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

EFFECTS OF FAVORABLE DOWNSTREAM PRESSURE GRADIENTS ON
SEPARATED SHOCK-WAVE/BOUNDARY-LAYER INTERACTIONS

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

Deepak Prem Ramaswamy,
Institute of Aerodynamics and Chair of Fluid Mechanics
RWTH Aachen University
52062 Aachen, Germany
d.ramaswamy@aia.rwth-aachen.deAnne-Marie Schreyer,
Institute of Aerodynamics and Chair of Fluid Mechanics
RWTH Aachen University
52062 Aachen, Germany
a.schreyer@aia.rwth-aachen.de

ABSTRACTsible effects induced by the downstream pressure gradient on
the low–frequency dynamics of the separation bubble have notyet been documented. The present study aims at closing thisgap in knowledge by jointly analyzing the experimental dataof a M∞ = 2.52, 24◦ compression-corner interaction followedby −24◦ expansion corner of Ramaswamy & Schreyer (2021)
with high-resolution large–eddy simulations (LESs).
We first computed the same compression corner–
expansion corner configuration as studied in the experiments.
This baseline computation is hereafter labeled as c24.0-e24.0.
Two additional computations with a single compression cornerwere also carried out. The first one uses the same 24◦ corneras the baseline case, while the corner angle has been reduced
to 21.7◦ for the second case to obtain the same separation location as in the c24.0-e24.0 case. These two computations arelabeled c24.0 and c21.7, respectively.

Shock-wave/boundary-layerinteractions(SWBLI)
strongly influence the aerodynamic behavior of many
aerospace- transportation systems. They have therefore beenstudied intensively – with different methods, configurations,and test facilities. Regions of favorable pressure gradientare often placed downstream of the SWBLI region. To beable to better understand their effect and compare cases fromvarying sources, a joint experimental–numerical study ofa supersonic compression/expansion corner flow is carriedout. A High–resolution long–time numerical simulation
complements particle-image velocimetry measurements, bothusing the same configuration and conditions. Derived setups
without the downstream expansion corner are also explored;
both for the same compression-corner angle and for a ramp
angle inducing a separation bubble of the same size as for theoriginal compression/expansion configuration. It is shown thatthe topology and low–frequency dynamics of the separation
bubble developing over the compression corner are alteredby the downstream expansion. POD analyses show that thegeometrical constraint downstream of the interaction region
results in a shortening of the “natural” modes but also in themerging of several modes, altering their time evolution.

EXPERIMENTAL AND NUMERICAL SETUP
Experiments
Experimental investigations were conducted in thetrisonic wind tunnel at RWTH Aachen University. Theintermittently-operating vacuum-storage facility achieves stable run times of 3–4 s. Air as the working fluid is supplied to asettling balloon. The air is dried to avoid condensation effects,
keeping the relative humidity below 6%. The 0.4 × 0.4 m2 test
section is optically accessible through two circular windows
on either side of the test section and one on the top wall. The
ambient conditions listed in Tab. 1 set the wind-tunnel stagnation conditions. The selected Mach number (M∞ = 2.52)
therefore determines the free-stream unit Reynolds number(Re∞ = 9.6 · 106 1/m).
For the two-component particle-image velocimetry (PIV)
measurements, we used a 532 nm Litron NANO-L pulsed
PIV laser with a maximum pulse energy of 200 mJ to illuminate the di-ethyl-hexyl-sebacate seeding particles in theflow. Two FlowSense EO 11M cameras (equipped with
4008 px×2672 px CCD sensors and Tamron SP AF 180mm
f/3.5 objectives) covered the same field of view, the complete
SWBLI region (−4.5δ ≤ x ≤ 4.5δ ; δ = 10.4 mm is the in-

INTRODUCTION
Regions of favourable pressure gradient are often located downstream of shock-wave / boundary-layer interaction (SWBLI), either by design for practical devices such as
air intakes or to prevent unstart of supersonic wind tunnels. If
the favourable pressure gradient is located in the close vicinityof a separated SWBLI, it may alter the size of the separation
bubble, as documented for instance experimentally by Grossman & Bruce (2018) or numerically by Duan et al. (2021).
Several studies have also documented the frequency content in
the favourable pressure gradient region, see for instance Grilli
et al. (2013) or Duan et al. (2021).
However, little attention has been paid to the frequency
link between the SWBLI and the favourable pressure gradient regions. Moreover, as far as the authors know, any pos-1

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

Flow parameters

LES
DNS Spalart (M~0, Reθ=1440)
Exp.

25

20

P0 (Pa)T0 (K)Reθδ (mm)

2.52

9.949×104

297

8225

10.4

y)
+

-1

uVD+

M∞

15
+

u

.4 1
=0

(
log

+5

.2

10

5

Table 2.

+

Simulation parameters

u =
+

0

100

y

101

y+

102

103

104

102

103

104

4

Case

Span

Cells

LF cycles

c24.0-e24.0

3.5δ

87 M

145

c24.0

7.4δ

181 M

55

c21.7

3.5δ

87 M

90

LES
DNS Spalart (M~0, Reθ=1440)
Exp.

+u’ Morkovin

3

2

1

0

-1

coming boundary-layer thickness), at a spatial resolution of
43 px/mm. The cameras were triggered alternately to increase
the overall acquisition rate to 6 Hz.

100

101

y+

Figure 1. Van Driest–transformed streamwise velocity profiles in wall unit (top) and velocity fluctuation profiles in
Morkovin Scaling (bottom) at location x/δ = −4.3

Large-eddy simulations
All computation were carried out using the FLU3M codewith second–order accuracy both in time and space. Time integration is implicit with a typical maximum CFL number of
11. The space scheme is hybrid centred–shock capturing discriminated by the Ducros’ sensor. The explicit subgrid scalemodelling relies on the selective mixed–scale model. A compressible variant of the synthetic eddy method (SEM) is used
to generate unsteady fully turbulent inflow boundary condition. The inflow boundary is located 16δ upstream of the separation corner for all computations in order to ensure a similar
development of the incoming boundary layer regardless of theextent of the interaction region.
For more details on the numerics see Jiang et al. (2017);
we derived the present mesh parameters from their study: forall computations, the inflow is located more than 10δ upstreamof the interaction region and the width of the computational
domain is set larger than 7 separation–bubble heights. The
size of the bubble varies between the cases, resulting in different extensions in the spanwise direction. The total cell counts
range between 87 and 181 millions, as shown in table 2.
The typical grid resolutions in wall units are set to ∆x+ =26.5 and ∆z+ = 10.5. In the wall-normal direction, the meshhas a typical stretching rate of 3%, starting from ∆y+wall = 0.7
with 170 cells within the boundary-layer thickness. The body–
fitted mesh is built using numerical conformal mapping.
All simulations have been run for 2.2 millions time steps,
thus providing data sampled over a physical duration of 0.22 s.
The corresponding number of typical low–frequency cycles
distributed over this duration varies, but is over 50 for all computations (see Tab. 2). This run time ensures good statisticalconvergence for all spectral estimators in all cases.

and skin friction differ by less than 3% from the experimental values and the overall profiles are in very good agreement(see Fig. 1). The main discrepancy is in the streawmise fluctuation profile above the log layer: the LES fails to reproduce
the secondary peak associated with very large structures thatis emerging in the experiments due to the high Reynolds number. The reason for this difference is that the inflow boundary
is located O(10δ ) upstream of the sampling station, i.e. not
far enough for structures with a larger length scale to sustain.
Next, we compare the PIV and the LES data for the reference case c24.0-e24.0. The global structure of the compression/expansion corner flow, visualized using the isolevels of
the streamwise velocity and streamlines, is shown in Fig. 2.
The LES accurately reproduces all flow features from the separation point to the relaxation region downstream of the reattachment – which is significantly altered by the expansion corner. The most noticeable discrepancy between experiment andsimulation is an underestimation by about 7% of the extent of
the separated region by the latter.
Such small differences in length can easily be compensated by rescaling the separated region with the actual interaction length, defined as the distance between the corner andthe extrapolation of the head shock down to the wall. The
relaxation region downstream of the reattachment, however,scales more naturally with the distance between the compression and expansions corners, which is identical for the experiments and the computation. The dual–lengthscale dependence is taken into account by considering a composite reference lengthscale for the computation: in the separated region,
the interaction length is considered. Downstream, from thereattachment point to the expansion corned, the length scale
is increased linearly to match the the experimental interaction
length at that location.
This way, the usual normalization is considered for the interaction region, and we are able to recover the same normalized location of the expansion corner in both the experiments
and the LES.
The normalized streamwise evolution of the mean andfluctuating velocity profiles is plotted in Fig. 3. It is clearly

Validation
The mean-velocity and Reynolds-stress profiles feeding
the SEM boundary conditions have been obtained from a preliminary LES of a M∞ = 2.52 flat–plate boundary layer undergoing transition. The location at which these profiles havebeen sampled has be adjusted so that the SWTBLI computation matches the boundary-layer parameters listed in Tab. 1 atlocation x/δ = −4.3. The resulting boundary-layer thickness

2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
0.8
0.7
0.6

y/L

0.5
0.4
0.3
0.2
0.1
0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.4

0.6

0.8

1

1.2

0.4

0.6

0.8

1

1.2

0.4

0.6

0.8

1

1.2

x/L

(a) u
0.8
0.7
0.6

y/L

0.5
0.4
0.3
0.2
0.1

Figure 2. Isocontour of streamwise velocity with superimposed streamlines. The red dashed line dénotes the 0-isolevelof streamwise velocity; PIV measurements (top) and LES simulation (bottom).

0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

x/L

(b) u′
0.8
0.7
0.6
0.5

y/L

visible that the LES accurately reproduces the flow developing
over the compression corner–expansion corner geometry. The
most noticeable difference occurs once again in the streamwise
velocity fluctuation profiles in Fig. 3(b), where the peak levelat the heart of the mixing layer developing over the separated
region is underestimated by about 15%. Such an underestimation was already observed in Jiang et al. (2017) and was traced
back through spectral analyses to the lack of very large structures in the incoming boundary layer. The same explanation
most probably holds here.
The validation process can be expanded to gain further information on the flow dynamics by comparing the Proper Orthogonal Decomposition (POD) of the 2D (u, v) velocity fields
obtained from the experiments and the LES. POD has been
computed using the snapshot method from 782 snapshots witha sampling rate of 6 Hz (experiments) and 88000 snapshotswith a sampling rate of 400 kHz (LES). Note that the highest available sampling rate has been used in the latter case to
make it possible to compute spectral estimates from the temporal modes over the full frequency range. Maps of normalized
2D turbulent kinetic energy are plotted in Fig. 4 for the six
most energetic modes found in the experiments (left column)
or the c24.0-e24.0 LES (center column).
The relative energy contents of the modes are also reported in Fig. 4. The modes obtained from the experiments
are carrying relatively more energy than their numerical counterparts. This difference may be explained with the differing
spatial extents of the domain over which the inner product iscomputed in the experiments and the numerical simulation (theupstream and downstream bounds of the experimental field ofview are denoted by dashed line onto the LES map). Moreover, levels of normalized turbulent kinetic energy (obtainedby dividing the unnormalized space modes by their respective
eigenvalues) are lower for the experiments than for the simulation (see Fig. 4). Hence, the unnormalized fluctuations associated with each mode are similar in absolute energy content.
The two first modes are similar in the experiments and the
LES, demonstrating that the dominant dynamics of the flow

0.4
0.3
0.2
0.1
0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

x/L

(c) v′
0.8
0.7
0.6

y/L

0.5
0.4
0.3
0.2
0.1
0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

x/L

p
(d) sign(u′ v′ ) × |u′ v′ |Figure 3. Streamwise evolution of the velocity profiles along
the interaction region: PIV measurements (◦) and c24.0-e24.0
LES (−−−−).

are well predicted by the simulation. The third modes sightly
differ in the location of the intermediate structure. This is mostprobably caused by the slight difference in size of the separated
region between the experiments and the computation, considering that the mode is driven both by the (variable) size of theseparation bubble and the (fixed) extent of the ramp.
Higher-order modes significantly differ. They occasionally bear a resemblance in the separated region, but differ further downstream. This behavior is possibly due to the difference in the separation length/ramp length ratio: compare for
instance modes 6 from the experiment and mode 5 from the3

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

Figure 4. Isocontours of the 2D turbulent kinetic energy for normalized POD modes obtained from snapshots of streamwise andwall–normal velocities: experiments (left), c24.0-e24.0 LES (middle) and c21.7 LES (right).
0.01

numerical simulation. We will show that the corner indeed induces compression as well as truncation in space for the modes
at the end of the next section.

0.008

c21.7
c24.0-e24.0
c24.0

Cf

0.006

0.004

RESULTS AND DISCUSSION
Equivalent compression corner flows

0.002

Comparing the c24-e24 computations with the two othercomputations, for which the downstream expansion corner has
been removed, helps assessing the influence of the downstream
favourable pressure gradient on the separated region. The
skin-friction and wall-pressure coefficient distributions are displayed in Fig. 5 for the three computations. They demonstratethat case c21.7 (reduced ramp angle) indeed results in the sameseparation location as the c24.0-e24.0 case, but that the lack of
a downstream favourable pressure gradient leads to a slightly
delayed reattachment. In contrast, the c24.0 case exhibits avery large separation region inducing a true pressure plateau.
The expansion corner in the nominal setup has indeed a significant impact on the mean development of the SWTBLI, leading
to a shrinking of the separation bubble by more than 30%.

0
-10

-5

0

5

0

5

x/δ
0.5

0.4

c21.7
c24.0-e24.0
c24.0

Cp

0.3

0.2

0.1

0
-10

-5

x/δ

Figure 5. Distribution of the skin friction coefficient (top)and wall pressure coefficient (bottom) for the three LESs.

Spectral analysis
On the basis of the scaling well acknowledged for turbulent SWBLIs (Dupont et al. (2006)), we expected that thec24.0-e24.0 and c21.7 cases would exhibit similar low frequency dynamics (associated with the breathing of the separated region), since they have the same interaction lengths.
Such dynamics are indeed evidenced by the wall pressure

spectrum near the separation point. Standard turbulent SWBLIs exhibit a clustering of the low–frequency power around a
Strouhal number based on the length of the interaction regionof StL ≃ 0.03, and both the c21.7 and c24.0 cases exhibit such4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
0.4

c21.7
c24.0-e24.0
c24.0

0.35

f * snormalised LF

0.3
0.25
0.2
0.15
0.1
0.05
0
10

-2

10

-1

10

0

10

1

StL

Figure 6. Normalized wall pressure power spectra at the beginning of the interaction region (x∗ = 0).

a clustering (see Fig. 6).
The c24.0-e24.0 case, however, has a narrower low frequency distribution whose centre is shifted up by 50% becauseof the disappearance of the energy content in the lower part ofthe low–frequency range. The lack of dynamic content at thelowest frequencies persists until the region downstream of the
expansion corner, as seen in the center plot of Fig. 7.
The streamwise distribution of the norm of the wall pressure cross–spectrum between the current location and the beginning of the interaction, normalised by the local pressure
variance to highlight the energetically prevailing linear coupling, is dominated by the frequency range over St = 0.03 (denoted by the horizontal dashed line). The single corner cases,on the other hand, exhibit significant linear coupling for frequencies below St = 0.03 downstream of the corner (locatedat x∗ = 1), as visible in the upper and lower maps of Fig. 7.
Figure 7. Streamwise distribution of the normalized crossspectrum between the local pressure and the pressure at location x∗ = 0 for the three LES computations.

Proper orthogonal decomposition
Although the low-frequency dynamics of the separated
region appear to be significantly affected by a downstream
favourable gradient exerted by the expansion corner, it is unclear how this gradient modifies the frequency–space structureof the interaction and downstream relaxation regions. Therefore, we computed time-resolved POD modes for the c21.7
case to complement the corresponding POD modes for thec24.0-e24.0 LES as a first attempt to relate flow entities coherent in space with given frequency distributions.
Turbulent kinetic energy maps for the six first modes obtained from the c21.7 computation are plotted in the right column of Fig. 4. The three first modes appear to be elongatedversions of the three first modes of the c24.0-e24.0 case, whilemodes 5 and 6 look similar to modes 5 and 4 in the experiments. It should, however, be considered that modes 2, 3 and
4 of the c21.7 case have to be associated with modes 1, 2 and 3of the c24.0-e24.0 case. This illustrates the difficulty to matchthe c21.7 and c24.0-e24.0 modes; it is challenging to infer thekind of distortion in space induced by the expansion corner.
To identify partial dynamics common to both cases, ajoint POD encompassing snapshots from both the c21.7 andc24.0-e24.0 LESs with the same full sampling rate has been
computed. As expected, modes obtained from the diagonalization of the time-correlation matrix are mostly dominated byfluctuations associated with one or the other of the single flow
cases and match the ones obtained when a single datadase isconsidered (see Fig. 4). Most often, however, flow patterns canalso be identified for the other case, yet with much lower energy. These “ghosts” are related to the flow patterns of anothermode, highlighting common dynamics.
Based on this methodology, modes 1 to 3 of the c24.0e24.0 case are more likely related to modes 2 to 4 of case c21.7,

Mode
8
Mode
9
Mode
10
Figure 8. Isocontour of 2D turbulent kinetic energy fromvarious modes of the joint POD: c24.0-e24.0 plane (left) andc21.7 plane (right).

especially for the latter pairs. Moreover, the approach allows
to clearly point out that the expansion corner causes modes 5
and 6 of the c21.7 computation (corresponding to modes 8 and
9 in the joint POD) to merge into the composite mode 4 (mode
10 in the joint POD), as evidenced in Fig. 8. “Ghosts” of casec24.0-e24.0 in joint modes 8 and 9 have the same structurein space as mode 10, whereas the “ghost” of the c21.7 LES
for mode 10 appears to be a blend of modes 8 and 9. Thesefindings may explain the previous observation that no mode
obtained from the experiments matches the c24.0-e24.0 mode
4, whereas c21.7 modes 5 and 6 have matching experimental
counterparts. The length scales of the latter two modes, whose
length naturally scales with the size of the separation bubble,
are compatible with the extent of the compression ramp in5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
80

frequency dominated modes or from a combination of the two.
Further refined analyses are required to clarify that point.

c24.0-e24.0 mode 1
c21.7 mode 1
c24.0-e24.0 mode 2
c21.7 mode 2

70
60

CONCLUSION AND FUTURE WORKS

Stl * s50

A joint experimental/numerical study has highlighted thata favourable downstream pressure gradient alters the lowfrequency dynamics of a separated SWBLI beyond the frequency scaling associated with a change in size of the separated region.
Comparison of PODs obtained for computations with andwithout the downstream expansion corner demonstrated thatthe corner induces distortions of the modes, either by contracting them or by merging several modes. It is still unclear, bywhich of these mechanisms the presence of the expansion corner results in the depletion of the very–low energy content.
Refined modal decompositions, more targeted towards
the frequency space, such as a linear stochastic estimation
built from the correlation between the wall pressure time series and the time coefficients of the POD modes or a spectral POD could help refine the analysis of the link between
the low-frequency breathing of the separated bubble and thedownstream relaxation region. Dynamic-mode decomposition
applied to the Schlieren-image sequences (see Schauerte &Schreyer (2018)) and a similar analysis applied to the LES
databases may also prove helpful.

40
30
20
10
0

10-2

10-1

100

StL
80

c24.0-e24.0 mode 2
c21.7 mode 3
c24.0-e24.0 mode 3
c21.7 mode 4

7060

StL * s

50
40
30
20
10
0

10-2

10-1

100

StL
80

c24.0-e24.0 mode 4
c21.7 mode 5
c21.7 mode 6

7060

StL * s50
40
30

ACKNOWLEDGMENTS

20This work was granted access to the HPC resources of
IDRIS under allocations A0062A01877, A0082A01877 andA0102A01877 made by GENCI.
This research was partially funded by the German Research Foundation DFG (Emmy Noether Programme, grant
SCHR 1566/1-1).

10
0

10-2

10-1

100

StL

Figure 9. Premultiplied power spectrum of the first POD
modes for the c24.0-e24.0 and c21.7 LES.
the experiments. For c24.0-e24.0, the slightly reduced lengthscale, due to the reduced separation length, is no longer compatible with the unchanged extent of the compression ramp,
and modes cannot be sustained without merging.
Once the links between modes have been identified, it ispossible to study their temporal behaviour by analyzing thepower spectra obtained from the temporal modes. Spectra obtained for the dominant modes are plotted in Fig. 9. The upper
plot shows that modes 1 for both computations have a similar frequency distribution and therefore cannot account for thedifference in the low-frequency region seen in Fig. 6. Modes
2 exhibit an upper phase shift between c21.7 and c24.0-e24.0
that could be compatible with what is seen in Fig. 6 but, as
mentioned previously c24.0-e24.0 mode 2 has more likely tobe related to c21.7 mode 3.
Indeed, as seen in the middle plot of Fig. 9, the frequency
range associated with c24.0-e24.0 mode 2 has a slightly better
match c21.7 mode 3 than with c21.7 mode 2, while the frequency ranges of the c24.0-e24.0 mode 3 and c21.7 mode 4
overlap almost perfectly. Lastly, the frequency distribution of
c24.0-e24.0 mode 4 is a mix of the distributions of the two
related modes of the c21.7 case. A net power loss results inthe low–frequency range for the c24.0-e24.0 single mode compared to the c21.7 pair of modes.
The lack of the lowest frequency range in the c24.0-e24.0
case thus appears to result from a shift of the POD modes
towards higher frequencies, from the disappearance of low–

REFERENCES
Duan, J., Li, X., X. Li, Xinliang & Liu, H 2021 Direct numerical simulation of a supersonic turbulent boundary layerover a compression–decompression corner. Physics of Fluids 33 (6), 065111.
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.
Grilli, M., Hickel, S. & Adams, N. A. 2013 Large-eddy simulation of a supersonic turbulent boundary layer over a compression–expansion ramp. International Journal of Heatand Fluid Flow 42, 79–93.
Grossman, I. J. & Bruce, P. J. K. 2018 Confinement effects on
regular–irregular transition in shock- wave–boundary-layerinteractions. Journal of Fluid Mechanics 853, 171–204.
Jiang, T., Schreyer, A.-M., Larchevêque, L., Piponniau, S. &Dupont, P. 2017 Velocity spectrum estimation in shockwave/turbulent boundary-layer interaction. AIAA Journal
55 (10), 3486–3498.
Ramaswamy, D. P. & Schreyer, A.-M. 2021 Control of shockinduced separation of a turbulent boundary layer using airjet vortex generators. AIAA Journal 59 (3), 927–939.
Schauerte, C. & Schreyer, A.-M. 2018 Design of a high-speed
focusing schlieren system for complex three-dimensionalflows. In Proceedings of the 5th International Conference
on Experimental Fluid Mechanics, pp. 232–237.

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