSE) (Herbert, 1997; Gudmundsson & Colonius, 2011). The
formulation considers solutions in the frequency domain ω asq(x, r, θ , ω) = q̂(x, r, θ , ω)ei1

Rxα(x1 )dx1(1)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
where both q̂ and α are slowly varying functions in the streamwise direction x, and q contains all flow quantities (specificvolume, pressure and the three velocity components). Insteadof assuming normal modes in θ , as is often the case in round
jets, we consider a base flow decomposed in a series of modes
in azimuth; thus, the streamwise velocity component of thebase flow (and equivalently for all other components) can be
written as

studied. Both resolvent analysis (Pickering et al., 2020) andspectral proper orthogonal decomposition (SPOD) (Nogueira
et al., 2019a) may be used to obtain these structures in turbulent jets. Both approaches are used here. SPOD modes wereobtained at vanishing frequency Stm → 0 from the streamwise
velocity PIV data presented in Nogueira et al. (2019a) for a
Mach M = 0.4 jet. Since these energetic structures were obtained in a limited domain, they have been extrapolated in theradial direction by means of fitting their radial structure witha Gaussian function at each streamwise step, which led to agood representation of the original data. Sample SPOD modes
for ms = 2, 4 and 6 are shown in figure 1. As also shown in
Nogueira et al. (2019a), streaky structures decrease in length
in the streamwise direction as ms is increased, a consequenceof their self-similar behaviour. For that reason, their peak also
approaches the nozzle for higher ms ; in these cases, as most of
the streak is located where the shear layer is thinner, they will
potentially have a stronger effect on the growth of the KH.
Resolvent analysis is based on an expansion of the NS
equations around a given base flow, which may be written in
an input-output form asi
As h
Ums (x, r)eims θ + U−ms (x, r)e−ims θ2(2)
where ms is the streak azimuthal wavenumber, Ums (x, r) defines the shape of the streak in space, and As is its amplitude.
This amplitude represents the maximum of the streak in (x, r)
as a percentage of the maximum jet Mach number. The proposed base flow includes the zeroth component (which is simply the azimuthally and temporally averaged mean flow fromexperiments) and the zero-frequency component of the flowat different azimuthal wavenumbers, which can be viewed as
its streaky component (Nogueira et al., 2019a). This base flow
will be used to study the effect of streaks and rolls in the development of the KH wavepacket. Using this base flow shape, theflow disturbances can be written as a sum of several azimuthal
wavenumbers (Sinha et al., 2016) as

U(x, r, θ ) = U0 (x, r)+−iωq′ = Lq′ + f′

where all variables q′ (x, r, ms , ω) are considered in thefrequency-azimuthal wavenumber domain, and the term f′
gathers all non-linear terms of the equations. Rearranging theterms of equation 5, one can writeNsq̂(x, r, θ , ω) =∑ qn (x, r, ω)ei(m+nm )θ
s

(5)

(3)

n=−Ns

q′ = (−iωI − L)−1 f′ = Rf′

where m is the central wavenumber of the disturbances. For
an originally axisymmetric wavepacket, this shape of solution
becomes a sum of the m = 0 mode and the respective streak
modulation components nms , which are truncated to a numberof harmonics Ns . Inserting equations 1, 2 and 3 into the linearized Navier-Stokes equations (NS), and gathering all termsof different powers of m + nms , we can write the equations inoperator form as−iω q̃ + Lt q̃ + Bt

∂ q̃
+ iαBt q̃ + imCt q̃ − m2 Dt q̃ = 0∂x(6)

A singular value decomposition of the resolvent operator
R leads to optimal forcing and response modes of the flow fora given frequency and azimuthal wavenumber, which are directly comparable to SPOD modes if the statistics of the forcing are white in space (Towne et al., 2018). Here, resolvent
modes were computed for St = 0 and ms = 6 as in Nogueira
et al. (2019b) using the same mean flow and reduced domain.
The mode shapes obtained from this method are given in figure 2, where the absolute value of all three velocity components are shown. The resulting structure looks like streaks (inthe streamwise velocity component) and streamwise vortices(or rolls, in the radial and azimuthal velocity components), inagreement with the lift-up effect (not shown here because of
space limitations). Comparing figures 1(c) and 2(a), it is clearthat resolvent modes are more extended in the streamwise direction, not reaching a peak within the domain studied here.
This is due to the fact that no eddy-viscosity model (or coloredforcing) was included in these computations, which leads toa stronger misalignment between resolvent and SPOD modes(Pickering et al., 2020, 2021). The differences in shapes ofthe streaks may also change the overall characteristics of themean flow given in equation 2, leading to a locally thinner
shear layer for the resolvent-streaks. These differences may
have an effect on the development of the KH mode.
With the shapes of the streaks defined by these methods,
their amplitudes As will be treated as a free parameter in thiswork. Modes resulting from resolvent analysis have fixed relative amplitudes between streaks and rolls, which removes thedependence on the rolls amplitudes in the present analysis.
Further details about the experiments and simulations can befound in Nogueira et al. (2019a,b); Schmidt et al. (2018).

(4)

where the vector q̃ = [q̂−N ... q̂0 ... q̂N ]T comprises all themodulation components. Following the PSE method, the second streamwise derivatives are neglected in the equations, andthe solution is marched from a position close to the nozzle
until the desired station downstream by discretizing the firststreamwise derivatives using first-order implicit Euler differences. The streamwise wavenumber is updated at each stepusing the normalisation condition outlined in Herbert (1994),
which minimizes the streamwise variation of the shape function. This formulation is similar to that developed by Sinha
et al. (2016), with the difference that only a single azimuthalmode (and its complex conjugate) is considered in the mean
flow. This considerably simplifies the analysis and also allows
us to evaluate the effect of isolated streaks in wavepackets.

Streaks in turbulent jets
The present formulation requires a mean flow and theshapes of the streaks for each azimuthal wavenumber to be2

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

from the PSE) is shown in figure 5. In contrast with the stabilisation effect seen in figure 4, this quantity is not strongly affected by the presence of streaks in the analysis. Small changes
in phase velocity are observed further downstream, with increasing reduction of c ph with increasing As and ms . Together,results from figures 4 and 5 suggest that streaks with moderatehigh amplitude and azimuthal wavenumber (within the range
studied herein) may be acoustically beneficial, reducing theamplitude of the noise-generating structures and possibly making them less acoustically efficient (Jordan & Colonius, 2013;
Cavalieri et al., 2019).

PSE results of streak-modulated KH modes were generated for Ns = 3, Reynolds number Re = 30000, and St = 0.4;
analysis of other frequencies led to similar results. We startwith a study of the modulation induced by streaks obtained
from SPOD. As mentioned earlier, SPOD was applied to thestreamwise velocity component, an thus the influence of the radial and azimuthal velocities (or the rolls) was neglected. Dueto the absence of data at the nozzle plane, the amplitude of thestreaks at that position was also assumed to be negligible, so
all modes start as axisymmetric in the spatial march.
Figure 3 shows the shapes of streak-modulated KH modes
for As = 6% and ms = 2 ,4 and 6, which illustrates the overall
effect of the streaks in the wavepacket shapes. The most clear
feature of these wavepackets is that, despite being initially axisymmetric, they have strong azimuthal non-uniformities associated with the streak azimuthal wavenumber. These structures
represent axisymmetric wavepackets that grow together withoblique waves, as in Nogueira & Cavalieri (2021). For m = 2,
the base flow that results from the superposition of streaks andthe axisymmetric mean flow has an almost elliptical shape; forthis case, the resulting wavepacket grows more strongly in the
direction perpendicular to the major axis of the ellipse, similar
to what is observed in elliptical jets (Edgington-Mitchell et al.,2015). For the other azimuthal wavenumbers (ms = 4 and 6),
chevron-like structures are observed to grow close to the jet
shear layer, which are followed by the appearance of tilted
structures further downstream. Qualitatively, these structuresare similar to the mushroom-like structures observed in previous locally parallel models (Lajús et al., 2019; Wang et al.,2021) and experiments/simulations (Alkislar et al., 2007; Violato & Scarano, 2011).
Here, we are interested in the changes induced by thepresence of steady streaks in the development of the KH; more
specifically, changes in the growth and phase velocity of the
KH are of particular interest, as these are expected to impact
the acoustic far-field. In order to evaluate how the streaks affect the growth of the wavepackets, an attenuation parameteris defined asξ (x) =uAs (x, r = 0)
uAs =0 (x, r = 0)

The same quantities were computed for resolvent-streaks,
which are shown in figure 6. If the rolls are neglected (figures 6(a,c)), results for moderate streak amplitude are qualitatively similar to the ones from SPOD-streaks, with a stabilisation region observed further downstream, which is also associated with a reduction of the phase velocity of the KH mode.
However, these trends are strongly amplified in this case, with
substantially higher reductions in both ξ and c ph . A small region of destabilisation is also observed around 1.5 < x/D < 4,
which may be associated with the fact that streaks start with
non-zero amplitudes in this case. Also, as shown in figure
7, the radial support of resolvent-streaks differs from SPODstreaks, with structures more concentrated across the shear
layer. These differences in spatial support leads to changes
in the mean flow that could be the source of the disparity
between figures 4, 5 and 6. This is exemplified in figure 7:
while SPOD-streaks induce smoother radial variation in themean flow, resolvent-streaks create stronger gradients acrossthe shear layer, which will have an effect on the stability of theflow.
The inclusion of rolls is evaluated in figures 6(b,d). Overall, the rolls seem to intensify the effect of the streaks on theattenuation for all x/D: very close to the nozzle, a slight stabilising effect is observed with increasing As , followed by astrong destabilisation close to the end of the potential core.
Similar to the previous cases, streaks stabilise the KH modeand reduce its phase velocity further downstream if they havemoderate amplitude. However, an increase in phase velocity
close to the nozzle is observed for the simulations includingrolls. The destabilisation effect close to the end of the potential core associated with an increase of the phase velocity
close to the nozzle observed for high roll amplitude may act tocounterbalance the acoustically beneficial trends observed for
SPOD-streaks.

(7)

which defines the amplitude ratio between the axisymmetricwavepacket and the streak-modulated ones at the centerline.
Figure 4 shows the attenuation induced by SPOD-streaks
for increasing As and ms = 2, 4 and 6. Since all SPOD-streaks
start with zero amplitude, the value of ξ is close to unity forthe first few diameters, indicating that these structures possibly
have too low amplitude to affect the KH mode in that region.
As the streaks grow spatially, they start to have a stabilising
effect on the KH. This effect increases monotonically with As
for all values of ms . It can also be observed that the start ofthe stabilisation region is usually associated with the regionin space where the streaks have higher amplitudes; thus, anincrease in ms could lead to an earlier stabilisation of the KH
mode, which is in line with the locally-parallel results of Lajús
et al. (2019). In fact, the present results suggest that higher ms
streaks may be more efficient in stabilising the KH mode, asa greater reduction of ξ is obtained for high streak azimuthal
wavenumber, keeping As constant.
The influence of the streaks in the phase velocity of theKH (defined as c ph = ω/α(x), with α(x) obtained directly

Non-trivial changes in the wavepacket shape are observed
for resolvent-streaks of higher amplitude, which can be induced by the strong dips in the ξ plot in figures 6(a,b). To
illustrate that, the shapes of the equivalent axisymmetric mode
(obtained by azimuthally averaging the resulting modes fromPSE) are shown in figures 8 (As = 6%) and 9 (As = 16%) for
SPOD-streaks (a), resolvent-streaks (b), and resolvent-streaks
and rolls (c). For low amplitude streaks without rolls (figure8(a,b)), the shapes of the KH mode are qualitatively similar toa non-modulated wavepacket (see, for instance Lesshafft et al.
(2019)), and the stabilisation can only be perceived in comparison with the axisymmetric PSE. If rolls are included, twofeatures become apparent: first, the wavepacket peak shifts
slightly upstream, which is in line with the stabilisation trend
observed in figure 6(b); at the same time, tilted structures areamplified across the shear layer, which are observed mostly
downstream. For the highest streak amplitude (figure 9), thesame tilted structure is observed, suggesting that the shapes of
the resolvent-streaks and the presence of the rolls triggers thedevelopment of these structures for lower As . In figure 9(b,c),3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Cavalieri, A. V. G., Rodrı́guez, D., Jordan, P., Colonius, T. &Gervais, Y. 2013 Wavepackets in the velocity field of turbulent jets. Journal of Fluid Mechanics 730, 559–592.
Edgington-Mitchell, D., Honnery, D. R. & Soria, J. 2015
Staging behaviour in screeching elliptical jets. International
Journal of Aeroacoustics 14 (7), 1005–1024.
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. The
Physics of Fluids 18 (4), 487–488.
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. Journalof Fluid Mechanics 689, 97–128.
Herbert, T. 1994 Parabolized stability equations. Special
Course on Progress in Transition Modelling (AGARD) 793,
1–34.
Herbert, T. 1997 Parabolized stability equations. Annual Review of Fluid Mechanics 29 (1), 245–283.
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet
noise. Annual Review of Fluid Mechanics 45 (1).
Lajús, F. C., Sinha, A., Cavalieri, A. V. G., Deschamps, C. J.
& Colonius, T. 2019 Spatial stability analysis of subsoniccorrugated jets. Journal of Fluid Mechanics 876, 766–791.
Lesshafft, L., Semeraro, O., Jaunet, V., Cavalieri, A. V. G. &Jordan, P. 2019 Resolvent-based modeling of coherent wave
packets in a turbulent jet. Phys. Rev. Fluids 4, 063901.
Nogueira, P. A. S. & Cavalieri, A. V. G. 2021 Dynamics ofshear-layer coherent structures in a forced wall-bounded
flow. Journal of Fluid Mechanics 907, A32.
Nogueira, P. A. S., Cavalieri, A. V. G., Jordan, P. & Jaunet, V.
2019a Large-scale streaky structures in turbulent jets. Journal of Fluid Mechanics 873, 211–237.
Nogueira, P.A. S., Cavalieri, A. V. G., Schmidt, O. T., Jordan,P., Jaunet, V., Pickering, E., Rigas, G. & Colonius, T. 2019b
Resolvent-based analysis of streaks in turbulent jets. In 25th
AIAA/CEAS Aeroacoustics Conference, p. 2569.
Pickering, E., Rigas, G., Nogueira, P. A. S., Cavalieri,
A. V. G., Schmidt, O. T. & Colonius, T. 2020 Lift-up,
Kelvin–Helmholtz and Orr mechanisms in turbulent jets.
Journal of Fluid Mechanics 896, A2.
Pickering, E., Rigas, G., Schmidt, O. T., Sipp, D. & Colonius,
T. 2021 Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets. Journal of Fluid
Mechanics 917, A29.
Schmidt, O. T., Towne, A., Rigas, G., Colonius, T. & Brès,
G. A. 2018 Spectral analysis of jet turbulence. Journal of
Fluid Mechanics 855, 953–982.
Sinha, A., Gudmundsson, K., Xia, H. & Colonius, T. 2016
Parabolized stability analysis of jets from serrated nozzles.
Journal of Fluid Mechanics 789, 36–63.
Tissot, G., Lajús Jr, F. C., Cavalieri, A. V. G. & Jordan, P. 2017
Wave packets and orr mechanism in turbulent jets. PhysicalReview Fluids 2 (9), 093901.
Towne, A., Rigas, G. & Colonius, T. 2019 A critical assessment of the parabolized stability equations. Theoretical andComputational Fluid Dynamics 33 (3), 359–382.
Towne, A., Schmidt, O. T. & Colonius, T. 2018 Spectral
proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. Journalof Fluid Mechanics 847, 821–867.
Violato, D. & Scarano, F. 2011 Three-dimensional evolutionof flow structures in transitional circular and chevron jets.
Physics of Fluids 23 (12), 124104–124104.
Wang, C., Lesshafft, L., Cavalieri, A. V. G. & Jordan, P. 2021
The effect of streaks on the instability of jets. Journal of
Fluid Mechanics 910.

it is shown that high amplitude resolvent-streaks strongly amplify the growth of these structures, which become dominant
over the axisymmetric wavepacket. The appearance of thesetilted modes may be associated with the mode-shifting phenomenon observed in the locally parallel framework (Lajús
et al., 2019), which may lead to a change in the mode being
marched by the PSE. They may also be correlated to the appearance of oblique waves by the presence of a streaky baseflow, as shown in Nogueira & Cavalieri (2021). The effect ofthese structures in the acoustic field is still unknown, and such
evaluation will be left as a future endeavour.

Conclusions
A PSE formulation is used to evaluate the changes in the
wavepacket hydrodynamic characteristics induced by the presence of streaks. The latter are obtained either from SPOD applied to PIV data, or from resolvent analysis, which allows us
to consider the influence of rolls in the development of the KH
mode. It is shown that, for SPOD-streaks (which are more radially spread across the shear layer), these structures have astabilising effect on the KH mode; increasing streak amplitude
As and azimuthal wavenumber ms further enhances that trend,
which may lead to a mitigation of the sound generated by theaxisymmetric wavepacket. It is also shown that the phase velocity of the resulting mode is slightly reduced further downstream due to the presence of these elongated structures. A
similar trend is observed for resolvent-streaks of low/moderate
amplitudes. However, since these structures are more radiallycompact, their presence induces stronger local shear, whichfurther enhances the signature of tilted structures in the velocity field. The overall effect of these streaks involves thedevelopment of regions of stabilisation and destabilisation depending on As .
It is important to highlight that, even though the present
formulation may be able to provide a first approximation of theeffect of streaks and rolls in the wavepacket characteristics, italso has some limitations. The first is that this formulation only
considers a one-way coupling between streaks and wavepackets, which means that streaks affect the KH mode, but the development of the KH does not affect the streaks. This assumption of “frozen” mean flow may be too stringent, as suggested
by Nogueira & Cavalieri (2021), and can only capture part ofthe dynamics. Secondly, it is well known that PSE struggles
to capture non-modal behaviour (Towne et al., 2019). Considering that high azimuthal wavenumber KH modes are active
closer to the nozzle, and the mode-shift observed downstream,
it is possible that PSE fails to closely capture the phenomenon
at higher x/D. A possible workaround is to solve the One-Way
Navier Stokes equations for the same problem, which mitigates the limitations of PSE. This will be performed in the next
steps of this project, which will also include a comprehensive
experimental campaign to confirm the trends observed herein.

REFERENCES
Alkislar, M. B., Krothapalli, A. & Butler, G. W. 2007 The effect of streamwise vortices on the aeroacoustics of a mach
0.9 jet. Journal of Fluid Mechanics 578, 139–169.
Cavalieri, A. V. G., Jordan, P., Colonius, T. & Gervais, Y. 2012
Axisymmetric superdirectivity in subsonic jets. Journal of
Fluid Mechanics 704, 388.
Cavalieri, A. V. G., Jordan, P. & Lesshafft, L. 2019 WavePacket Models for Jet Dynamics and Sound Radiation. Applied Mechanics Reviews 71 (2), 020802.

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 1.

Figure 2.

Absolute value of streamwise velocity fluctuations obtained from SPOD data for ms = 2 (a), 4 (b) and 6 (c).

Absolute value of streamwise (a), radial (b) and azimuthal (c) velocity fluctuations from resolvent analysis for ms = 6.

Figure 3. Isocontours of 30% of the maximum streamwise velocity from PSE (real part) for St = 0.4 and ms = 2 (a), 4 (b) and 6 (c).
Streaks obtained from SPOD with amplitude chosen as As = 6%.
1.1
1
0.9
0.8
0.7
0.6
0

2

4

6

0

2

4

6

0

2

4

6

Figure 4. Attenuation of the KH mode at the centreline for increasing streak amplitude. Streaks obtained from SPOD with amplitudes
varying in the interval As = [0, 16%] for ms = 2 (a), 4 (b) and 6 (c) and St = 0.4. Dashed lines indicate the As = 0 case.
0.5

0.4

0.3

0.2
0

2

4

6

0

2

4

6

0

2

4

6

Figure 5. Phase velocity of the KH mode as function of the streamwise coordinate for increasing streak amplitude. Streaks obtained
from SPOD with amplitudes varying in the interval As = [0, 16%] for ms = 2 (a), 4 (b) and 6 (c) and St = 0.4. Dashed lines indicate the
As = 0 case.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
1.5
1
0.5
0

0.4
0.3
0.2
0.1
0

2

4

6

8

0

2

4

6

8

Figure 6. Attenuation (a,b) and phase velocity (c,d) of the KH mode as function of the streamwise coordinate for increasing streak
amplitude without (a,c) and with (b,d) rolls. Streaks and rolls obtained from resolvent analysis with amplitudes varying in the interval
As = [0, 16%] for ms = 6 and St = 0.4. Dashed lines indicate the As = 0 case.

Figure 7.

Mean streamwise velocity build using SPOD (a) and resolvent (b) streaks at x/D = 6, ms = 6 and As = 16%.

Figure 8. Real part of streamwise velocity fluctuations obtained from PSE for St = 0.4, ms = 6 and As = 6%. Streaks obtained fromSPOD (a) and resolvent without (b) and with (c) rolls.

Figure 9. Real part of streamwise velocity fluctuations obtained from PSE for St = 0.4, ms = 6 and As = 16%. Streaks obtained fromSPOD (a) and resolvent without (b) and with (c) rolls.

6