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

LARGE-EDDY SIMULATION OF A HYPERSONIC TURBULENTBOUNDARY LAYER OVER A CONE IN SUPPORT OF FOCUSED LASERDIFFERENTIAL INTERFEROMETRY (FLDI) MEASUREMENTS

Takahiko Toki
Victor C. B. Sousa
School of Mechanical Engineering
School of Mechanical Engineering
Purdue University
Purdue University
West Lafayette, Indiana 47906, USA West Lafayette, Indiana 47906, USAttoki@purdue.edu
vsousa@purdue.edu

Yongkai Chen
Giannino Ponchio Camillo
School of Mechanical Engineering
German Aerospace Center (DLR)
Purdue University
Institute of Aerodynamics and Flow Technology
West Lafayette, Indiana 47906, USAGöttingen, Germany
chen1305@purdue.edu
giannino.ponchiocamillo@dlr.de

Carlo Scalo
Alexander Wagner
School of Mechanical Engineering
German Aerospace Center (DLR)
Purdue University
Institute of Aerodynamics and Flow Technology
West Lafayette, Indiana 47906, USAGöttingen, Germany
scalo@purdue.edu
alexander.wagner@dlr.deder the evaluation of such flow fields valuable.
Convection velocities of over 2 km/s are present in hypersonic boundary layers in atmospheric flight conditions. Furthermore, in a turbulent state, relevant energy levels cascade
to flow field structures smaller than the boundary layer height,
yielding frequencies in the order of MHz. This set of conditions motivates the need for time-resolved high-fidelity flow
field simulation, and defines the requirements for flow diagnostic techniques.
Among the few techniques with such bandwidth for experimental evaluation of hypersonic turbulence is Focused
Laser Differential Interferometry (FLDI), a focusing line-ofsight technique capable of fine temporal and spatial resolution in the vicinity of the center plane of the probing volume(Parziale et al., 2012). The downside of this focusing ability
is the added complexity to the interpretation of the data, stemming from a sensitive length, which is dependent on the sizeof the convecting disturbances in the flow field. This way, theconversion of FLDI values into flow field density quantities
requires flow field information.
A promising approach to overcome this challenge is tomake use of a computational equivalent of the experimental FLDI procedures, as demonstrated by Lawson & Austin(2021). An accurate simulation of FLDI measurements, referred to as computational FLDI, or cFLDI, is possible by applying a ray-tracing algorithm to high-fidelity, instantaneous
three-dimensional flow data.
In this study, a high-fidelity large-eddy simulation is carried out to provide instantaneous data for future cFLDI studies,
virtually applying the experimental setup described in Camillo& Wagner (2022) to a fully turbulent hypersonic boundary

ABSTRACT
Large-eddy simulations (LES) of a hypersonic turbulent
boundary layer over a cone are carried out to provide instantaneous data for future computational Focused Laser Differential Interferometry (cFLDI) calculations. First, a precursor
Reynolds-Averaged Navier-Stokes (RANS) simulation is performed to provide mean-flow velocity and temperature profiles
as inlet boundary condition for the LES. A rescaling method
is used to achieve fully turbulent conditions as early as possible in the domain, resulting in an approximate recovery lengthof 5 cm. The LES simulations are based on a new QuasiSpectral Viscosity (QSV) closure of the filtered Navier-Stokes
equations Sousa & Scalo (2022a). Time series of local density fluctuations collected from two points taken 2 mm apart,mimicking typical FLDI measurements, show a strong correlation, with the data extracted downstream exhibiting the expected convective delay. Observation of instantaneous density
fluctuations in wall-parallel planes reveals that the density fieldin the buffer layer has a streaky structure, with large streamwise coherence, which disappears further away from the wall.
This results in a density power spectrum focused on lower frequencies near wall, with a broader energy distribution, including higher frequencies, in the outer layer. Future work will
focus on simulating higher Reynolds numbers and performing
cFLDI simulations on the LES data.

INTRODUCTION
The state of the boundary layer on high-speed flight vehicles is a paramount factor in their design and performance.
The extreme conditions imposed by near-wall turbulence ren-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1. List of referred experimental flow conditions (Wagner, 2014).

Rem [1/m]

4.1 × 106

p∞ [Pa]2129

T∞ [K]268

ρ∞ [kg/m3 ]0.0276u∞ [m/s]

2422

M∞

7.4
Figure 2. Schematic of the present QSV-LES of a hypersonicboundary layer over a cone. Contours in wall-parallel andcross-flow planes show streamwise velocity fields. A contour
in a side plane shows magnitude of density gradient.
by the Taylor-Maccoll inviscid solution (Taylor & Maccoll,1933), and those at the inlet are given by combining the TaylorMaccoll inviscid solution with a viscous solution for the coneboundary layer (Lees, 1956). At the wall, an isothermal andno-slip boundary condition is imposed with the wall temperature of 300 K. At the outlet, a homogeneous Neumann condition is imposed for all flow quantities. In addition, sponge
layers are used at the inlet, outlet, and upper boundaries. The
length of the sponge layers at the inlet and outlet are 3% of thetotal computational domain extent in the streamwise direction.
That at the upper boundary is 5% of the wall normal extent.
The number of grid points are Nx × Ny × Nθ = 4096 × 128 × 6.
Spalart-Allmaras (SA) model (Spalart & Allmaras, 1992) isused as RANS model. To simulate a boundary layer with turbulent transition, a trip is located at x = 0.3 m. The trip location is decided based on the experiment by Wagner (2014).
Only a part of the RANS computational domain is simulated using QSV-LES. Figure 2 shows the domain length forthe present QSV-LES, and figure 3 depicts the relative positioning between the present RANS and QSV-LES. The inlet of
the QSV-LES domain is located at x = 0.6 m, and its domain
length in streamwise direction Lx is 0.31 m. The azimuthal
angle is extended to 18 degrees. The number of grid points
are Nx × Ny × Nθ = 1280 × 128 × 112. The measurement pointof FLDI in the experiment is located at x = 0.814 m, and thecomputational domain for QSV-LES is designed to include themeasurement point. To generate a realistic inflow turbulence at
the inlet, the rescaling method is used (Urbin & Knight, 2001).
The inlet profiles of mean quantities are given by the RANS results at x = 0.6 m. Turbulent fluctuations for 0.83 ≤ x ≤ 0.85
are extracted, and the fluctuations are scaled and imposed upon
the mean quantities for 0.6 ≤ x ≤ 0.62. The distance between
the recycling box and the inlet is 0.22 m. This length is equivalent to 51δ99 at x = 0.6 m, where δ99 is 99% of boundary layer
thickness.

Figure 1. Sketch of the present hypersonic boundary layer
configuration.

layer.

PROBLEM FORMULATION
Computational setup
The numerical study performed in this work is basedon experiments conducted in the DLR High Enthalpy Shock
Tunnel Göttingen, HEG (Wagner, 2014; Hannemann et al.,2018). The present configuration is a hypersonic boundary
layer over a 7 degree half angle cone, and test condition of
Rem = 4.1 × 106 m−1 is chosen. The same flow was simulated by Sousa et al. (2019) but in an axisymmetric fashion
only. In table 1, Rem ≡ ρ∞ u∞ /µ∞ is the Reynolds number per
meter. The subscript ∞ indicates free stream conditions. Intable 1, p is pressure, T is temperature, ρ is density, u is velocity, and M is Mach number. Figure 1 provides a sketch of
the present configuration. In the experiment, a conical shock is
attached to the tip of the cone and a boundary layer is formedon the cone surface under the shock. The laminar boundary
layer starts from the tip and transition to turbulence happens
further downstream on the surface. In this study, the turbulent
region under the shock is simulated using a new LES technique called the Quasi-Spectral Viscosity (QSV) Large-Eddy
Simulation, described in Sousa & Scalo (2022a).
To impose physical turbulent inlet boundary conditionsin the QSV-LES, a preliminary axisymmetric ReynoldsAveraged Navier-Stokes (RANS) simulation is carried out.
The inlet of the computational domain is located at x = 0.045
m from the cone tip, where x is a streamwise coordinate. The
domain length in streamwise direction Lx is 1 m. The domain
length in the wall-normal direction Ly is 0.0022 m at the inlet and 0.05 m at the outlet. These lengths are decided sothat the upper domain is located under the shock. The azimuthal extension is 1.5 degrees. At the inlet and the upperboundary, a Dirichlet boundary conditions are imposed. The
flow properties at the upper boundary are analytically derived

Numerical method
In the QSV-LES technique, Favre-filtered Navier-Stokes
equations are solved via a six-order compact finite difference
code originally developed by Nagarajan et al. (2003) and now
under continued development at Purdue. The QSV approachby Sousa & Scalo (2022a) is used for turbulence closure, andthe present QSV-LES is the first practical case where this approach is used. The approach was also developed to be appli-2

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

Figure 3.

Positional relationship of computational domains between the present RANS and LES.

2.5
Normalized heat flux

cable to unstructured grids (Sousa & Scalo, 2022b). The time
integration is carried out by a four-stage third order strong stability preserving (SSP) Runge-Kutta scheme (Gottlieb, 2005).
In order to ensure time stability, the conservative variables
are filtered using sixth-order compact filter described by Lele
(1992). Its filter coefficient is 0.495.

RESULTS
Flow statistics
Since the mean quantities at the inlet are given by the
RANS results, a recovery process from the RANS profiles tothe real turbulent boundary profiles should happen from the inlet. To confirm that a fully developed boundary layer is formed
after the recovering process, flow statistics are examined.
Figure 4 compares wall heat flux profiles of the present
RANS and LES with the experiment (Wagner, 2014). The wallheat fluxes are normalized by their laminar value at x = 0.232
m. The heat flux from the QSV-LES has a sharp peak aroundx = 0.62 m and thereafter a plateau develops for 0.65 ≤ x ≤0.85. In the plateau region, the QSV-LES heat flux is comparable with the experimental one. The sharp peak around x = 0.62
m implies the recovering process happens around this region,and the plateau implies the real turbulent boundary profiles areformed after x = 0.65 m.
To investigate whether the flow field reaches the real turbulent profiles after x = 0.65 m, figure 5 shows mean profilesof velocity and temperature, and figure 6 depicts velocity fluctuation correlations. The mean profiles and fluctuation correlations are calculated by averaging in the azimuthal direction
f indicates Favre average; the notation (·)′′ indiand time; (·)
cates fluctuations about the Favre average. The velocity fluctuation correlations are computed asu′′ rms =q′′ u′′ , v′′
ug
rms =

q
q
′′ v′′ , w′′′′ w′′ .
vg
=
wg
rms

2
1.5
1
RANS
QSV-LES
Experiment0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x[m]

0.5

1

Figure 4. Comparison of wall heat flux profiles between
RANS, LES and experiment (Wagner, 2014). The black
dashed and dash-dotted lines represent turbulent and laminarheat flux correlations respectively.
collapse on to each other. The agreements of the velocity andtemperature implies that the boundary layer does not evolve
convolutedly in the present simulation. However, the transformed velocity profile reveals that the data at x = 0.625 m
deviates from those at the other locations. The deviation at
x = 0.625 m indicates that self-similarity does not perfectly
hold, and the recovery process is not completed at this point.
On the other hand, the profiles after x = 0.7 m agree well witheach other, indicating that the flow field becomes self-similar
after this point and therefore the real boundary layer profiles
are formed at x = 0.7 m and thereafter. This consideration
is supported by the velocity fluctuation correlations shown infigure 6. The velocity fluctuation correlations in streamwise
and azimuthal directions at x = 0.625 m are slightly smaller
than the other locations around y/δ99 = 0.1. The smaller correlations indicate that turbulence is not fully developed at thispoint, and this result is consistent with the transformed velocity profiles. In addition, the correlations after x = 0.7 m agree
well in all directions and thus the boundary layer turbulence isfully developed after x = 0.7 m.

(1)

The mean velocity is normalized by the boundary layer edge
value, and the mean temperature is normalized by the wallvalue. Figure 5 also includes transformed velocity profiles
by Trettel & Larsson (2016) transformation law. The velocity fluctuation correlations in figure
p 6 are normalized by thesemi-local friction velocity u∗τ ≡ τw /ρ, where τw is wall
fluid shear stress, and (·) indicates averaging in the azimuthal
direction and time.
The mean velocity monotonically increases from the wallat any x−location, and all profiles collapse to each other. The
mean temperature increases from the wall to y/δ99 ≈ 0.05 andfurther decreases to the far field. All temperature profiles also

Local density fluctuations
To discuss computational FLDI signal, local density fluctuations are extracted from the present LES flow field. Figure
7 shows time history of local density fluctuations at different ylocations; y =1, 2 and 3 mm. The density fluctuations are collected at the center in azimuthal direction and at two different
x locations; x = 0.8139 and 0.8161 m. The distance between

3

1

1

0.8

0.8
0.6

0.2

0.2
0.5
ũ/ue

0

1

0.2

0.4

0.6
y/δ99

0.8

1

1.2

0

0.2

0.4

0.6
y/δ99

0.8

1

1.2

0

0.2

0.4

0.6
y/δ99

0.8

1

1.2

0.5

1.5

1 1.5 2 2.5 3
T̃ /T wx = 0.625[m]x = 0.65[m]x = 0.70[m]x = 0.75[m]x = 0.80[m]

1
0.5
0

Figure 6. Velocity fluctuation correlations in streamswise(top), wall-normal (middle), and azimuthal directions (bottom). The legend is the same as figure 5.

u+T L

40
35
30
25
20
15
10
5
0

0

0

1
0

′′

0

′′

0.4

vrms /u∗τ

0.4

5
4
3
2
1
0

1.5

wrms /u∗τ

0.6

′′

1.2
urms /u∗τ

1.2

y/δ99

y/δ99

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

0.015

x = 0.8139[m]x = 0.8161[m]

1

10

y∗

100

ρ′ [kg/m3]

0.01

1000

0.005
0
-0.005-0.01

Figure 5. Mean profiles of streamwise velocity (upper left),temperature (upper right) and Streamwise velocity transformed by Trettel & Larsson transformation (Trettel & Larsson, 2016) (bottom).
ρ′ [kg/m3]-0.015ρ′ [kg/m3]

the two locations is around 2 mm, and this value is comparablewith the distance between the two independent FLDI system inthe experiment (Camillo & Wagner, 2022).
The comparison of the density fluctuation histories between the two different x−locations shows that their time histories are similar to each other, and the density fluctuation at
x = 0.8161 m is slightly delayed. This similarity is observed at
all y locations. The clear similarity of their time histories indicates that cross-correlations between the different x−locations
can provide an accurate estimate of the time lag. Therefore,
this result implies that local velocity can be measured by analyzing density fluctuation signals at two different points inexperiments.
On the other hand, the comparison of the density fluctuation histories between the different y−locations reveals thatdensity fluctuates more rapidly further away from the wall. To
elucidate the reason of the different frequency, instantaneous
snapshots of density fluctuations are shown in wall-parallels infigure 8. The density fluctuation at y = 1 mm exhibits streaky
structures. The structures are long in streamwise direction andthe density slightly varies in the streaks. Thus the time scaleof the density fluctuation becomes long at y = 1 mm. The contour at y = 2 mm also has streak structures, but their lengths
are shorter than those at y = 1 mm. Furthermore, streaky structures no longer exist at y = 3 mm, and the density is finely fluctuating in the streamwise direction. Since the time scale of thedensity fluctuation accompanies the structures of the density

0.02
0.015
0.01
0.005
0
-0.005-0.01-0.015
-0.02
0.02
0.015
0.01
0.005
0
-0.005-0.01-0.015
-0.02

0

5

10

15
t[µs]

20

25

30

0

5

10

15
t[µs]

20

25

30

0

5

10

15
t[µs]

20

25

30

Figure 7. Time history of local density fluctuations at different y locations; y = 1 mm (top), y = 2 mm (middle), and y = 3
mm (bottom).
field, the time scale also becomes shorter further away fromthe wall.

Conclusion
A large eddy simulation of a hypersonic boundary layerover a 7 degree half angle cone has been performed to compare
its results with the FLDI setup described in Camillo & Wag-4

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

(a)

(b)

(c)
Figure 8. Instantaneous snapshot of density fluctuations in wall parallel planes at y =1 mm (a), 2 mm (b), and 3 mm (c). X denotes acoordinate from the tip of the cone along its axis.
ner (2022). Reynolds number per meter Rem is 4.1 × 106 m−1 .
To impose reasonable turbulent inlet boundary conditions, therescaling method described by Urbin & Knight (2001) is used.
This paper focuses on computational results in the present
LES.
The velocity profiles with the transformation by Trettel &Larsson (2016) and velocity fluctuation correlations become
self-similar 0.1 m downstream from the inlet. The measurement point of FLDI in the experiment is located 0.214 m downstream, and thus the self-similarities indicate that real turbulent
boundary profiles are formed at the FLDI point.
The time histories of local density fluctuations at two
points around 2 mm apart reveal their strong correlations, withthe data collected in the downstream side being slightly delayed. This result implies that local velocity can be measured
by analyzing cross-correlations between these density fluctuation signals. In addition, The comparison of the time histories between the different y− locations shows that the density
fluctuate more rapidly further away from the wall. Examination of wall-parallel planes reveals that the density fluctuate
slowly near the wall because the density field has long streaky
structures in the streamwise direction. On the other hand, thestreaky structures do not persist away from the wall, and thedensity largely varies in the streamwise direction. The difference of the density flow field induces the different frequencyof the density fluctuation. The results of the present QSVLES will be compared with density fluctuations and convection velocity estimates obtained with the FLDI setup describedin Camillo & Wagner (2022) in future research.

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