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

SWEEP EFFECTS ON A CANONICAL SHOCK WAVE/TURBULENTBOUNDARY LAYER INTERACTION

Thomas Bergier(*), Romain Gojon, Jérémie Gressier, Stéphane Jamme and Laurent Joly
ISAE-SUPAERO, University of Toulouse, France(*)thomas.bergier@isae-supaero.fr

ABSTRACTThis study focuses on the effect of a sweep angle between
the upstream mean flow direction and the shock plane normal
in a canonical oblique shock wave/boundary layer interaction.
We study a configuration based on the experiments of Bo et al.
(2012): an incident shock impinges on a turbulent boundary
layer developing at free stream Mach number M = 2.7 anda Reynolds number Reθ = 3200. The sweep angle is implemented by adding a spanwise velocity component (crossflow)at the inflow and enforcing periodicity conditions on the sideboundaries. Several sweep angles are investigated and compared to a canonical unswept case. A viscosity is selected foreach case in an effort to keep a constant Reθ across the whole
range of sweep angles. The mean flow structure is analyzed,
showing a strongly skewed flow around the separation zonethat increases in size compared to a reference unswept case.
Pressure loads on the wall surface are also higher for swept
cases; and the characteristic low-frequencies of the flow separation also slightly increase.

Figure 1: Schematic representation of a swept configura-tion of SBLI and associated computational domain. Side
view (top), top view (bottom).
INTRODUCTION
Shock/boundary layer interaction (SBLI) is a phenomenon commonly found in supersonic airflows. Several different configurations have already been studied (Clemens &Narayanaswamy (2014)) over the last decades among which
the interaction of an oblique shock impinging on a turbulent boundary layer (TBL) developing on a flat plate. This
setup is known to lead to the formation of a separation bubble, due to the adverse pressure gradient created by the presence of the shock, as well as a “reflected” shock upstream ofthis bubble. This reflected shock/bubble system exhibits low
frequency streamwise oscillations which can be the cause of
detrimental pressure and load fluctuations. Efforts have been
made to reduce the amplitude of these oscillations using control devices, among which micro-vortex generators. These devices and their impact on impinging SBLI configuration havebeen the focus of a recent work (Grébert et al. (2018)).

NUMERICAL SETUP
The present LES were performed with the IC3 solver,which solves the spatially filtered compressible Navier-Stokes
equation using, in our case, spectral difference methods
applied on an unstructured mesh for the spatial discretization.
Spectral difference methods (SD) are based on a polynomial
representation of the solution inside the cell and of thenumerical flow at the interface between cells (Liu et al.
(2006)). Moreover, each control volume of the mesh isdivided into several “spectral points” (also called “degrees of
freedom”, or “nDofs”) where the solution is given and whose
number depends on the order of the scheme used. A 4rd order
spectral scheme is used in the present work for the spatial part,
meaning that each control volume (cv) contains 4 spectral
points in each direction, giving a total of 64 solution points
by cv. The temporal integration uses a 3rd order Runge-Kutta
method and no subgrid scale model is used. In order to deal
with discontinuities, we use a limiter specially developed forhigh-order compact numerical schemes such as SD methods,
named SWeP (Spatially Weighted Projections). This limiter isbased on a reduction of the order of the polynomial used todescribe the solution at the cells affected by discontinuities(Lamouroux et al. (2016)).
The size of the computational domain in our simulations
corresponds to Lx × Ly × Lz = 59.6δi × 17.95δi × 8.54δi , δibeing the boundary layer thickness at the inlet of the domain

Even though SBLI and different control strategies havealready been studied , most of these previous works were doneon “canonical” setups, i.e cases not accounting for potential
three-dimensional effects such as curvature of the wall or asweep angle of the shock plane with respect to the incoming flow. These would however be more representative of apractical configuration of SBLI featuring vertical fins or swept
ramps for instance. The objective of the present work is to address this question by investigating the effect of a sweep angle
as shown in figure 1 using Large Eddy Simulations (LES).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 2: Parameters for the different swept case upstream of
the interaction.

Table 1: Physical parameters of the simulation

M∞

T∞

P∞

u∞

ψ

2.7

300 K

101 kPa

937 m/s30.3◦

β (◦ )w∞ (m/s)µ∞ (Pa · s)Reθ (x = Xint )

Mx

0

0

4.5 × 10−5

3371

2.7

251.274.681 × 10−5

3367

2.7

341.324.812 × 10−5

3381

2.7

15

(δi = 1.12 × 10−3 m). The grid is uniform along spanwise (zcoordinate) and longitudinal (x coordinate) directions, except
in the region near the interaction where the longitudinal sizeof the cells is reduced to better handle the strong adverse
pressure gradient. With this meshing strategy, we have+∆+x = [11.5; 16.8] and ∆z = 10.6. The grid in the wall-normal
direction (y coordinate) is gradually refined towards thebottom wall with ∆+y = [0.6; 72.1]. As we do not use any wallmodel, this level of precision is needed near the wall to be able
to resolve accurately the viscous sub-layer of the boundary
layer. The mesh results in 1384 × 216 × 260 nDofs in x, y andz directions respectively, giving a total of around 77.7×106
nDofs.

20

topology of the interaction area depending on the value ofthe sweep angle (Padmanabhan et al. (2021)). For low sweep
angles, a cylindrical symmetry is expected, i.e separation
and reattachment lines are parallel, whereas for higher sweep
angles, conical symmetry appears, meaning that the separation
and reattachment lines are no longer parallel. As previously
mentioned, periodic boundary conditions are imposed in
the simulations presented in this work, preventing conicalsymmetry of the interaction area, and making it irrelevant tostudy these cases with our current numerical setup. According
to Padmanabhan et al. (2021), the limit between cylindricaland conical symmetries for their configuration, which is quite
close to ours, is 24.7◦ . This is why we chose to study therange of sweep angles presented in Table 2, all belonging
to the cylindrical symmetry. It can also be noted that theMach number given in Table 2 is the axial Mach number(Mx = u/c0 ) : the added spanwise velocity component has noinfluence on the shock itself as its plane is aligned with the
spanwise direction and tangential velocity is conserved. Thus,
the incident shock features (angle ψ and flow deviation) are
the same regardless of the sweep angle.
It should also be noticed that as upstream velocity magnitude
is increased to reach the desired sweep angle, the viscosityof the fluid is also increased, with the aim to keep a constantReynolds number (as well as constant axial Mach number)
across our sweep angle range. This can be verified in Table
2 that reports very close values for Reθ measured at theinteraction location between our different cases.

The SBLI configuration selected in this work is the one
studied by Grébert et al. (2018) and is based on Bo et al.
(2012) experiments. The physical parameters used in the inletof the computational domain, as well as the shock features(angle of shock ψ), are summarized in Table 1. Regarding
the boundary conditions (see figure 1 for their location), an
adiabatic wall is used on the bottom boundary, a symmetry
condition is used at the shock generator so that no boundary
layer develops here and, on the upper boundary, value ofthe primitive variables of the flow are imposed. For theoutlet, in the supersonic area (outside of the boundary layer)
variables are extrapolated. For the subsonic area (inside theboundary layer), the value of static pressure is imposed equal
to the mean of the pressure retrieved at the outlet for thesupersonic area. At the inlet, the production of turbulenceis achieved thanks to a “digital filtering” technique basedon a filtering operation applied to statistical data (Touber& Sandham (2009)). It was first introduced by Klein et al.
(2003) and then modified by Xie & Castro (2008) and Touber& Sandham (2009). This method presents the advantage
of not introducing any artificial low-frequency mode, as
opposed to recycling or synthetic turbulence techniques. Thisis important in the case of SBLI as this phenomenon is known
for displaying low-frequency dynamics. However, turbulence
obtained thanks to digital filtering needs a certain length ofdevelopment, around [15-20δi ], to reach a physical state.

Boundary layer validation
Before carrying out the swept SBLI computations, thestate of the upstream turbulent boundary layer has to beassessed. Indeed, we have to make sure that the synthetic
turbulence created at the inlet has developed well and that
it is consistent between our different cases. The turbulent
boundary layer characteristics are shown in figure 2 forthe different sweep angles and compared to DNS data or
correlations from the literature. The value of incompressible
friction coefficient C fi is obtained thanks to the van Driest II
transformation (Van Driest (1956)) :RESULTS
The aim of these simulations is to compare the behaviourof the mean flow, as well as the unsteady behaviour of theinteraction system for different sweep angles. The sweep
angles (β ) considered in this work are given in Table 2. The
simulations are performed with what is called in the literature“infinite sweep”, meaning that sweep is obtained introducing
a spanwise (crossflow) velocity component for the incoming
flow and using periodic conditions at the spanwise boundaries.
Thus, in our computational domain, the shock plane stays thesame as in the unswept case but the upstream flow is modified(figure 1). It should be noted that in the flow configurationof a swept shock impinging on a boundary layer, two flow
configurations can be observed experimentally regarding the

Tw /T∞ − 1arcsin2 α
Tw /T∞ − 1and α = p
Tw /T∞ (Tw /T∞ − 1)

C fi = Fc C f with Fc =(1)

with Tw the wall temperature. This allow to compare
our data, obtained from compressible simulation, withincompressible data or correlations, like Karman-Schoenherr

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
(KS) or Blasius (BL) (Hopkins & Inouye (1971)) :C fKS =

1
17.08 log(Reθi )2 + 25.11 log(Reθi ) + 6.012
0.026µ∞
C fBL =
, with Reθi =
Re
1/4
µw θ
(Reθi )

(2)

µw being the dynamic viscosity at the wall.

Figure 3: Time-averaged skin-friction lines superimposed toa colormap displaying the gap between the reference angle
and the actual computed angle arctan(< w > / < u >) at thebottom wall. From top to bottom : β = 0◦ , β = 15◦ , β = 20◦ .

Figure 2: Top : Time and spanwise-averaged incompressible friction coefficient, based on longitudinal andspanwise velocity components (u and w) over the bottom
wall, upstream of the interaction. Bottom : Reynolds
stress tensor components (spanwise-averaged), values
are made dimensionless by taking into account u and w.

The shift in Reθi observed on the upper image of figure 2
when the sweep angle increases can be explained by the factthat as β goes up, µe increases more than does µw (which is
obtained from Tw with a power law). The development lengthof the turbulence can clearly be seen looking at the undershootof C fitot near the inlet of the domain. More downstream, allcurves can be found at reasonable C fi levels compared to thecorrelation, and pretty close to each other. On the bottom figure, the Reynolds stress tensor components are globally close
to the DNS results. Differences can however be observed, particularly for the < u0 v0 > component which is overestimatedover the whole profile compared to the reference data.

Figure 4: Top : Evolution of C p = (Pw − P∞ )/(γP∞ Mx2 /2),

time-averaged pressure coefficient. Bottom : evolution of
C fx , time-averaged x-wise skin friction coefficient aroundthe separated area. The horizontal dashed line corresponds
to C fx = 0, used to delimit the extent of the separation
bubble.

the prescribed sweep angle and the actual angle of the flow
with respect to the longitudinal (x-wise) direction. This measure was chosen because it provides a clear visualization of theseparated area. This zone seems to grow in size as the sweep
angle increases. This observation is comforted by the study ofthe x-wise skin friction coefficient (defined using the longitudinal component of the wall shear stress) along the bottom wall
which shows a greater separated area (zone where C fx < 0)

SBLI results and sweep effects
Regarding the SBLI simulations, the mean flow is studied first, especially near the interaction zone. Time-averaged
skin friction lines are presented in figure 3, where they are superimposed to a colormap displaying the difference between

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
by theses curves that also show a complete reversal of the flow
direction (obtained when arctan(< w > / < u >) > 90◦ ) inside
the separated region for the layers of the flow the closest to thewall.

Figure 5: Maps of time-averaged longitudinal velocity. Incident
and reflected shocks locations (extrapolated inside the boundary
layer) are marked with dashed lines. Mean separation location is
located with white circle and mean reattachment point with whitesquare. Scale is the same for the 3 maps and ranges from u = 0
to u = u∞ . From top to bottom : β = 0◦ , β = 15◦ and β = 20◦ .

Figure 6: Streamlines for different wall-normal altitudes, ranging
from the wall (skin-friction lines are plotted in that case) to y =δi . Top : β = 15◦ , bottom : β = 20◦

when the sweep angle increases, as seen in figure 4 (bottom).
A possible explanation for this phenomenon, also presented
by Larsson et al. (2022), is that the wall being adiabatic andthe overall Mach number increased by the added swept, thewall temperature at a given longitudinal location is higher fora higher sweep angle. This induces a decreased near-wall density, and thus decreased near-wall inertia, leading to an upstream shift of the separation line. The different separation
lengths and wall average temperatures (given for x = 35δi , ata location just upstream of the interaction) are specified in Table 3. As it stands, we cannot ensure that temperature alone
plays the dominant role in the growing of the separated area.
Indeed, a study involving constant wall temperature across thewhole sweep angle range would be needed to discriminate thedifferent possible effects (see Larsson et al. (2022) for sucha study). In addition to the longitudinal size of the separation bubble, one can take an interest in the wall-normal size of
said bubble. Figure 5 shows maps of time-averaged longitudinal velocity (u) in the x − y plane around the interaction zone.
The separation bubble is found to grow slightly in size as thethe sweep angle increases, even if the differences remain verysmall and are not easy to discriminate on the figure.
In order to investigate the behaviour of the flow awayfrom the wall, we present in figure 6 the streamlines at foury-locations distributed across the thickness of the inlet boundary layer (i.e. for 0 < y < δi ) for the the swept cases β = 15◦
and β = 20◦ . It can clearly be seen that the flow is highly deviated near the separation and reattachment lines, and that thecloser to the wall, the more important the deviation. This leads
to a strongly skewed flow around the separation zone, whichmakes it prone to develop flow instabilities in this region. Further downstream, the flow tends to recover towards the expected post-shock direction, beyond x/δi = 50, even thougha full recovery is not reached before the end of the computational domain. Figure 7 displays the streamwise evolution of
the angle of the flow with respect to the x axis (i.e. arctan(<w > / < u >)). The above-mentioned remarks are confirmed

Figure 7: Flow angle with respect to x axis (given by arctan(<w > / < u >)) for different wall-normal altitudes, ranging fromthe wall (angle of skin-friction lines are plotted in that case) toy = δi . The horizontal full line situated at 90◦ allow to see where
the flow is reversed. Horizontal dashed line shows the upstream
imposed sweep. Top : β = 15◦ , bottom : β = 20◦ .

As seen on figure 4 (bottom), both the separation andreattachment points move (upstream and downstream respectively) when sweep is added. Moreover, the difference of Lsep4

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 3: Comparison between separation length and temperature upstream of interaction

β (◦ )

Lsep /δi

Tw (x = 35δi )/T∞

0

4.84

2.18

15

5.01

2.25

20

5.24

2.31

is greater between 20 and 15 degrees than the one seen between 0 and 15 degrees, despite a similar increase in wall temperature in both cases.
In addition to the effect on mean flow properties, it can be
expected that the introduction of a sweep angle could also havean impact on the unsteady dynamics of the interaction. Figure 8 shows maps of spanwise-averaged Power Spectrum Densities (PSD) of wall pressure fluctuations. On the upper figure, the characteristic spectrum of a canonical unswept SBLI
can be observed with low frequencies near the separation line.
These frequencies are typically found to be two orders of magnitude below the ones associated with the upstream turbulent
boundary layer. According to Piponniau et al. (2009), who developed a model for the low frequency unsteadiness of SBLI,a value of Strouhal StlLsep = f · Lsep/U∞ = 0.03 should beobserved for a flow with an upstream Mach number above 2.
Downstream of the interaction, intermediate frequencies tend
to dominate and are linked with shear layer vortices developingat the edge of the separation bubble. These observations are inline with several previous numerical and experimental studies(Dupont et al. (2006), Touber & Sandham (2009), Priebe &Martn (2012), Grilli et al. (2012)). When a spanwise velocity
is added, peaks associated with higher frequencies appear near
the average separation location. As seen on figure 8, values of
StlLsep of 0.1, 0.18 and 0.2 exhibits high energetic content. For
β = 20◦ , we can moreover see a shift of the lowest-frequency
peak from StlLsep = 0.03 to StlLsep = 0.05.
The intensity of the fluctuating pressure loads at the wallaround the interaction area is also of particular importancein practical applications. To quantify this, we use a metric
introduced by Grébert et al. (2018), defined as follows:q
IF0 S (x) =FS02 (x)hFS i (x)Zwith FS (x,t) =z

Pw (x, z,t)ds

Figure 8: Spanwise-averaged PSD spectra of wall pressure fluctuations. From top to bottom : β =R 0◦ , β = 15◦ , β = 20◦ .
PSD are normalized by local integral PSD(x, f ) d f . xsep andxr stands for the mean locations of separation and reattachment respectively. The horizontal dashed lines are located at
StlLsep = 0.03 and StlLsep = 0.06

(3)

Figure 9 shows the evolution of IF0 S (x) along x axis forthe 3 sweep angles studied. It can be observed that adding a
spanwise velocity component induces higher intensity of pressure loads, even though the maximum level presented here forthe β = 15◦ case is higher than the β = 20◦ one right after
the reflected shock (around x/δi = 38). Further downstream,
the higher the sweep angle, the higher the IF0 S (x). The effectof a crossflow is thus potentially detrimental for the structuralloading of the surfaces involved in this type of configurations.
Figure 9: Streamwise evolution of the pressure loads intensity
IF0 S̄ (x) for the 3 sweep angles.

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

ference method for unstructured grids i: Basic formulation.
Journal of Computational Physics 216 (2), 780–801.
Padmanabhan, Sathyan, Maldonado, Jorge C, Threadgill,
James AS & Little, Jesse C 2021 Experimental studyof swept impinging oblique shock/boundary-layer interactions. AIAA journal 59 (1), 140–149.
Piponniau, S., Dussauge, J. P., Debive, J. F. & Dupont, P.
2009 A simple model for low-frequency unsteadiness inshock-induced separation. Journal of Fluid Mechanics 629,
87108.
Priebe, Stephan & Martn, M. Pino 2012 Low-frequency unsteadiness in shock waveturbulent boundary layer interaction. Journal of Fluid Mechanics 699, 149.
Touber, Emile & Sandham, Neil D 2009 Large-eddy simulation of low-frequency unsteadiness in a turbulent shockinduced separation bubble. Theoretical and Computational
Fluid Dynamics 23 (2), 79–107.
Van Driest, Edward R 1956 On turbulent flow near a wall.
Journal of the aeronautical sciences 23 (11), 1007–1011.
Xie, Zheng-Tong & Castro, Ian P 2008 Efficient generation of
inflow conditions for large eddy simulation of street-scale
flows. Flow, turbulence and combustion 81 (3), 449–470.

In this work, several LES simulations have been
conducted in order to compare swept and canonicalshock/boundary layer interaction configurations. Results showthat the sweep angle affect both the average flow and the unsteady low-frequency dynamics of the SBLI. The size of separated flow grows with the sweep angle, which can be linked
to an increased wall temperature due to the added transverse
velocity and the adiabatic nature of the wall. Around the separated bubble, the flow is highly skewed for the swept cases,
which is favorable to the development of instabilities. It thengently recovers the expected post-shock direction. Pressureloads on the wall surface are higher for swept cases, which canbe detrimental in real-life configurations such as supersonic
air-intakes. The well-known unsteadiness of SBLI configurations is also altered by the presence of sweep, its characteristic
low-frequency being slightly shifted to higher values.

ACKNOWLEDGEMENTS
This research was supported by the French Ministryof Defense through the financial support of the DGA(grant #2020-650026). This work was performed using resources from GENCI [CCRT-CINES-IDRIS] (Grant 2021[A0102A07178] and 2022-[A0122A07178]).

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