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

LINEAR AND NON-LINEAR DYNAMICS OF STREAK BREAKDOWN IN A
HYPERSONIC BOUNDARY LAYER

Clément Caillaud
Guillaume Lehnasch
Peter Jordan
DFTC-2AT
DFTC-2AT
DFTC-2AT
Institut Pprime / CEA-CESTA
Institut Pprime - ENSMA
Institut Pprime - CNRS
Chasseneuil du Poitou, France
Chasseneuil du Poitou, France
Chasseneuil du Poitou, France
clement.caillaud@isae-ensma.fr guillaume.lehnasch@isae-ensma.fr peter.jordan@univ-poitiers.fr

Eduardo Martini
DFTC-2AT
Institut Pprime / CEA-CESTA
Chasseneuil du Poitou, France
eduardo.martini@univ-poitiers.fr

Eric Goncalves
Ludovic Hallo
DFTC-2AT
CESTA
Institut Pprime / ENSMA
CEA-DAM
Chasseneuil du Poitou, France
Le Barp, France
eric.goncalves@ensma.fr
ludovic.hallo@cea.fr

ABSTRACTthat arise due to the lift-up mechanism.
While the evolution and breakdown of the hypersonicboundary layer for selected modal instabilities have been well
documented (Franko & Lele, 2013), the path to transition via
non-modal mechanisms lacks a clear understanding at high velocities. Recent studies by Paredes et al. (2017b, 2019) haveprovided a description of the linear growth of selected modes
supported by the streaky boundary layer using PSE analysis. These studies demonstrated the co-existence of modes
destabilised by the streaks and the usual second mode (Mackmodes) and presented an overview of the linear amplificationof selected modes for increasing streak amplitudes. Specifically, at 𝑀∞ = 5.3 on a sharp cone, the second Mack-modes
family was found to dominate for streak amplitudes up to 40%
of 𝑢 ∞ . At hypersonic speed, the aforementioned PSE study
also demonstrates stabilising effect of streaks on the second
Mack mode growth for streak amplitudes around 10% of 𝑢 ∞ .
However, investigation of the different modes receptivities andnon-linear effects has not been addressed. Therefore, to complete these previous studies, the current work aims at providing
insight into the receptivity, growth and non-linear dynamics of
instabilities on a streaky base flow.
This study is organised as follows. After having defined
the specifics of the computational framework, the base flows
and forcing are described. Then the obtained disturbance fields
from the DNS and LDNS are discussed. We first present
an analysis of the linearly dominant modes at the end of thedomain. Then, we decompose the disturbance fields using
Floquet-SPOD and discuss the unstable modes. Finally, we
track the dominant modes within the DNS domain and discuss
the dependence of the transition scenario on the streak amplitude.

Hypersonic vehicles often present curved geometries andrough surfaces, directly affecting the boundary layer topology.
An important feature of the boundary layer in such conditions
is a system of streaks. These streaks induce a flow distortionand lead to a new set of unstable modes. We study the effect of this boundary layer modification using Direct Numerical Simulation (DNS) of a flow at 𝑀∞ = 6.0 over a flat plate.
Optimal disturbances of increasing amplitudes are used to design a set of three stationary base flows containing growing
streaks. We then perform DNS and linearised DNS of thesebase flows subjected to white-noise forcing. The forcing leads
to the emergence of an ensemble of modes, which we identify and track with a flow decomposition using SPOD and Floquet theory. The transition scenarios for the streaky flows arestudied. Specifically, we observe the early dominance of lowfrequency modes in contrast with the Blasius boundary layer
and previous PSE studies where the second Mack mode wasgiven as dominant. Finally for this specific case, we highlight the importance of taking into account broadband forcing
to study the streaky baseflows in order to appropriately trackthe evolution of relevant unstable modes.

Introduction
The geometry of hypersonic vehicles often resembles aslender geometry starting with a blunt forebody to reduce thestagnation temperature at the nose-tip. It has been demonstrated by Stetson (1983) that the transition dynamics can bestrongly related to this bluntness (curvature radius) and thatfor large bluntness, transient growth of streaks is an important
ingredient in the transition process (Paredes et al., 2017a). Additionally, to alleviate the substantial heat fluxes, the thermal
protection systems of such vehicles are designed to be ablative,
progressively inducing the appearance of distributed roughness on the walls. This significantly increases the complexityof the transition process through modal growth and mechanisms issued from the deformed base flow associated with theroughness (Schneider, 2008). This may result in steady streaks

Simulation framework
A Direct Numerical Simulation (DNS) code:
CurviCREAMS has been developed and is used to analyse thelinear and non-linear stages of the hypersonic boundary-layertransition. The code solves the compressible Navier-Stokes
equations for a mixture of thermally perfect gases in gener-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
0.0010alized coordinates. The governing equations are computed
in a finite-difference framework using 8th-order schemes
and an adaptive shock capturing methodology consistent
with the curvilinear framework. Additionally, to compare
the linear and non-linear dynamics of disturbances evolving
in the boundary layer, a discrete linearization procedure
is implemented to perform Linearised DNS (LDNS). The
construction of the LDNS is done in the spirit of Fosas dePando et al. (2012) and is described hereafter. Consider thediscrete non-linear operator of the Navier-Stokes equations:
N (𝑞𝑞 ), for a set of initial and boundary conditions. The
dynamical evolution of any state vector 𝑞 (𝑥, 𝑡) reads:𝜕𝑞𝑞= N (𝑞𝑞 ).
𝜕𝑡Gmax (β)/Reδ∗ 20.00080.0000
0.00

0.20

0.25

0.30

0.35

This way, optimal vortices are found to have a spanwise
wavenumber 𝛽𝑆 = 0.31 similar to Andersson et al. (2001);
Paredes et al. (2019) and the optimal disturbance gain evolution with 𝛽 = 2𝜋/𝜆 is given in Fig. 1. This spanwise wavelength serves as a parameter to define the spanwise extension of the DNS domain. We compute four streak periods(𝑧 ∈ [0, 4𝜆 𝑆 ]) to account for the subharmonic secondary instability (Andersson et al., 2001) and eventual instabilities of
longer wavelength.

(2)

(3)

The linear optimal forcing is non-linearly propagateddownstream with the DNS. We choose to study three cases of
streak amplitudes 𝐴𝑠𝑢 (𝑥) = 12 max 𝑦,𝑧 𝑢 0 (𝑥) − min 𝑥,𝑦 𝑢 0 (𝑥) ,named A1, A2 & A3. These are respectively constructed by increasing the initial optimal disturbance amplitude 𝐴0 . The associated streak amplitude evolution along the domain is given
in Fig. 2a. These cases span the streak amplitudes studied by
Paredes et al. (2019), with further emphasis on high-amplitude
streaks (A3). We show the final shapes of the steady streaks 𝑞¯modulating the base flow in Fig. 2b.

Setup and base flows

As we aim to understand which mode dominate, without
selecting specific modes a priori, we introduce a white noise
made of pressure disturbances in a volume centred at 𝑥 = 5.0.
We generate this noise following the procedure developed byHader & Fasel (2019), slightly adapted to our case. We chosethe forcing amplitude at 𝐴 𝑓 = 5 × 10−3 to trigger transition in
the most sensitive case of the A3 streaks. Using this forcing,
we perform, for each streak amplitude, DNS and LDNS simulations. The DNS allows us to get insights in the transition
scenario, whereas the LDNS supports the study by capturing
the linearly most amplified modes from the forcing.

We consider the spatial development of a boundary layerover an adiabatic flat plate at 𝑀∞ = 6.0, 𝑅𝑒 𝛿 ∗ = 28000 and𝑇∞ = 273.15𝐾. This boundary layer is developed from selfsimilar profiles injected at the inlet, from which we take thedisplacement thickness 𝛿∗ as the reference length. The domain
is of size 𝐿 𝑥 × 𝐿 𝑦 × 𝐿 𝑧 = 350 × 15 × 4𝜆 𝑆 , with 𝜆 𝑆 the streaks
spanwise wavelength. Accounting for the sponge zones, weuse a grid of 𝑁 𝑥 × 𝑁 𝑦 × 𝑁 𝑧 = 4130 × 335 × 300. On this baseflow, the streaks are generated by a volume of stationary forcing close to the inlet and centred at 𝑥 𝑓 = 5.0. This stationary
forcing consists of optimal vortical disturbances obtained witha 1D locally parallel, spatial transient growth analysis for abase flow profile at 𝑥 = 5.0. The spatial transient growth analysis is carried out at a frequency 𝑓 = 0.0 for a range of increasing
spanwise wavenumbers to find the most amplified disturbanceat the end of the domain. We ensure that upstream-travellingmodes are filtered out from the spectrum before computing the
spatial transient growth (Jordan et al., 2017). To obtain optimal disturbances, we seek to maximize the gain 𝐺 (𝑥 = 350)
between an initial disturbance 𝑞 0 and a final response 𝑞 𝑓 , defined as follows, with the norm || • || 𝐸 taken as the compressible energy norm (Hanifi et al., 1996).

||𝑞 0 || 𝐸

0.15

Figure 1: Rescaled transient growth gain 𝐺 as a functionof the spanwise wavenumber

duced in Eq. 2 can be used either to extract the jacobian matrix
L for a global stability analysis or to compute the linear time
evolution of a chosen disturbance around 𝑞 0 . The latter approach is employed here. Therefore, the linear problem is timemarched using the same schemes and boundary conditions as
the non-linear solver for the same computational cost, allowing rigorous comparisons between LDNS and DNS results.

||𝑞 𝑓 || 𝐸

0.10

(1)

Then, depending on its form, the disturbance vector 𝑞 0 intro-

𝐺=

0.05

β

with 𝜀 a small value. This yields the linear dynamics of a
disturbance vector 𝑞 0 around 𝑞 0 :𝜕𝑞𝑞 0= L (𝑞𝑞 0 )𝑞𝑞 0 .
𝜕𝑡

0.00040.0002

Considering that the DNS code can compute N (𝑞𝑞 ) for any
given 𝑞 , the tangent linear operator L at a fixed point 𝑞 0 (𝑥)can then be extracted from the non-linear code by a first-orderapproximation,
N (𝑞𝑞 0 + 𝜀𝑞𝑞 0 ) − N (𝑞𝑞 0 )
L (𝑞𝑞 0 )𝑞𝑞 0 =,𝜀0.0006For each simulation, we perform a time sampling of thesolution to build a discrete state vector 𝑞 . We decompose thisvector in frequency-wavenumber space to clarify the transition process. This analysis is supported by the results of a
spatial 1D LST (Linear Stability Theory) analysis of a nonstreaky boundary layer at the same flow conditions. We use
this LST study to determine the frequency of the most amplified Mack mode in the domain. This mode, being the instability of highest frequency, guides us in selecting a proper
Nyquist frequency. Obtaining a second Mack mode frequencyof 𝑓 𝑀 ≈ 0.28 close to the inlet, we use a Nyquist frequencyof 𝑓𝑠 = 1.3 to be able to capture this mode and its first harmonic. The sampling is performed over 1300 convective times
𝑡 𝑐 = 𝛿∗ /𝑢 ∞ , resulting in 1300 snapshots of the flow evolution
presented in next section.

(4)2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
0.5
A3
A2
A1

0.4

Asu

0.3
0.2
0.1
0.0

0

100

200

300

x

(b) Streaks velocity disturbance 𝑢 0 .

(a) Streaks amplitude 𝐴𝑠𝑢

Figure 2: Streaks amplitude evolution and final streaks shape at 𝑥 = 350

Figure 3: Top : normalised contours of axial velocity from the DNS at 𝑦 = 1.5𝛿 ∗ . Bottom : associated LDNS axial

disturbance velocity 𝑢 0 at 𝑥 = 350. Dashed lines : contours of the base flow axial velocity.
Linear and non-linear disturbance fields

low-speed streaks. This superposition suggests a competition
mechanism between the modes triggered by the white-noise
forcing that can be made clear by the decomposition of thesolution fields.

A first investigation is made with the DNS response to
the white-noise forcing and the associated linear disturbances
fields from the LDNS. These fields are presented in Fig. 3
for the DNS velocity field and the LDNS disturbances fieldobtained with Eq. 3. To help identify the growing modes, we
also performed a 2D locally parallel LST at 𝑥 = 350 for the
streaky base flows. Out of conciseness, we only present therelevant unstable modes eigenfunctions on the left side of Fig.
5 for the first mode family (FM), second Mack mode family(MM) and streak modes family (SM). These modes serve as areference for the identification of observable flow structures inFig. 3.
The DNS velocity fields show various secondary structures as a function of streak amplitude. The A1 case shows ashort streamwise wavelength in the velocity field that has noclear symmetry. This symmetry being defined as the symmetry of the axial disturbance velocity distribution about the lowspeed streaks centerline. The associated LDNS disturbanceslook to comprise a superposition of many modes under thisbroadband forcing. For the A2 case, a spanwise motion of
the streaks with a varicose organisation is noticed. Associated
LDNS fields depict a marked anti-symmetric and subharmonic(𝜆 = 2𝜆 𝑆 ) disturbance distribution on the top of the boundary
layer, suggesting the dominance of a single linear subharmonicand sinuous mode. Finally, the A3 case undergoes transition toturbulence before the end of the domain, following a sinuous
motion of the streaks. The associated LDNS disturbance fieldat the end of the domain shows a superposition of at least twomodes. One is spanwise harmonic (𝜆 = 𝜆 𝑆 ) and the other issubharmonic, but both are anti-symmetric with respect to theSparse L-periodic SPOD formulation
We decompose the disturbances vector 𝑞 𝑖0 for a case 𝐴𝑖
in an optimal orthogonal basis using SPOD (Schmidt & Colonius, 2020). We point out that some special care should begiven to the decomposition in our case. Considering the absence of an homogeneous direction in 𝑧 for our base flow
(i.e. 𝜕𝑞¯ /𝜕𝑧 ≠ 0), the usual decomposition into single spanwise modes to reduce the size of the problem cannot be performed. Nonetheless, given the L-fold symmetry of the streak
pattern in the z-direction, with 𝐿 = 4 here, we can make use of
the Floquet theory to reduce the size of the analysis. Hence,the spanwise direction of 𝑞 can be decomposed in three Floquet components for the 𝜆 = [𝜆 𝑆 , 4𝜆 𝑆 , 2𝜆 𝑆 ] spanwise periodic
modes given respective Floquet exponents 𝑀 = [0, 1, 2], which
respectively refers to harmonic, 4-streaks and subharmonic
spanwise periodicity. Using the spanwise Fourier transform( •˜ ), we obtain a complete basis of spanwise wavenumbers indexed by 𝑙 for each Floquet exponent 𝑀,
q 0 (𝑥, 𝑦, 𝑧) =

∑︁ ∑︁
𝑀

q̃ 0𝑀 −𝐿𝑙 (𝑥, 𝑦)𝑒 𝑖 ( 𝑀 −𝐿𝑙) 𝑧 .

(5)𝑙Following initial work on the SPOD formulation using Floquet theory by Rigas et al. (2019), where the use of spanwisemodes sparsity was not made to reduce the SPOD problem

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Low-frequency
FM & SM
High-frequency
MM

Figure 4: First SPOD eigenvalue energy at 𝑥 = 350. Black : full spanwise modes base. Colors : spectrum by Floquet

exponent. Blue : harmonic ; green : subharmonic ; red : 4-streaks spanwise periodicity.
LDNS spectra at x=350

size. We devise a complete development in the frequencywavenumber domain to reduce the SPOD cost by only using
relevant spanwise wavenumbers with respect to the streak periodicity considered. These developments are summarised inwhat follows.
We start with the usual cross-spectral density formulation
Sff , with 𝑄ˆ the blocks matrix whose columns are made of a selected frequency issued from the time Fourier transform of 𝑞 0 ,using blocks of size 𝑁fft = 128. Noting the hermitian transpose𝐻𝑄ˆ and the space metrics 𝑑𝑉, we write :∫Sff =Q̂ 𝐻 WQ̂dV.

Using the framework developed above, we compute the
SPOD of 𝑞 0 for the LDNS at 𝑥 = 350. We give in Fig. 4 an
overview of the energy spectrum of the first eigenvalue 𝜎0 of
the SPOD along with identified eigenfunctions in Fig. 5, thenotations are defined below. A white noise forcing of identical amplitude for all streak cases was used. By looking at
the black markers, summing all Floquet components in the zdirection, we notice at first sight a strong increase in the amplitude levels reached by the most amplified instabilities as 𝐴𝑠𝑢
increases, showing how increasing the streak amplitude considerably amplify the linear growth mechanisms. Additionally,
we notice a separation between low-frequency ( 𝑓 ≈ 0.1) andhigh-frequency modes ( 𝑓 > 0.22). Using the 2D-LST results
mentioned above, we identified the modes at 𝑓 = 0.1 to belong
to the first mode family (FM) and streak-modes family (SM),whereas the modes at 𝑓 > 0.22 were identified to belong to thesecond Mack-mode family (MM). Specifically, for the highfrequency content, we notice the progressive separation of thehigh-frequency peak in the A1 case into two peaks of frequencies 𝑓 ≈ 0.25 and 𝑓 ≈ 0.32 for cases A2 & A3. This frequency
separation has not been observed before and reveals two subfamilies of the Mack mode, which we shall describe later as
being symmetric/varicose (MM𝑠 ) and anti-symmetric/sinuous(MM𝑎 ) Mack modes. To further help the detailed interpretation of these spectra in what follows, we make use of theconvention (•) 𝑘 for modes of wavelength 𝜆 = 𝑘𝜆 𝑆 . E.g. MM2𝑠
is the subharmonic sinuous second Mack-Mode as depicted inFig. 5e.
Whereas for the A1 case, the low and high-frequency
peaks are of a similar order of magnitude, with increasing
streak amplitude (case A2 & A3), the low-frequency mode
dominates: increasing the streak amplitude thus enhances thelinear growth mechanism associated with the first mode. The
strong amplification of the low-frequency content contrasts
with the PSE results of Paredes et al. (2019) at similar valuesof Mach number and streak amplitude, where the introduced
Mack mode was found to dominate. This difference underlines
the importance of accounting for the receptivity process in theanalysis of the transition on such distorted boundary layers.
Going further in the interpretation of the LDNS spectra,
we look at the coloured lines of Fig. 4 representing the Floquet
decomposition of the SPOD energy spectrum of the full spanwise base depicted in black, using Eq. 7. The blue, green andred curves give the frequency energy spectrum of respectively,
harmonic (𝜆 = 𝜆 𝑆 ), subharmonic (𝜆 = 2𝜆 𝑆 ) and four-streaks
modes (𝜆 = 4𝜆 𝑆 ). Each energy peak of the coloured curves
has been identified as an independent 2D-LST mode. This decomposition highlights a rich modal content for the streaky-(6)

𝑉

W is the matrix form of the compressible energy norm, and it isa function of the non-homogenous base flow 𝑞¯ . This inhomogeneity of the base flow is what complicates formulating thesparse SPOD, in contrast to the homogeneous case, where justthe 𝛽 = 0.0 component of the base flow can be used. Specifically, a sparse SPOD should only take the relevant spanwise
modes associated with the particular streak periodicity considered (harmonic, subharmonic, ...) to alleviate the computational cost of decomposing the DNS database. Hence, starting by from Eq. 6 in frequency-space domain, both the matrices W and Q̂ in Eq. 6 have to be Fourier transformed in thez-direction and only the relevant modes retained. Therefore,
with ( •˜ ) the Fourier transform in 𝑧, and using the property of
the Fourier transform, we can write the sparse expression of𝑆ff for a non-homogenous base flow within the Floquet framework,
∫
Sff =

˜ 𝐻W
g
Q̂
Q̂dV,

𝑉

∫
=



˜ 𝐻 W̃ ∗ Q̂
˜ dV,
Q̂

(7)

𝑉

with (∗) the convolution operator in the transformed direction. Using Eq. 7, we obtain a physically meaningful formulation of Sff now in the frequency-wavenumber domain for
non-homogeneous base flows. In the initial formulation ofRigas et al. (2019), the database is first Fourier transformedin 𝑧 to filter only the relevant spanwise modes, then Fouriertransformed back in space to perform the SPOD in the physicaldomain. Our formulation allows us to take full advantage of
the Floquet decomposition of 𝑞 0 by using only a few relevant
wavenumbers and keeping the SPOD analysis in wavenumber
domain with significant computational savings. Finally, Eq. 7
leads to a cost reduction by a factor 𝐿 to the SPOD computation.

4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022of the domain. Both amplified DNS modes follow the amplification rate of their LDNS counterpart, especially the MM1𝑠
mode closely follows the LDNS up to 𝑥 > 200. We can notice that the receptivity in the non-linear regime shows a slight
discrepancy with a delayed growth of the low-frequency modewith respect to the LDNS. Suggesting that some non-linear
receptivity mechanism might be at play. Comparing these observations to the non-streaky LDNS, we note an overall delaying effect of the streaks on the linear receptivity and growthof these unstable modes. We also remark, as noted by Paredes et al. (2019), a reduction in the Mack mode amplification
rate, confirming the stabilising effect at this low streak amplitude of 𝐴𝑠 𝑢 ≈ 0.1. On the other hand, the first mode shows anincreased amplification.
The dynamics of case A2 are driven by the FM2𝑎 modeon the second half of the domain. This mode has a strong
amplification rate compared to the non-streaky case, showing increased instability. On the other hand, the Mack-mode
MM1𝑠 shows a fast initial growth, followed by a stabilisation. A
second interesting feature is a discrepancy between the initial
growth of the DNS and LDNS Mack modes. The former starts
to grow strongly around 𝑥 = 80 where the LDNS mode amplification shows a considerable delay. This early non-linear
growth suggests the presence of a non-linear receptivity mechanism for the MM1𝑠 mode. Furthermore, after the strong initial
growth, the Mack mode observes a damping resulting in an
amplitude decay from 𝑥 = 150. Although delayed, similar behaviour is observed for the LDNS. Then, the MM1𝑠 from the
DNS shows an almost constant growth rate. Its LDNS counterpart shows a stronger growth at 𝑥 = 200 up to 𝑥 = 280 where
its starts getting damped again. This discrepancy could be explained by the non-linear base-flow modification due to thevaricose motion observable in Fig. 3 at 𝑥 ≈ 280, supporting
the instability of the DNS MM1𝑠 mode

Figure 5: Comparison of LST-2D (left) and SPOD (right)

eigenfunctions at 𝑥 = 350. Normalised contours of thereal part of the axial velocity disturbance. Modes shapefor A2 are not shown as they are the continuation of A1.
(a) : A1 𝑓 = 0.08, FM2𝑎 ; (b) : A1 𝑓 = 0.26, MM1𝑠 ; (c) :A3 𝑓 = 0.09, SM1𝑎 ; (d) : A3 𝑓 = 0.26, MM2𝑠 ; (e) : A3
𝑓 = 0.33, MM2𝑎boundary layers.
The modes FM2𝑎 at 𝑓 = 0.08 (Fig. 5a) and MM1𝑠 (Fig.
5b) drive the dynamics of case A1. As the streak amplitude
increases, for case A2, the low-frequency mode FM2𝑎 becomesdominant. Concurrently, we note in Fig. 4 a decrease of 70%
in the amplitude of the mode MM1𝑠 , suggesting a stabilizing
effect also noted in Paredes et al. (2019). We also note theemergence of the secondary peak at 𝑓 = 0.33 for the subharmonic component MM2𝑎 (Fig. 5e). Finally, for case A3, thelow-frequency mode that has here become dominant switches
to a harmonic spanwise periodicity (SM1𝑎 ) (Fig. 5c). Concurrently, the decaying harmonic Mack mode switches to a subharmonic Mack mode (MM2𝑠 ) (Fig. 5d), strongly amplified
compared to cases A1 & A2. This amplification suggests now
a destabilising effect of the high amplitude streaks.

Finally, we investigate the transitional scenario of case
A3. At first sight, the non-linear dynamics seem to be driven
by two low-frequency modes, which are the subharmonic FM2𝑎and the harmonic SM1𝑎 modes. This latter mode is noted as a
streak mode because it hase be identified to unstable only forhigher streak amplitude by 2D-LST. The FM2𝑎 mode has an
early receptivity and a constant growth throughout the domain,
while the SM1𝑎 mode starts being linearly unstable only after𝑥 = 120 but quickly outgrows the FM2𝑎 mode as the streaks
amplitude increases. From this spectrum, it seems that bothmodes leads to the onset of turbulence around 𝑥 = 300. However, looking at Fig. 3 we observe an harmonic sinuous motion related to mode SM1𝑎 symmetry, favouring the hypothesisof the final stages of transition being dominated by the strong
growth of this latter mode for this specific case. Although, thisresult does not rule out the possibility of a transition driven
by the earlier amplification of the subharmonic instability ina noisier environment. As for the high-frequency content, we
now observe the strong amplification of the MM2𝑠 mode. Thisamplification exceeds the Mack modes growths of cases A1 &A2, suggesting an increased instability of the Mack modes forlarge streaks amplitude.

Non-linear dynamics
Having assessed the most amplified instabilities originating from the white noise forcing. We compare and track theseidentified modes in the non-linear simulation. Using Eq. 7,
the growth of the main modes retained from Fig. 4 can befollowed in Fig. 6 for cases A1, A2 & A3. For further comparison, we also compute the LDNS of the white noise forcingof a Blasius boundary layer, i.e. without streaks, at exactly thesame flow conditions. We add the most amplified first modeand Mack mode issued from this non-streaky LDNS to the figures to quantify the streaks effect. Hence Fig. 6 contains the
DNS and LDNS results for streaky base flows with respectively thick and dashed lines, and it also contains LDNS results for the non-streaky base flow, shown as thin dotted lines
with markers. As cases A1 & A2 follow a similar trend, withincreased dominance of FM2𝑎 , the A3 case reveals more intricate non-linear dynamics requiring the tracking of additionalmodes.
The DNS of case A1 does not reach turbulence by the end

These combined observations for all three cases reveal thestrong effect of the streaks distortion on the boundary layer receptivity and stability. The main result here is the effect of thestreaks on the Mack-mode receptivity and the first mode amplification. It shows that even for low streaks amplitude, thedominance of the second Mack mode observed for the Blasius
case is no longer verified. Furthermore, these results suggestthat the family of the low-frequency modes is actually dominant at 𝑀∞ = 6.0 for all the observed streaks amplitude here.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
A1
10

−1

10

−3

A2

A3
SM1a , f = 0.09
MM2s , f = 0.26
MM2a , f = 0.33

Blasius LDNS FM, β = 0.09
Blasius LDNS MM, β = 0.0
FM2a , f = 0.09

σi

MM1s , f = 0.09

10−5

10−7
0

100

200

300

0

100

200

x

x

300

0

100

200

300

x

Figure 6: Non linear evolution of the main SPOD modes energy (full), compared to the rescaled LDNS dynamics (dotted).

Conclusion

streaks. Journal of Fluid Mechanics 428, 29–60.
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In this study, we have investigated the effects of steadystreaks on the boundary layer receptivity, stability and transition. In order to get a further understanding of these effects, we
devised a DNS framework to study the linear and non-linear
dynamics of disturbances triggered by white noise. We apply this forcing on three steady base flows consisting of growing optimal streaks of three different amplitudes spanning previously studied amplitudes from the PSE analysis of Paredes
et al. (2019) at a similar Mach number.
To decompose the dynamics in time and space, we developed an SPOD formulation using Floquet theory, which allows
us to split the growing modes into different families according
to their periodicity with respect to the streaks. This method allows us to capture the emergence of a larger number of distinct
modes in the streaky flow compared to the Blasius boundary
layer. By comparing the linear growth of these modes to previous linear analyses, we highlight the role of the forcing andreceptivity in the evolution of the modes. We show a strong
amplification and dominance of low-frequency modes for all
streak amplitude considered in contrast with the previous PSE
results.
Going further in the analysis by tracking the non-linear
evolution of the most amplified modes, the DNS shows the various effects of the streaks on the mode receptivity and growthcompared to the streaky-linear and the non-streaky, Blasius
boundary layers. In short, the Mack modes are mainly dampedby the streaks and their receptivity is also affected. Also,
the low-frequency modes of subharmonic spanwise periodicity
dominate at low amplitude, whereas the strong amplificationof the harmonic component is leading to turbulence at highamplitude. This set of results serves as a first estimation ofnatural transition dynamics in a streaky boundary layer. Finally, this study underlines the sensitivity of the receptivity,
modal growth and non-linear stages of transition process toslight changes in the boundary layer topology. Further analysis
will now be necessary to understand which underlying physical mechanisms drive these changes in the flow dynamics.

Acknowledgements
This work is part of the project TRANSITION supported by Région Nouvelle Aquitaine under grant 20181R10220. Additionally, this work was granted access to
the HPC ressources of IDRIS under the allocations 2020A0092A10868 and 2021-A0112A10868 made by GENCI.

REFERENCES
Andersson, Paul, Brandt, Luca, Bottaro, Alessandro & Henningson, Dan S. 2001 On the breakdown of boundary layer

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