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

NON-LINEARITIES IN TRANSITIONAL SHOCK WAVE / BOUNDARYLAYER INTERACTIONSMariadebora MaurielloAix-Marseille Univ, CNRS, IUSTIMarseille, France
mariadebora.mauriello@univ-amu.fr

Lionel Larchevêque
Aix-Marseille Univ, CNRS, IUSTIMarseille, France
lionel.larcheveque@univ-amu.fr

Pierre Dupont
Aix-Marseille Univ, CNRS, IUSTIMarseille, France
pierre.dupont@univ-amu.fr

ABSTRACT

thors propose to correlate this unsteadiness with the dynamics
of the separated region (Wu & Pino Martı́n, 2008), or witha resonance mechanism in the interaction region (Pirozzoli& Grasso, 2006). Piponniau et al. (2009) developed a simple model to explain the low–frequency unsteadiness. It isbased on the entrainment characteristics of the shear layer andrelates the mass recharge within the separated bubble to theflapping dynamics occurring near to the reattachment point.
However, a new consensus paved the way of uniting both
upstream and downstream mechanism as contributors to theemergence of the low–frequency unsteadiness, but the effectof the incoming turbulent boundary layer diminishes for increasing interaction strengths (Clemens & Narayanaswamy,2014). Less attention has been devoted to interactions between
shock waves and an incoming laminar boundary layer and only
recently the scientific community has taken steps in that direction. The TFAST project carried out both numerical (Sansica et al., 2016; Larchevêque, 2016) and experimental (Diopet al., 2019) studies on these transitional interactions. Theirstudies confirmed that unsteadiness comes up when transition
to turbulence occurs. For example, Sansica et al. (2014) revealed that as the breakdown of the turbulence occurs, broadband disturbances travel upstream in the separated region of
the boundary layer with a corresponding response near the
separation point. Further, temporal measurements revealed
several frequencies being amplified along the interaction withthe low–frequency unsteadiness of the separation region being almost one order of magnitude higher than that found in
the turbulent counterpart (Diop et al., 2016). Sansica et al.
(2016) demonstrated that the region of reattachment might be
the origin of the low–frequency unsteadiness and Larchevêque(2016) showed that their amplitude vary significantly with thesize of the separated region and the location of the transition toturbulence. However, all of these studies agree to associate thelow–frequency unsteadiness with a feedback originating fromthe reattachment region, although a lot of open questions areleft unanswered.

Large–eddy simulations (LES) of a transitional shock
wave boundary layer interaction (TrSBLI) are carried out toinvestigate the origin of the low–frequency unsteadiness in an
incident–reflecting shock configuration. The separated regionis characterised by low–frequency breathing and is associatedwith a fluidic feedback originating from the vicinity of thereattachment point, in agreement with previous investigations.
The non–linear spectral analysis reveals a quadratic coupling
between the low–frequency feedback and the unstable modes
developing within the separated shear layer. The chronologicalsequence of the phenomena is retrieved in the physical space
through the bicorrelation statistical tool.

INTRODUCTION
Shock wave boundary layer interactions (SBLIs) havebeen the subject of extensive investigations in the last 70
years (Dolling, 2001). Much of the interest has been devoted
to interactions between shock waves and turbulent boundary layers and the necessity of understanding such flows has
motivated numerous experimental, numerical and theoreticalstudies. A large variety of geometrical configurations havebeen covered such as normal shock interactions, incidentreflecting interactions, compression corner, over–expanded
nozzle, etc., and investigations spanned a wide range of Machand Reynolds numbers. The global organisation in terms ofspace and time dynamics have been depicted and the qualitative mean organisation of the flow is currently well understood. In the case of turbulent separated SBLIs, evidence
showed that the interaction is highly unsteady with very low–frequency motions of the separated region (Dolling, 2001;
Ganapathisubramani et al., 2009; Dupont et al., 2006). Thisphenomenon is often reported in the literature as breathing of
the bubble and its origin is still under debate dividing the scientific community into different strands of thought. Multiple
studies claim the upstream mechanism as a natural source ofthe low–frequency unsteadiness of the system such as broadband fluctuations (Touber & Sandham, 2011), or the existence of superstructures in the incoming turbulent boundary
layer which are required to induce the flapping of the separated region (Ganapathisubramani et al., 2009). Other au-

The present work aims to better characterise the physicalorigin of the low–frequency unsteadiness leveraging the non–linear spectral analysis, a tool that suits well when dealing withquadratic interactions.

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1.

Flow parameters.

4

LES reference
LES enlarged
LES refinedExperiments

3.5
3

Reimp

θ

∗
L/δimp1.68

0.65 × 106

4.96◦

4.02

2.5

y/δ99

Mach

2

1.5
1

NUMERICAL SETUP

0.5

The datasets of the present numerical simulations are
based on a transitional shock wave boundary layer interaction (TrSBLI) investigated in the experiments carried out in the
TFAST project and performed at the IUSTI laboratory in Marseille (Diop et al., 2019). The flow parameters of the incidentreflecting interaction are reported in table 1. Simulations were
performed using the FLU3M code from ONERA. The code
relies on a finite volume discretisation in space and an implicit Gear scheme for the temporal discretisation, both being
second–order accurate. To minimise the numerical dissipation,
the space scheme is modified by adding the dissipative part of
the Roe scheme to a centered scheme in regions where strong
compressibility/low vorticity occurs, as identified by meansof Ducros sensor (Ducros et al., 1999). The time integration is performed with a maximum Courant-Friedrichs-Lewy
number of 11, and the non-linear system is solved through 7
sub-iterations resulting in a reduction of the residuals of morethan three orders of magnitude in the laminar and transitional
regions and about two orders of magnitude in the turbulent
ones. The LES modeling is built from an implicit grid filtering
coupled with an explicit subgrid modeling through the selective mixed-scale model. The reference meshes are ∆x+ ≲ 30,+y+1 ≲ 1.25 and ∆z ≲ 15 and the width of the computational
domain is 7 times larger than the separation bubble height. Additional meshes refined by 40%, 20% and 25% in the x, y andz directions have been built in order to perform a grid convergence study. A mesh enlarged in the spanwise direction
by 100% has also been tested. The wall is treated as adiabatic, and periodicity is used in the spanwise direction. Nonreflecting boundary conditions based on characteristics formulation are set at the inflow, outflow and upper boundaries. The
inflow conditions are obtained by superimposing flow perturbations of very low amplitude and large bandwidth to the compressible Blasius profile by means of a compressible variant of
the Synthetic Eddy Method (SEM). However, the amplitude ofthese perturbations were not known from the experiments. The
strategy followed was to adjust them iteratively such that the
final computations resulted in the same separation length as
the experiments. The simulations were run long enough to encompass 70 cycles of the typical low–frequency unsteadinessallowing the extraction of spectral quantities.

0

-60

-50

-40

-30

-20

-10

0

10

(x-ximp)/δ99

Figure 1. Comparison of the mean streamwise velocity
profiles between experiments and computations at different
streamwise locations.

strating that the procedure described in the previous section
regarding the adjustment of the inflow perturbations amplitude is successful. In particular, experimental and numerical
data show excellent agreement up to the apex of the recirculation bubble. Nonetheless, experiments tend to yield transition downstream of the apex. The transition phenomena occurs more abruptly over a shorter distance for the numerical
data, resulting in quicker relaxation towards a canonical turbulent boundary layer downstream of the interaction region.
It is unclear if this discrepancy is due to either the different
physical nature of the perturbations in the incoming boundary
layer or an alteration of the transition mechanism within the
bubble due to the LES modelling. The fact that no difference
is observed between the base computations and the ones relying on the refined meshes, even downstream of the apex, is anindication that the former hypothesis is more likely.

LINEAR SPECTRAL ANALYSISThe spectral analysis is considered to identify the key features of the shock motion and the characteristic temporal scales
are extracted through spectral decomposition of wall pressure
fluctuations.
Figure 2 shows the streamwise distributions of the premultiplied, normalised wall pressure spectrum. The horizontal axis displays the frequency normalised by the interaction
length Lint , defined as the distance between the impingement
location ximp of the incident shock and the beginning of theinteraction region xint , and the free–stream velocity U∞ , corresponding to the Strouhal number StL = f Lint /U∞ . The vertical axis shows the streamwise evolution normalised by theinteraction length Lint , as defined above. In this way the interaction region ranges from 0 to 1. The spectrum shows thatthe beginning of the interaction region, near x∗ = 0, is dominated by power spectral density (PSD) content spreading at
low frequencies, with typical Strouhal number of StL ≃ 0.055.
This value is close to the one found in the DNS by Sansica
et al. (2016) and LES by Larchevêque (2016), but it is significantly lower than the StL ≃ 0.1 value seen in the experiments
by Diop et al. (2019). Conversely, the work of Bonne et al.
(2019) yields an intermediate values of StL ≃ 0.08. Such discrepancies in the low–frequency Strouhal numbers may be related to different physical forcing strategies of the incoming
boundary layer/initial mixing layer adopted by the aforementioned works. Moving in the streamwise direction towards theimpingement point x∗ = 1, the typical frequency increases up
to StL ≃ 0.6 . Once past this point, a spot of high PSD content
emerges. It is localised in the region of the flow downstream

VALIDATION
Computations have been compared to experimental measurements obtained through LDV technique and documented
in Diop et al. (2019). The validation accounts for refined
meshes in all of the directions as described in the previous section, and a mesh enlarged in the spanwise direction has also
been validated. Figure 1 plots the mean streamwise velocity
profiles at different streamwise locations. The streamwise coordinate is defined as the distance between the impingement
location of the incident shock ximp and the current locationx and it is normalised by the incoming boundary layer thickness δ99 . The wall–normal coordinate is normalised by using
the same reference length. All LES profiles collapse demon-2

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

Figure 3. Si indicates the location of the sensors used for thephase analysis as well as for the bispectral analysis. The contours show the exponentially–spaced filled isolevels of turbulent kinetic energy and the mean dividing streamline is indicated with white dashed line.

Figure 2. Streamwise evolution of the premultiplied, normalised wall pressure spectrum. White horizontal dashed lines
mark separation (x∗ = 0) and impingement (x∗ = 1) locations while the vertical one correspond to the Strouhal number
StL ≃ 0.055. The white circle denotes the high PSD content
associated with the transition and the breakdown of obliqueinstability modes.

denotes the complex conjugate. In this work, the streamwise
momentum cross–spectrum has been computed and figure 3
shows the location of the sensors where ρu has been sampled.
Si correspond to the end of the ascending mixing layer (S1), thecenter of the descending mixing layer (S2) and three locations
distributed on a horizontal plane fairly close to the wall, either
upstream of, at, or downstream of the reattachment point (S3,S4 and S5 respectively). The reference point for the cross–
spectrum is the position where the breakdown of the obliquemodes occurs (S1) and all the possible linear links between
streamwise momentum signals along a plane crossing sensors
S3, S4 and S5 are obtained.
From the cross–spectrum, phase information is extracted
for three regions, namely the separation point, inside the separation bubble and near the reattachment point, based on theircharacteristic Strouhal numbers, i.e. StL ≃ 0.055, 0.11 <StL < 0.33 and StL ≃ 2.1 respectively. Figure 4 plots thestreamwise evolution of the normalised phase associated withρu at the fixed aforementioned Strouhal numbers. Note thatthe normalisation of the streamwise coordinate is the sameused for figure 2, but it is shifted from the wall. In this wayx∗ = 0 corresponds to the point where the plane considered forthe analysis crosses the upstream mixing layer, while x∗ = 1 isfurther downstream of the reattachment point (see horizontal
straight white line in figure 3). Moreover, the phase data arenormalised by the frequency and unwrapped so that jumps of
2π are removed, and shifted vertically such that phase is zeroat the origin x∗ = 0 of the reference frame. The inverse of theslope gives information regarding the phase velocity between
the reference ρu at S1 and the selected streamwise momentum signals along the plane as well as the direction of travel.
It is important to point out that the phase velocity is associated with a fixed frequency, but is contaminated by all thestreamwise wavenumbers kxi . Not having selected a particular wavenumber, one can assume that rather than the phasevelocity, what is observed is the average velocity at which thegroup of signals characterised by a given frequency is traveling, i.e. the group velocity Vg . By assuming VΦ ∼ Vg ∼ Uc
where Uc is the convection velocity, figure 4 provides the following information. For StL ≃ 2.1, the phase increases linearlythroughout the streamwise direction. The convection velocity
is Uc ≃ +0.5U∞ , meaning a motion from upstream towards
downstream regions with respect to the location on the planeof sensor S1 indicated by the vertical dashed black line in the
figure. However, for lower Strohual numbers an abrupt change
in phase is observed at x∗ ≃ 0.95. In particular, up to x∗ ≃ 0.95,

the impingement point at x∗ ≃ 1.1 and the associated Strouhal
number is StL ≃ 2.1 (see the white circle in figure 2). The
presence of this high PSD spot has already been observed in
the work of Larchevêque (2016) who investigated transitional
SBLIs, but no evidence have been reported for the turbulent
counterpart. Its presence is explained below. In the first part of
the recirculating region, structures develop in the mixing layerzone and grow as they move downstream. After some distance, these structures are shed into the downstream flow, possibly leaving their trace at the wall. However, previous studies
demonstrated that they remain at a certain distance from thewall, making the previous statement weak as the only explanation. Moreover, the mechanism just described is also common in turbulent interactions, where the emergence of such
localised high PSD spot might be hidden by the turbulencefrom the incoming boundary layer. Therefore, other features
have to be taken into account. The position x∗ ≃ 1.1 is associated with the location where transition to turbulence occurs(results are not shown here). So, rather than a footprint of themixing layer structures, it could represent the trace of the transition process. In addition, this spot could be related to thebreakdown of oblique first–mode waves. In fact in this region,
high disturbance amplitude is reached through linear growththat occurs along the mixing layer. Thus, oblique modes enter in the non–linear stage and transition soon follows. These
instability modes originating from the initial mixing layer andpropagating downstream to the reattachment point play a fundamental role in explaining the origin of low–frequency unsteadiness, as will be demonstrated in the next section.
From the previous analysis it emerges that the reattachment region is characterised by high–frequencies scales (StL ≃2.1) associated with the breakdown of the oblique first–modewaves and transition. Conversely, the separation region isdominated by low–frequency scales (StL ≃ 0.05) associatedwith the unsteadiness of the head shock. A question that arises
now is: are the high–frequencies scales a local process or theyaffect all the other regions? To give insight into the question, amore deep investigation has been conducted by resorting to thephase information from the two–points spectrum. The cross–
spectrum is a complex valued quantity defined as Sg1 ,g2 ( f ) =G1 ( f ) G∗2 ( f ), where G1 ( f ) and G2 ( f ) are the Fourier transforms of the time series g1 (t) and g2 (t) respectively, while ∗3

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

Figure 5.

Figure 4. Streamwise evolution of the normalised phase associated with ρu for the Strouhal numbers i.e. StL ≃ 0.055,
0.11 < StL < 0.33 and StL ≃ 2.1. The vertical dashed line indicates the streamwise location shifted from the wall of sensor
S1.

Bicoherence map between S1 × S1 ↔ S2.

teractions. Such interactions can be highlighted in the spectral space from the bispectrum, which is a measure of thequadratic non–linearities at the bifrequency ( f1 , f2 ) (Tynan
et al., 2001; Cui & Jacobi, 2021). One way to normalise thecomplex–valued bispectrum is the bicoherence between timeseries g1 (t), g2 (t) and g3 (t), which reads:

the phase decreases nearly linearly while it increases linearly
everywhere else. Before the jump, upstream propagation velocities are found which correspond to Uc ≃ −0.24U∞ and
Uc ≃ −0.41U∞ for the low and the medium frequencies respectively. After the jump, downstream propagation speeds
are observed.
From this scenario it is possible to infer that the sourceof the slow–upstream motion is located at x∗ ≃ 0.95, that at
the wall corresponds to the position where the high PSD spot
previously observed emerges (see figure 2). This high PSD
content associated with the transition and oblique mode breakdown has a global effect. In fact, it affects the upstream region
through a fluidic motion that starts from the position where
the high PSD spot emerges and moves upstream at a convection velocity that decreases as the Strouhal number decreases.
Moreover, the position x∗ ∼ 0.95 that corresponds at x∗ = 1.1
at the wall, can be associated with the location of the source ofthe low–frequency unsteadiness.

Bicg1 ,g2 ,g3 ( f1 , f2 ) =G1 ( f1 ) G2 ( f2 ) G∗3 ( f1 + f2 )2

∥G1 ( f1 ) G2 ( f2 )∥2 ∥G3 ( f1 + f2 )∥2(1)

where G( f ) denotes the Fourier transform of the time series g(t). A value of Bicg1 ,g2 ,g3 that differs significantly fromzero is associated with a (partial) phase relationship between
g1 × g2 and g3 being stable over time, and it is indicative of
possible non–linear energy exchange between the frequencies
under consideration. In the present work, the analysed variable is the streamwise momentum ρu since it may experience quadratic interactions through the convective term of theNavier-Stokes momentum equation. The locations where ρuhas been sampled has already been shown in figure 3. Figure 5 plots the bicoherence map between S1 × S1 and S2 andthe presence of strong bicoherence indicates those frequency
modes that most strongly contribute towards the quadraticphase coupling (QPC). The linearly unstable modes develop
in the ascending mixing layer and enter the non–linear regime
near the apex of the bubble. Indeed, the positive quadrant of
the bicoherence map is typical of a transitioning mixing layer.
It displays an emerging direct turbulent cascade involving triadic interaction between frequency ranges being integer multiples of the original instability wave range at location S1, centered around StL ≃ 2. On the contrary, quadrant defined by
StL1 < 0 and StL2 > 0 displays a quadratic link towards lower–
frequencies with |StL1 | ≃ |StL2 |. Spots of high values of bicoherence for very low values of Strouhal numbers are observed.
For triadic interactions between band-limited signals suchas the ones centered around StL ≃ 2, bicoherence is rather easy
to analyse. However, such an analysis may become trickier
for more broadband signals since a single frequency can be
involved in multiple triadic interactions. One way to overcome
this difficulty is to use the summed bicoherence (Tynan et al.,2001), defined as:

NON–LINEAR ANALYSIS
Two–points spectral analysis revealed that the origin of
the low–frequency breathing of the separation bubble corresponds with the location where the breakdown of the obliquemodes emerges. Moreover, band-pass filtered pseudoschlieren movies shows the occurrence of a fluidic feedback
starting from the reattachment point for StL ≲ 1. It progressively restricts to a narrower low–frequency band while moving upstream in the separated region. From the movies, highly
energetic fluctuations are also visible in the vicinity of the reattachment point for much higher frequencies. Therefore, multiple mechanisms are at play in this region, including transition and quadratic interactions that might involve different frequency bands. However, second–order statistics are limited
to describing linear processes. To detect and quantify non–linearities in time series, higher–order statistics are required.
In that context, bispectrum and bicoherence information areused to propose a scheme explaining how low-frequency motion appears near the reattachment point before being convected upstream, then inducing the breathing of the bubble.
The region of the flow around the reattachment point is dominated by small–scale motions whose frequencies, as seen on
the spot at x∗ ≃ 1.1, StL ≃ 2.1, are much higher than the low–frequency range associated with the upstream fluidic feedback.
Because of the frequency difference, both phenomena cannot
be linearly linked but they can be related through quadratic in-Σbicg1 ,g2 ,g3 ( f ) =2∑

G1 ( f1 ) G2 ( f2 ) G∗3 ( f1 + f2 )∑∥G1 ( f1 ) G2 ( f2 )∥2 ∥G3 ( f1 + f2 )∥2| f1+f2 |= f
| f1+f2 |= f(2)4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
0.04
0.035

S2
S4

0.03

Σbic(f)

0.025
0.02
0.015
0.01
0.005
0 -2
10

10-1

100

101

StL
0.04
0.035

Figure 7. Streamwise evolution of the normalised biphasealong the horizontal line crossing sensors S3 to S5. For each
StL3 spanning from low to high values, the three combinations
resulting in the highest values of the bicoherence are plotted.

S3
S5

0.03

Σbic(f)

0.025
0.02
0.015

ference between part of the g3 series considered at frequency
f1 + f2 that is quadratically linked with part of the product between g1 and g2 , considered at frequencies f1 and f2 respectively. The streamwise evolution of the biphase between the
S1 × S1 and sensors distributed over the horizontal line crossing sensors S3 to S5 is plotted in figure 7. High–frequency
pairs whose sums correspond either to the various typical lowfrequency Strouhal fluctuations identified in the separated region (StL3 < 1) or to the oblique modes and their harmonics(StL3 ≃ n × 2.1, n ∈ N) have been selected. Among the various frequency pairs of a given sum that have been computed,
the three ones resulting in the largest bicoherence levels havebeen retained. In that sense, the streamwise phase evolution
being plotted is representative of the strongest quadratic contributions to the low-frequency range. Comparison of figure 4
and figure 7 confirms that in the initial region of the interaction up to x∗ ≃ 0.95 the phase evolution in both cases isvery similar for the frequency range under consideration. As
a consequence, the group velocity associated with the motionof the fluctuations quadratically linked with the oblique modeis roughly the same as the one found from the linear two–
point analysis. This, coupled with the rather large values ofbicoherence found in that region, confirms that the upstream–
convected streamwise momentum fluctuations are for a largepart induced by non–linear coupling with the mixing layer
oblique modes. Since such an upstream motion has been associated with the low-frequency breathing of the separated region, it is therefore strongly suggested that the breathing is induced by non–linear beating in the vicinity of the reattachment
point of the high–frequency oblique modes having developed
within the mixing layer.
Up to now all the information have been extracted fromthe spectral space. However, the chronological sequenceof the phenomena can be observed in the physical spaceby computing the inverse Fourier transform of the bispectrum, i.e. the bicorrelation that reads Rg1 ,g2 ,g3 (τ1 , τ2 ) =g1 (τ1 ) g2 (τ2 ) g∗3 (τ1 + τ2 ). It is a measure of the time shift of
the two time series g1 (τ1 ) and g2 (τ2 ) with respect to the third
one g3 (τ1 + τ2 ). Figure 8 shows the bicorrelation map computed for the streamwise momentum with S1 × S1 ↔ S3. The
first two sensors are located in the region where the breakdownof the oblique modes occurs, S1, while the destination sensor is
located in the separated region, S3. Note that also in this casethe range of significant bicorrelation corresponds to a low values as a consequence of the normalisation. High bicorrelation
values are observed for negative time delay suggesting that thenon–linear coupling between the oblique unstable modes and

0.01
0.005
0 -2
10

10-1

100

101

StL

Figure 6. Summed bicoherence with S1 × S1 for sensors located within the descending shear layer (upper panel) and near
the wall, both upstream and downstream of the reattachment
point (lower panel).

Summed bicoherence plots between the “source” signal
S1 × S1 and streamwise momentum at different locations are
displayed in figure 6. Note that the low values associatedwith the summed bicoherence and reported in the y–axis area consequence of the adopted normalisation (see equation 2).
The cascade process for sensors S2 and S4 that are locatedat the center of, and near to the wall of the descending mixing layer is shown in the upper panel of figure 6. The energy
transfer towards higher harmonics of the original instability
wave at StL ≃ 2 seen in the positive quadrant of figure 5, ismostly achieved when approaching the vicinity of the reattachment point (see sensor S2). Once the reattachment location isreached, the turbulence is close to be fully developed and only
short–range triadic interactions are observed (see sensor S4).
However, an energy transfer towards the low–frequency range
takes place within the separation bubble, as sensor S3 shows in
the lower panel of figure 6. For this range of low–frequency,
the linear analysis revealed an upstream motion as pointed out
with figure 4. No such process is identified downstream of
the reattachment point as sensor S5 shows. It is worth notingthat the relative energy content at low–frequency is increasing when moving upstream within the bubble. Based on theseresults, it can be postulated that the upstream low–frequency
feedback is driven by quadratic interactions between high–frequency oblique modes found in the mixing layer developing
over the separated region.
Bicoherence spectra convey information on the quadratic
link between various regions of the flow, but do not provide any data regarding motion within these regions. However, information on the convective or stationary nature of thequadratic link can be recovered from the biphase spectrum using the same methodology developed for the cross–spectrum.
The biphase spectrum correspond to the argument of thecomplex-valued bispectrum G1 ( f1 ) G2 ( f2 ) G∗3 ( f1 + f2 ). For
a given frequency pair ( f1 , f2 ), it is a measure of the phase dif-5

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

Figure 8.

dynamics of a mach 1.6 shock wave/transitional boundary layer interaction using a rans based resolvent approach.
Journal of Fluid Mechanics 862, 1166–1202.
Clemens, N. T. & Narayanaswamy, V. 2014 Low-frequency
unsteadiness of shock wave/turbulent boundary layer interactions. Annual Review of Fluid Mechanics 46, 469–492.
Cui, G. & Jacobi, I. 2021 Biphase as a diagnostic for scaleinteractions in wall-bounded turbulence. Physical Review
Fluids 6, 014604.
Diop, M., Piponniau, S. & Dupont, P. 2016 On the length andtime scales of a laminar shock wave boundary layer interaction. In 54th AIAA Aerospace Sciences Meeting.
Diop, M., Piponniau, S. & Dupont, P. 2019 High resolution lda
measurements in transitional oblique shock wave boundary
layer interaction. Experiments in Fluids 60, 57.
Dolling, D. S. 2001 Fifty years of shock-wave/boundarylayer interaction research: What next? AIAA Journal 8,
1517–1531.
Ducros, F., Ferrand, V., Nicoud, Weber, F. C., Darracq, D.,
Gacherieu, C. & Poinsot, T. 1999 Large-eddy simulation of
the shock/turbulence interaction. Journal of Computational
Physics 152, 517–549.
Dupont, P., Haddad, C. & Debiève, J. F. 2006 Space and timeorganization in a shock-induced separated boundary layer.
Journal of Fluid Mechanics 559, 255–277.
Ganapathisubramani, B., Clemens, N. T. & S., Dolling D.
2009 Low-frequency dynamics of shock-induced separation
in a compression ramp interaction. Journal of Fluid Mechanics 636, 397–425.
Larchevêque, L. 2016 Low- and medium-frequency unsteadinesses in a transitional shock–boundary reflection with separation. 54th AIAA Aerospace Sciences Meeting .
Piponniau, S., Dussage, J. P., Debiève, J. F. & P., Dupont
2009 A simple model for low-frequency unsteadiness inshock-induced separation. Journal of Fluid Mechanics 629,
87–108.
Pirozzoli, S. & Grasso, F. 2006 Direct numerical simulation of
impinging shock wave/turbulent boundary layer interaction
at m=2.25. Physics of Fluids 065113.
Sansica, A., Sandham, N. D. & Hu, Z. 2014 Forced responseof a laminar shock-induced separation bubble. Physics of
Fluids 26, 093601.
Sansica, A., Sandham, N. D. & Hu, Z. 2016 Instability
and low-frequency unsteadiness in a shock-induced laminar
separation bubble. Journal of Fluid Mechanics 798, 5–26.
Touber, E. & Sandham, N. D. 2011 Low-order stochastic
modelling of low-frequency motions in reflected shockwave/boundary-layer interactions. Journal of Fluid Mechanics 671, 417–465.
Tynan, G. R., Moyer, R. A., Burin, M. J. & Holland, C. 2001
On the nonlinear turbulent dynamics of shear-flow decorrelation and zonal flow generation. Physics of Plasmas 8,
2691–2699.
Wu, M. & Pino Martı́n, M. 2008 Analysis of shock motion inshockwave and turbulent boundary layer interaction using
direct numerical simulation data. Journal of Fluid Mechanics 594, 71–83.

Bicorrelation map S1 × S1 ↔ S3.

the low–upstream motion occurs after the apex of the separated
region. Moreover the periodic pattern has been found to correspond with the period of the oblique mode being ∆τ ∗ = StL−1 ,with StL = 2.1. This result confirms that the oblique modes
and their harmonics are mostly involved in the non–linear interactions.

CONCLUSIONSLES computations of a TrSBLI with separation have been
carried out and compared with experimental databases fromthe TFAST project (Diop et al., 2019). All computations havebeen validated resulting in a good agreement between experimental and numerical data. The higher–order statistical tools
in the spectral space have been used to explore the linear andquadratic interactions responsible for the origin of the low–frequency unsteadiness. The two–points linear spectral analysis confirmed the occurrence of a slow–upstream convection
motion originating from the vicinity of the reattachment point
where a spot of high PSD content emerges. It has been shownthat it globally affects the dynamics of the flow. The low–frequency range in the separated region was found to be significantly quadratically linked with the oblique mixing layer
modes of much higher frequencies, as pointed out from bicoherence spectra. The biphase diagnostic then confirmedthat such a long–range quadratic link is indeed related to thedownstream-to-upstream fluidic feedback and from the bicorrelation analysis it is shown that is characterised by a negative
time delay.

ACKNOWLEDGMENTS
This work was supported by TEAMAero Horizon 2020
research and innovation program under grant agreement
860909. This work was granted access to the HPC resourcesof CINES and IDRIS under the allocations A0022A01877,
A0062A01877 and A0102A01877 made by GENCI.

REFERENCES
Bonne, N.and Brion, V., Garnier, E., Bur, R., Molton, P., Sipp,
D. & Jacquin, L. 2019 Analysis of the two-dimensional

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