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

UNSTEADINESS IN A 2D CYLINDER WAKE FLOW AT MACH 6

Premika S. Thasu
Department of Aerospace Engineering
Indian Institute of Science
Bengaluru 560012<premikat@iisc.ac.in>

Vaisakh SasidharanDepartment of Aerospace Engineering
Indian Institute of Science
Bengaluru 560012<vaisakhs@iisc.ac.in>

Subrahmanyam Duvvuri
Department of Aerospace Engineering
Indian Institute of Science
Bengaluru 560012<subrahmanyam@iisc.ac.in>

ABSTRACT

gradient, and hence it is natural to expect that the boundary
layer would not be prone to separation. However, the downstream reattachment shock wave, which turns the flow parallel to the free-stream direction, causes a pressure jump. Theelevated pressure is communicated upstream through the subsonic region and the boundary layer, thereby causing flow separation and the formation of a free shear layer. According toDewey Jr (1965) the static pressure along the surface just upstream of the separation point is below the base pressure level,
and then subsequently rises to the base pressure value throughthe self-induced pressure gradient that accompanies separation. Since the pressure rise required to separate the boundary
layer is modest, the separation shock wave generated in theprocess is weak.
The dependence of near-wake features on the governing
flow parameters, i.e. Mach and Reynolds numbers, was investigated in more recent studies and empirical correlations
for the same were obtained (Park et al., 2016; Hinman & Johansen, 2017; Grasso & Pettinelli, 1995; Lamb & Oberkampf,1995). While the mean (temporal) flow structure of the nearwake region of a circular cylinder has been characterized indetail (McCarthy Jr & Kubota, 1964; Bashkin et al., 2002;
Park et al., 2016), the unsteady character of the wake has received surprisingly little attention, with the work of Schmidt& Shepherd (2015) seemingly being the only exception. Experiments by Schmidt & Shepherd (2015) in a Mach 4 flow revealed periodic oscillations in the wake, with a Strouhal number behaviour over a decade range in the Reynolds numbersthat suggests the free shear layer length to be the governing
length scale for the oscillations. The present work aims tobuild on these observations and obtain a more detailed understanding of the unsteady flow dynamics in the near-wake of acircular cylinder at high speeds.

Flow oscillations in the near-wake of a 2D circular cylinder at Mach 6 are experimentally investigated. Time-resolved
schlieren and wall-pressure measurements are analyzed to infer a coherent spatial structure and frequency (Strouhal number) for the oscillations. The findings presented here lend support to a universal scaling of Strouhal number proposed in recent literature for high-speed cylinder wakes.

INTRODUCTION
Blunt-body wake flows are a class of fluid dynamics
problems that have received close research attention over thepast several decades. A large proportion of the research effort has been in the low-speed (incompressible) flow regime,with the canonical 2D cylinder wake being a classic example (see for instance Williamson, 1996). Whereas the numberof high-speed (compressible) wake flow studies is somewhat
limited. Early efforts to understand the near-wake region in
high-speed flows employed experimental wall-pressure measurements and hot wire anemometry (McCarthy Jr & Kubota,
1964; Dewey Jr, 1965), and were accompanied by development of low-order models (Chapman et al., 1958; Reeves &Lees, 1965).
A primary understanding of the flow structure in thebase and near-wake regions of a circular cylinder at hypersonic Mach numbers emerged from the experimental studies
of McCarthy Jr & Kubota (1964) and Dewey Jr (1965). The
schematic in figure 1 illustrates the key flow features in thewake of a 2D circular cylinder in a high-speed flow, and also
shows a time-averaged schlieren intensity map obtained fromthe present experiments at Mach number M∞ = 6 (detailed in
the next section). The flow downstream of the leading bow
shock wave, in the vicinity of the fore stagnation point, issubsonic. The fluid from this region accelerates as it moves
around the cylinder, and attains supersonic Mach numbers
downstream of the sonic line. A set of expansion fans in the aftregion result in a further increase in Mach number as the flow
curves around the cylinder. Unlike in an incompressible flow
scenario where an adverse pressure gradient gets established in
the aft region, the expanding flow yields a favourable pressure

EXPERIMENTAL SET-UPExperiments with a 2D circular cylinder in a Mach 6 flow
were performed in the Roddam Narasimha Hypersonic WindTunnel (RNHWT) at IISc. RNHWT is a pressure-vacuum
type 0.5 m diameter enclosed free-jet facility (see Sasidharan& Duvvuri, 2021, for further details on the wind tunnel). In1

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

Figure 1: Key features of hypersonic flow past a 2D circular cylinder. The time-averaged schlieren intensity map, which

was experimentally obtained, indicates density gradients in the direction normal to the free-stream flow.
RESULTS AND DISCUSSION

these experiments the flow stagnation pressure and temperature were set to P0 = 9.7 bar and T0 = 460 K respectively. A
test model of diameter D = 50 mm and span L = 120 mm wasused in this study with provisions to install miniature pressuretransducers on the span-wise mid-plane at the top and bottom
separation points and the centerline base point; these locations
are marked in figure 1. The Reynolds number with respect tothe diameter of the cylinder (D) is defined as

ReD =U∞ D,ν∞The standard deviation map of temporal fluctuations inschlieren intensity shown in figure 2b highlights regions of theflow that have a high level of local unsteadiness. It is seenthat the neck region, i.e., the region where the top and bottom shear layers meet, and reattachment shock waves are unsteady. A small degree of local unsteadiness is also noted in
the free shear layer close to the separation point. Schmidt &Shepherd (2015) analyzed the Fourier spectrum of local intensity fluctuations in the neck region and found that a single
frequency can be associated with the unsteadiness. Here weuse the modal analysis technique of spectral proper orthogonal
decomposition (SPOD) to extract a timescale and associatedspatial structure for the flow from the time-resolved schlieren
data. The technique is briefly outlined below and results arepresented.

(1)

where U∞ and ν∞ are free-stream velocity and kinematic viscosity respectively. ReD for the present study is 4.29 × 105 .
High-speed schlieren data recorded at 100,000 frames per
second was obtained in the near wake region of the cylinder;
the schlieren set-up is similar to the one used earlier by Sasidharan & Duvvuri (2021). The schlieren “knife-edge” was set
in horizontal position to capture density gradients normal to
the free-stream flow direction, i.e. ∂ ρ / ∂ y. A high valueof knife-edge cut off was used to obtain sufficient sensitivity to density gradients. This brings about a non-linearity inlight intensity that breaks the symmetry between corresponding bright/dark bands in the top/bottom halves of the wake.
This aspect is seen in figure 2a which shows an instantaneous
schlieren snapshot of the flow. The top/bottom symmetry in
the image is missing, with flow features that are clearly visible
in the bright band (top half) not being captured equally wellby the saturated dark bands (bottom half). However, given thephysical symmetry, the flow is statistically similar in the top
and bottom halves, and essential flow elements are containedin both the halves. It is noted that key flow features – separation shock wave, free shear layers, re-attachment shock wave
– can be identified in the instantaneous schlieren snapshot of
the flow. In addition to schlieren visualization, time-resolvedpressure measurements were made at the top and bottom separation points and the base point using piezo-resistive transducers (Kulite XCE-093-5A) that were flush-mounted to the surface of the cylinder model. These transducers have a resonancefrequency of 150 kHz, thereby limiting the usable bandwidth
to about 30 kHz. The 30 kHz bandwidth however easily covers the frequency range of interest in these experiments. Thetransducers have a sensing diameter of 2.4 mm, and hence thepressure output from the transducers are to be interpreted as
spatial averages over the corresponding circular area.

Spectral proper orthogonal decomposition
SPOD is the frequency domain form of proper orthogonal
decomposition (POD) and is suited for extracting physicallymeaningful spatio-temporal coherent structures in statisticallystationary flows (Towne et al., 2018). SPOD finds an optimal orthogonal basis for the data such that a subset of the obtained modes capture a larger fraction of the total energy (variance) than any other orthogonal basis. The distinction between
SPOD and standard POD is that SPOD modes are orthogonal
under a space-time inner product, unlike POD modes that are
only spatially orthogonal. Mathematically, SPOD modes are
the eigenvectors of a cross-spectral density tensor at each frequency.
Consider a single time series consisting of N snapshots sampled at sampling frequency Fs broken into L(k)blocks/realizations, each consisting of M snapshots. Let q j
be the kth realization of the vector of observations at discrete
time t j . The discrete temporal Fourier transform for each realization is written as(k)M−1(k)

qbm = ∑ q j+1 e−i2π jm/M for m = −j=0M
M+ 1, ..., . (2)22The corresponding power spectral density (PSD) for each re-2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022(a) An instantaneous schlieren snapshot with normalized intensity, which indicates density
gradients in the direction normal to the free-stream flow.

(b) Map of the normalized standard deviation of temporal fluctuations in schlieren intensity.
Figure 2: Schlieren of the near-wake region downstream of the cylinder.
(k)(k)
alization, denoted by Pbm , then follows from qbm as

1 (k)
(k)
Pbm = 2 qbm
N2.

(3)

The rest of the analysis is performed at individual freb (m) of size P × L corresponding toquencies. The data matrix Q
any one frequency is constructed such that each column corresponds to the PSD vector along P spatial points. The cross(m)correlation matrix, b
S , is written as(m)bb (m) Qb (m) T ,S =Q

Figure 3: The modal energy spectrum of the first fourmodes obtained from SPOD. The dominant frequency
corresponding to the oscillations can be identified as
StD = 0.36.

(4)

and the eigenvalue decomposition of the cross-correlation matrix is carried out as(m)bS Φ (m) = Φ (m) Λ (m) .

ergy spectrum λ of the first four modes thus obtained. The frequency f is shown in the form of a non-dimensional Strouhal
number based on the cylinder diameter,

(5)
(m)

Here Φ (m) is a size P × P matrix whose columns φ p(p = 1, 2, ..., P) are the SPOD modes at a particular frequency
fm = mFs /M, and Λ (m) is the corresponding diagonal matrix(m)whose elements are the eigen values λ p .

StD =fD.
U∞(6)Mode 1 is clearly seen to be more energetic in comparison
with the other modes, and hence any coherent spatial structures
present in the flow are expected to be captured by this mode.
Further, a peak in the spectrum is observed at StD = 0.36 in
the mode 1 spectrum, and spectra of other modes also reflectincreased energy at the same Strouhal number. This analysis

Strouhal number and wall-pressure signature
SPOD was performed with schlieren data with L = 13,
M = 2048, and Fs = 100, 000. Figure 3 shows the modal en-3

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

Figure 4: Amplitude map of SPOD mode 1. The red-colored bounding box highlights the upstream portion of the shear

layer where a signature of energetic activity is visible.

suggests that the dominant oscillations in the near-wake region occur at a Strouhal number StD of 0.36; figure 4 shows
the spatial mode shape at this Strouhal number. For reference,
StD = 0.36 corresponds to a physical frequency of 6.49 kHz.
The qualitative similarities between intensity maps shown figures 2b and 4 also suggests that the frequency corresponding to
StD = 0.36 is a significant driver of the overall flow unsteadiness. It is interesting to note that mode 1 shows a signature offluctuations along the entire shear layer length; the upstream
portion of the shear layer is highlighted with a bounding box
in figure 4.

Schmidt & Shepherd (2015) had proposed a universal
Strouhal number based on the length of the shear layer S(marked in figure 1),

StS =fS,U∞(7)and found StS to be approximately 0.48 over the Reynolds
number range 2 × 104 < ReD < 5 × 105 at M∞ = 4. Fromthe present experiment we find StS = 0.47. The close match
between the Strouhal numbers lends support to the universal
scaling proposed by Schmidt & Shepherd (2015).

BRIEF CONCLUSIONS
The dynamics of the near-wake for a Mach 6 flow past a
2D circular cylinder was investigated experimentally throughtime-resolved schlieren and wall-pressure measurements (onthe cylinder aft surface). A Strouhal number for coherent
oscillations in the wake and associated spatial structure wasobtained by applying spectral proper orthogonal decomposition to the schlieren data. The Strouhal number is close to
the universal scaling proposed by Schmidt & Shepherd (2015)
from experimental observations at M∞ = 4. Schmidt & Shepherd (2015) hypothesize that propagation of acoustics waves
between the separation points and the neck through the subsonic portion of the shear layer sustains the wake oscillations.
The signature of oscillations seen at the separation points inthe present experiments are consistent with the hypothesis. A
more detailed study with stability calculations is needed to obtain a full understanding of the oscillations and the mechanisms involved in sustaining the same.

b of non-dimensional
Figure 5: Power spectral density (P)pressure (p/P0 ) fluctuations at the top and bottom separation points and the base point.

Based on the SPOD results, we expect to see a footprintof the oscillation frequency at the separation points. Figure 5
shows the PSD curves of the non-dimensional pressure fluctuations p/P0 measured at the top and bottom separation points
and the base point. A peak is indeed seen at StD = 0.36 in PSD
curves of both top and bottom separation points. Interestingly,
there appears to be no activity at that Strouhal number in thebase region. The phase difference in the pressure signals between the top and bottom separation points at StD = 0.36 was
extracted from a cross power spectral density (CPSD) calculation, and was found to be very close to π radians. This suggeststhat disturbances from the top and bottom shear layers are outof phase as they arrive at the neck, and this perhaps is a key
feature of the overall mechanism that governs the near-wakeoscillations.

ACKNOWLEDGEMENTS
Support from GoI Ministry of Education PhD Scholarship (P.S.T.), SERB National Postdoctoral Fellowship (V.S. –file no. PDF/2020/001278), and Saroj Poddar Trust (S.D.) is
gratefully acknowledged.

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