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

SHOCK-WAVE AND SHEAR-LAYER OSCILLATIONSOVER A DOUBLE CONE IN HIGH-SPEED FLOW

Vaisakh SasidharanDepartment of Aerospace Engineering
Indian Institute of Science
Bengaluru 560012, Indiaăvaisakhs@iisc.ac.inąGaurav Kumar
Department of Aerospace Engineering
Indian Institute of Science
Bengaluru 560012, Indiaăgauravkumar2@iisc.ac.iną

Akshaya G. Kumara
Department of Aerospace Engineering
Indian Institute of Science
Bengaluru 560012, Indiaăakshayag@iisc.ac.iną

Subrahmanyam Duvvuri
Department of Aerospace Engineering
Indian Institute of Science
Bengaluru 560012, Indiaăsubrahmanyam@iisc.ac.iną

(b)

(a)
y

Small-amplitude shock-wave oscillations in a canonicaldouble-cone geometry are investigated by wind tunnel experiments and large-eddy simulation (LES) at Mach 6 flow conditions. The second cone angle is fixed at 90˝ while the first coneangle and the ratio of the slant heights are varied as parameters.
Two distinct type of flow oscillations are observed and characterized using time-resolved schlieren data from the experiments. The Strouhal number for the oscillations is obtained
through spectral proper orthogonal decomposition of schlieren
data. The LES results, which are found to match well with theexperiment, clearly show the three-dimensional nature of flow
unsteadiness. Key results from this work and a discussion arepresented here.

100
x

l2
θ1

M=6

Λ (log scale)

ABSTRACT

Pulsation

Steady10-1

Oscillation

θ2 = 90o10
Two geometricparameters: θ1 and Λ = l2/l1

Steady-21525
θ1 (degrees)

35

Figure 1. (a) A schematic of the axisymmetric double-conemodel with θ2 “ 90˝ . (b) θ1 –Λ parameter space with classification of flow states as per Sasidharan & Duvvuri (2021).
Data markers in the oscillation regime denote individual experiments from the present work.

INTRODUCTION
High-speed flow over a double cone geometry is a canonical problem that exhibits interesting unsteady shear-layer andshock-wave dynamics. A concise account of the general
double-cone problem is provided by Hornung et al. (2021).
Recent experimental work by Sasidharan & Duvvuri (2021)
showed two disparate states of shock-wave and flow unsteadiness for a particular class of double-cone geometries where
the second-cone angle is fixed at 90˝ . Here the flow is governed by two geometric parameters – the first cone angle (θ1 )and ratio of slant lengths of the two cones (Λ) – illustrated
in figure 1a. A certain portion of the two-parameter space(see figure 1b) was found to exhibit large-scale shock-wave
unsteadiness, termed as pulsation, which is driven by periodic
unsteady growth and collapse of the separated flow region that
forms over the conical surface. Whereas in an other portion,
distinct small-amplitude shock-wave oscillation is observed,
and this type of unsteadiness is simply termed as oscillation.
The flow was found to be steady outside the pulsation and oscillation regions. Based on qualitative observations of timeresolved schlieren data, the oscillations were hypothesised as
being driven by instabilities in the shear layer that forms overa separated region of the flow (Sasidharan & Duvvuri, 2021).

The present study is aimed at obtaining a more detailed understanding of the small-amplitude shock-wave oscillations andassociated flow unsteadiness. To that end, new experiments
and large-eddy simulation (LES) of the flow were performed.
Results from these efforts and discussion are presented here.

EXPERIMENTAL AND COMPUTATIONAL
SET-UP
Mach 6 flow experiments were performed at IISc in the
Roddam Narasimha Hypersonic Wind Tunnel, a 0.5 meter diameter enclosed free-jet facility that uses dry air as the working fluid in a pressure-vacuum mode. The free-stream unit
Reynolds number was set to 9.9 ˆ 106 m´1 with stagnationpressure P0 “ 11.1 bar and stagnation temperature T0 “ 455
K. Axisymmetric double cone models with fixed θ2 “ 90˝
were used, with θ1 and Λ “ l2 {l1 being varied across experiments such that their range remained restricted to the oscillations regime of the governing parameter space. It is noted1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
that each data marker in figure 1b in the oscillation region denotes an individual experiment. In all experiments high-speed
schlieren data was acquired with camera framing rates in therange 48,000 to 140,000 frames-per-second with an effective
frame exposure time of 10 ns. The total number of frames(or snapshots) in each experiment range from 8000 to 25000.
This experimental data provides good spatio-temporal resolution for detailed analysis of flow features.
Large-eddy simulation (LES) of the flow was carried out
for a particular combination of rθ1 , Λs “ r15˝ , 0.20s, whichcorresponds to one of the experimental data points in figure 1b,
to probe the flow in a more detailed manner than allowedby experimental data. A set of low-pass filtered conservation equations – mass, momentum, energy – are solved using a density-based compressible-flow solver that utilizes anunstructured finite volume formulation in OpenFOAM® (seeKumar & De, 2021; Greenshields, 2018). Flow behavior at thesmall scales is lumped into a single term in the governing equations (low-wavenumber-pass filtering) and is cast as a turbulent
stress tensor. The wall adapting local eddy-viscosity (WALE)
model of Nicoud & Ducros (1999) is used at the sub-gridscales to close the turbulent stress tensor term in the numerical simulations. The double cone surface is modelled as anadiabatic no-slip wall with sufficient grid resolution to maintain the average grid size at the wall close to 1 viscous wall
unit. The free boundaries away from the surface are modelled
as non-reflecting boundaries. The free-stream flow is treated to
be uniform with nominally steady velocity, pressure, temperature corresponding to the wind tunnel experiments. The fluctuations in the wind tunnel free-stream are not accounted for in
the LES since they are deemed to not have a significant effecton the flow phenomenon that is of present interest. The numerical discretization methods that are used provide an overall 2nd
and 3rd order spatial and temporal accuracy respectively in the
numerical solution. The time step size for numerical integration of the governing equations is around one nanosecond, andthat corresponds to a Courant–Friedrichs–Lewy (CFL) number of 1. The simulation was carried out on a 3D multi-block
structured grid with 12 million cells.

Reattachment
shock wave

Separation
shock wave
Conicalshock wave
Shear layer

Figure 2. An instantaneous schlieren snapshot of the flow
with rθ1 , Λs “ r25˝ , 0.11s.

interacts with the downstream separated flow that is inducedby the blunt aft-body. The separated flow generates a shock
wave, i.e. the separation shock wave, and a shear layer develops between the separated flow region and the supersonicflow over it. A reattachment shock wave is generated at theaft-body as the shear layer impinges the second cone aroundthe shoulder region.
In the present experiments a distinct type of unsteadyshock-wave motion associated with oscillations was observedat θ1 “ 15˝ . For purposes of differentiation from the previously reported oscillation type (Sasidharan & Duvvuri, 2021),
this flow state is labeled as “anchored-oscillation.” And
the previously reported oscillation type is now labeled “freeoscillation.” These distinct oscillation types, which are an interesting feature of the flow, are discussed below in detail.

Free- and anchored-oscillations
We consider the case of rθ1 , Λs “ r15˝ , 0.08s as representative example of free-oscillation. Figures 3a and 3b show
instantaneous schlieren snapshots of this flow at two different
time instants in the free-oscillation cycle, separated in time byapproximately half a period of oscillation. The point of flow
separation, which is also the location of the separation shock
wave foot, is marked as ‘S’ and ‘S1 ’ respectively in the two images. From a comparison of the location between ‘S’ and ‘S1 ’,
it is straightforward to see that the separation point executes
back-and-forth translation along the conical surface. This unsteady motion of the separation point is due to the periodic
expansion and contraction in the size/volume of the separation region, which is brought about by the impingement of theunsteady shear layer on the second cone. The key featuresof free-oscillation are schematically illustrated in figure 3c,where the extreme locations of the separation point within one
cycle are labeled ‘S’ and ‘S1 ’. The periodic motion of the separation point is seen to also bring about a continuous backand-forth shift in the spatial location where instabilities in theshear layer begin to manifest. This location is identified a ‘ˆ’marker and labeled ‘T’ and ‘T1 ’ in figures 3a and 3b respectively. From the schlieren images it is noted that flow disturbances arising from the shear layer instability interact with theseparation shock wave and bring about undulations in its structure. Experiments in which free-oscillations were observed are
indicated by a ‘‚’ marker in figure 1b.
Anchored-oscillation flow state was observed forrθ1 , Λs “ r15˝ , 0.20s; this location in the parameter space ismarked by a ‘▲’ marker in figure 1b. Figures 4a and 4b show

RESULTS AND DISCUSSION
Consider the θ1 ´ Λ parameter space shown in figure 1b.
For any fixed θ1 , as Λ is increased starting from Λ “ 0, a separation bubble emerges in the vicinity of the cone base, nestedby the corner. The bubble gives rise to a region of flow overit with large shear, i.e. a shear layer. The extent of the bubble
along the conical surface increases with Λ, thereby leading tolonger development lengths for the shear layer. Beyond a certain critical development length, rapid spatial growth of instabilities in the shear layer results in the onset of oscillations. At
this point the flow transitions from a steady regime into the oscillations regime. A further increase in Λ, beyond another transition point, moves the flow into the pulsation regime. Here abow shock wave forms around the shoulder of the second cone
and interacts with the leading separation shock wave, and theresulting dynamics bring about a drastic change in the overall
flow behavior in comparison to the oscillation state (Sasidharan & Duvvuri, 2021). Since the present focus is on oscillations, flow pulsations are not discussed further.
Figure 2 shows an instantaneous schlieren image of theflow for rθ1 , Λs “ r25˝ , 0.11s as a representative example ofthe flow in the state of oscillations. (Note that only the top halfof the flow field is shown in all the schlieren images presented
here.) The fore-body generates a conical shock wave, which212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022(a)instantaneous schlieren snapshots of the flow at two different
time instants in the anchored-oscillation cycle, separated intime by approximately half a period of oscillation. At thislarger Λ (relative to free-oscillation case discussed above),the separation bubble envelopes the entire cone surface.
With that the separation point ‘S’ is pushed to cone noseand remains anchored at that location throughout the entire
oscillation cycle. The periodic expansion/contraction cyclesof the separation bubble lead to a larger scale distortionin the separation shock wave structure, in comparison
with free-oscillation. The overall motion of the separation
shock-wave traces out an approximately bi-convex shape;
this aspect is schematically illustrated in figure 4c. Local
undulations in the shock-wave structure that are induced byshear layer disturbances accompany the larger scale coherent
motion. Significant variation in the local angle (inclination)of the separation shock wave near the cone nose brings
about unsteady variation in the downstream shear layer Mach
number distribution. This is expected to continually alter the
shear layer stability characteristics. Direct evidence for thesame is seen in the schlieren images where the location of‘T’ (marked by ‘ˆ’) shows significant displacement. This isperhaps an important aspect of the mechanism that sustains
the overall flow unsteadiness.
While only the free-oscillation flow state was found forθ1 “ 35˝ and θ1 “ 25˝ , a reduction in θ1 to 15˝ revealed thedistinct anchored-oscillation flow state at sufficiently large Λ.
Hence, in the θ1 ´ Λ parameter space the first appearance ofanchored-oscillations with reducing θ1 is expected to occur atsome θ1 between 25˝ and 15˝ . Another interesting flow feature observed at θ1 “ 15˝ is the blurred nature of the boundary
between free- and anchored-oscillations as Λ is increased from0.08 (figure 3) to 0.20 (figure 4). Flows at the three experimental data points between Λ of 0.08 and 0.20, marked by ‘‚’ infigure 1b (Λ values of 0.10, 0.12, 0.14), displayed a mix of features described above for free- and anchored-oscillation states.
The primary observation at these three values of Λ is that the
separation point ‘S’ remains anchored at the nose for only aportion of the oscillation cycle, and translates along the conesurface for the remaining cycle duration.

T

S
(b)

Shocklets

Separation
shock wave

T'

Conicalshock wave

Separated shear layer

S'(c)Shear layer
Shock wave
T
S

T'

S'

Figure 3. (a) and (b): Schlieren snapshots at two different
time instants within a cycle of free-oscillation for rθ1 , Λs “
r15˝ , 0.08s. (c) A schematic illustration of free-oscillations.
(a)T
Reference lineS(b)Strouhal number scaling
Following the characterization of physical features, weseek to now extract a timescale and associated spatial structure for oscillations. The modal decomposition technique ofspectral proper orthogonal decomposition (SPOD; see Towne
et al., 2018) is applied to the schlieren dataset. SPOD is afrequency-domain variant of the conventional proper orthogonal decomposition (POD) technique, and is well suited forextracting spatio-temporal coherent structures in statisticallystationary flows. An ensemble of realisations of schlieren intensity field qpx,tq “ tq j px,tq, j “ 1, 2, . . . , nu, where n, x, andt are the total number of realizations, spatial coordinate vector,
and time respectively, is first transformed to power spectral
density (PSD) field q̂px, f q “ tq̂ j px, f q, j “ 1, 2, . . . , nu in thefrequency ( f ) domain. Subsequently, q̂px, f q is decomposed
into orthogonal modes at individual frequencies through POD:

T'

S
(c)

Shock wave

Shear layer
T'
TS

Figure 4. (a) and (b): Schlieren snapshots at two different time instants within a cycle of anchored-oscillation
for rθ1 , Λs “ r15˝ , 0.20s. (c) A schematic illustration ofanchored-oscillations.

Q̂px, f q “nÿ

a j p f qψ j px, f q.

(1)

j“1

This essentially means that SPOD decomposes the ensembleof realisations q̂px, f q into orthogonal modes ψ j px, f q at different frequencies, and Q̂px, f q represents the optimal recon-312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 5. Modal energy spectra from SPOD normalized by maximum energy of the leading mode (top row) and the corresponding
mode shape of the leading mode (bottom row). (a, c) Free-oscillation for θ1 “ 15˝ and Λ “ 0.08. (b, d) Anchored-oscillation forθ1 “ 15˝ and Λ “ 0.20.
0.04

struction of the flow data set that contains the maximum energy(variance) of the ensemble. This optimization problem leads
to an eigen value decomposition problem, and the eigen values
corresponding to these modes at different frequencies are represented by λ j p f q “ a2j p f q. With the present data set, PSD of
the schlieren snapshot matrix was calculated to obtain convergent estimates of spectral density as per the recommendationof Towne et al. (2018). The SPOD modes are ranked based on
the corresponding λ j values, and then the leading mode canrepresent the dominant coherent structure in the flow.
Figure 5 shows SPOD results for the two specific oscillation cases discussed in the previous section. In both flow
scenarios the leading mode is seen to be much more energetic
in comparison with the other modes, and hence it provides agood representation of any coherent spatial structure present
in the flow. A peak in the spectra can be clearly identified inboth cases, which indicates the presence of a dominant timescale that characterizes flow unsteadiness, i.e. a frequency for
oscillations. Non-dimensional frequency St (Strouhal number)is defined here as

St “

f l2
,
U8

St

0.03

3 = 15/
3 = 25/
3 = 35/

0.02
0.01
0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

$

Figure 6. The Strouhal number (St) defined based on dominant frequency ( f ) obtained from SPOD for the θ1 ´ Λ parameter space exhibiting oscillations.

and synthetic schlieren images from LES at four different instances within one oscillation time period (T ). Phase locking
between the experimental and computational data was done
simply by visual assessment. From the figure it is seen that thelocation/shape of shear layer and shock wave are captured by
the LES with reasonable accuracy (with reference to the experiment). It is noted that the synthetic schlieren was generatedusing vertical-direction density gradients on a single azimuthal
plane, i.e a 2D slice of 3D data. Whereas the experimental
schlieren contains artefacts of flow non-uniformity along the
span (direction of schlieren light beam propagation). Hence a
close match between the two is not expected. For quantitative
validation of the LES, static pressure p is probed at six different locations in an azimuthal plane. These locations, which
are marked in figure 8, are in the separation bubble and shear
layer regions of the flow. All the PSD curves of static pressurein figure 8 clearly show a peak at St “ 0.035, which matchesvery well with the value obtained from the corresponding experiment.
LES data allows for a detailed inspection of the 3D flow
field. Figure 9a shows a 3D visualization of instantaneous
streamwise-direction density gradient in the flow. Figures 9b
to 9e show the same data at four different planes normal to the(2)where U8 is the free-stream velocity. From the spectra plots
the Strouhal numbers for these free- and anchored-oscillations
cases are inferred to be 0.014 and 0.035 respectively. SPOD
analysis outlined here was also performed for the rest of thedata points shown in figure 1. The results are summarized infigure 6 in the form of a Strouhal number plot.

3D nature of flow unsteadiness
The results of LES that was carried out for the caserθ1 , Λs “ r15˝ , 0.20s are discussed here. As seen above, thiscombination of θ1 and Λ results in anchored-oscillations. Figure 7 shows a qualitative comparison between experimental

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 7. Comparison between the experimental (top half of images) and synthetic (bottom half of images) schlieren images from
LES for a double cone configuration with θ1 “ 15˝ and Λ “ 0.20 at four different time (t) instances within an oscillation time period(T ). The position/shape of the separation shock wave and the shear layer are qualitatively similar. A quantitative match is foundbetween the two for the oscillation time period.

Figure 8. Power spectral density (PSD) of non-dimensionalpressure (p{P0 ) obtained from LES for θ1 “ 15˝ and Λ “ 0.20
at six different probe locations (marked at the top). The probes
are placed at three equally-spaced streamwise planes xi {L =0.33, 0.67, 0.99. Here L “ l1 cos θ1 .

streamwise direction, centered on the cone axis. The threedimensional nature of unsteadiness in the flow is evident froma visual inspection of the figure; density-gradient iso-contours
reveal clear azimuthal variation in the flow structure. These
results indicate that any stability analysis of the flow aimed
at explaining the oscillation phenomenon will be required toaccount for non-zero wavenumbers in the azimuthal direction.

BRIEF CONCLUSIONS
Shock-wave oscillations were investigated in a doublecone geometry with θ2 “ 90˝ . Experimental data revealed two
distinct types of oscillations, and they were termed as free- andanchored-oscillations. The latter oscillation type has not beenpreviously reported, and calls for further investigation. Present
data suggests that the region of the θ1 –Λ parameter space that
supports anchored-oscillations would grow with reducing θ1 ,i.e. the boundaries of the anchored-oscillation regime would
span a larger range of Λ at lower θ1 . SPOD analysis shows

Figure 9. A visualization of instantaneous streamwisedirection density gradient (normalized) generated from LES
data θ1 “ 15˝ and Λ “ 0.20. (a) Azimuthal plane cut-awayview. (b-e) Sectional views at four different streamwise planes.
Here x “ 0 is cone nose and L “ l1 cos θ1 .

5

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

that both oscillation types can be described by a dominant timescale. LES was performed for a particular combination of θ1and Λ where the flow exhibits anchored-oscillation, and theoscillation time scale obtained from it matched well with experimental results. The simulation results clearly showed thethree-dimensional nature of flow unsteadiness. LES data will
be probed further to understand the physical behavior of theseparation bubble and the shear layer, with a view toward developing a mechanistic model to explain the oscillation phenomenon.

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Turbulence and Combustion 62 (3), 183–200.
Sasidharan, V. & Duvvuri, S. 2021 Large- and small-amplitude
shock-wave oscillations over axisymmetric bodies in highspeed flow. Journal of Fluid Mechanics 913, R7.
Towne, A., Schmidt, O. T. & Colonius, T. 2018 Spectral
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ACKNOWLEDGEMENTS
Support from SERB National Postdoctoral Fellowship(V.S. – file no. PDF/2020/001278; G.K. – file no. PDF/
2021/001358), GoI Ministry of Education MTech Scholarship(A.G.K.), and Saroj Poddar Trust (S.D.) is gratefully acknowledged.

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