††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Polarization Criteria in Targeted SETI Observation

Jian-Kang Li  Department of Astronomy, Yunnan University, Kunming 650500,
China  Department of Astronomy, Beijing Normal University, Beijing 100875,
China  Yu Chen  Institute for Frontiers in Astronomy and Astrophysics, Beijing
Normal University, Beijing 102206, China  Department of Astronomy, Beijing
Normal University, Beijing 100875, China  Zhen-Zhao Tao  Institute for
Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing
102206, China  Department of Astronomy, Beijing Normal University, Beijing
100875, China  Xiao-Hang Luan  Institute for Frontiers in Astronomy and
Astrophysics, Beijing Normal University, Beijing 102206, China  Department of
Astronomy, Beijing Normal University, Beijing 100875, China  Bo-Lun Huang
Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal
University, Beijing 102206, China  Department of Astronomy, Beijing Normal
University, Beijing 100875, China  School of Physics and Astronomy, University
of Glasgow, Glasgow, G12 8QQ, UK  Sun Xiao-Hui  Department of Astronomy,
Yunnan University, Kunming 650500, China  Tong-Jie Zhang<EMAIL_ADDRESS>Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal
University, Beijing 102206, China  Department of Astronomy, Beijing Normal
University, Beijing 100875, China  Vishal Gajjar  Breakthrough Listen,
University of California Berkeley, Berkeley, California, 94720, USA  SETI
Institute, 339 N Bernardo Ave Suite 200, Mountain View, CA 94043, USA

###### Abstract

In this letter, we propose a new method for distinguishing extraterrestrial
intelligence (ETI) signals from the radio frequency interference (RFI) by
exploiting the polarization features. The linearly polarized component of
Stokes parameters should vary with the parallactic angle in sinusoidal form
for ETI signal, while such linearly polarized component should remains
relatively stable for terrestrial RFI. To witness such sinusoidal variations,
usually, at least 4-8 hours of observation time is required. Polarization in
search for extraterrestrial intelligence (SETI) also allow us to study the
radio stellar bursts of M-type stars, which is relevant to the habitability of
exoplanets. Compared with the frequency drift method, polarization method can
effectively reduce the observation time required for signal identification,
and also improve the process of signal identification.

††preprint: APS/123-QED

##  I Introduction

In radio SETI observation, there are two main method: sky survey and targeted
observation[1]. The sky surveys are carried out in the early stages, such as
SERENDIP, SETI@home [2] and Five-hundred-meter Aperture Spherical radio
Telescope (FAST) SETI backend [3]. In recent years, targeted SETI observations
are becoming popular among the main SETI projects such as Breakthrough Listen
(BL) initiative (e. g. [4, 5, 6, 7]), Allen Telescope Array (ATA) [8, 9] and
FAST targeted SETI observation [10, 11]. Most of the observations only focus
the total intensity instead of the polarization components. On the other hand,
most of the radio telescopes (arrays) have been equipped with orthogonal dual-
polarization receivers, allowing us to carry out the full measurement of
polarization.

Our technologies can also produce polarized radio emissions. Many of them are
used for communication. Therefore, it is reasonable to infer that ETI signals
are possible to be polarized with certain polarization features, and we can
use these possible features to identify the potential ETI signal. In this
letter, we propose a new method to identify ETI signal by its polarization
features. The Stokes parameters, the parameters used to characterized
polarization, of the ETI signal should vary with the parallactic angle due to
the rotation of Earth. In SETI observation, polarization also allows us to
investigate radio stellar bursts of M-type stars, which is relevant to space
weather and habitability of exoplanets.

##  II Polarization Features of Polarized ETI Signals

The radio telescopes are capable of detecting three signal types, namely, ETI
signals, natural astrophysical phenomena, and radio frequency interference
(RFI). All these three types of signals may presents particular polarization
features. For example, some commonly seen natural astrophysical radio sources
such as pulsars, masers, stars can be polarized [12]. The statistical analysis
of data from the Five-hundred-meter Aperture Spherical radio Telescope (FAST)
has also revealed that terrestrial RFI signals are predominantly polarized
[13]. Since most of the polarized astrophysical radiations in nature are
broadband signals, we can easily distinguish them by the frequency ranges. We
are more concerned about the differentiation between ETI signal and RFI. It is
generally assumed that ETI signals are narrow-band as most of the natural
occurring signals are known to be wideband ($>$ 1 kHz)[14], while many RFIs
are also narrow-band signals. RFIs can be categorized into different types. In
this study, we classify them into terrestrial RFI and satellite RFI. In order
to distinguish ETI signal from RFI, we propose to apply the variations of
Stokes parameters into signal identification.

The all-Stokes observation can be measured by a telescope with two orthogonal
linear polarization X and Y, which can produce auto-correlation data XX and YY
along with cross-correlation data XY and YX via Fourier transform within the
integration time. The Stokes vector $\mathbf{S}$ can be written by

$\mathbf{S}=\left[\begin{array}[]{c}I\\\ Q\\\ U\\\
V\end{array}\right]=\left[\begin{array}[]{c}XX+YY\\\ XX-YY\\\ 2XY\\\
-2YX\end{array}\right]$ (1)

where $I$ represents the total intensity, $Q$ and $U$ is the linearly
polarized component and $V$ the circularly polarized component. In real
observation, the field of view with respect to the observer should rotate with
Earth, and the Stokes parameters we receive should be

$\displaystyle I^{\prime}$ $\displaystyle=$ $\displaystyle I,$ (2a)
$\displaystyle Q^{\prime}$ $\displaystyle=$ $\displaystyle Q\cos 2q+U\sin 2q,$
(2b) $\displaystyle U^{\prime}$ $\displaystyle=$ $\displaystyle-Q\sin 2q+U\cos
2q,$ (2c) $\displaystyle V^{\prime}$ $\displaystyle=$ $\displaystyle V,$ (2d)

where $q$ is the parallactic angle defined as the angle between the great
circle through a celestial object and the zenith, and the hour circle of the
object, which can be calculated by [15]

$\tan q=\frac{\sin h}{\cos\delta\tan\varphi-\sin\delta\cos h}.$ (3)

where $h$ is the local hour angle, $\delta$ is the declination of the source
and $\varphi$ is the latitude of the telescope. Equation (2b) and (2c) can be
rewritten into a uniform sine or cosine form by the auxiliary angle formula,
and there is a phase difference of 45∘ between $Q^{\prime}$ and $U^{\prime}$.
Different declination of targets and latitudes of telescopes can cause
different changes of parallactic angles, therefore, whether the phase
difference can be witnessed also depends on $\delta$ and $\varphi$. Typically,
telescopes located at higher latitudes and observing targets at lower
declinations require longer observation times compared to those located at
lower latitudes and targeting objects at higher declinations (See Fig 1).

Figure 1: The change of parallactic angle with hour angel for observation
targets at different declinations and telescopes at different latitudes,
respectively. The black horizontal dashed line represents the case that
$\Delta q=45^{\circ}$.

To witness such phase difference of 45∘ between $Q^{\prime}$ and $U^{\prime}$,
at least 4-8 hours observation time are required and recommended. For
terrestrial RFIs, their Stokes parameters $Q^{\prime}$ and $U^{\prime}$ should
not change with $q$, since terrestrial RFI sources are spatially stable. In
targeted SETI observation, the satellite RFIs can be detected when the
satellite is orbiting exactly in the observation area, and disappears when the
satellite orbits out of the observation area.

In SETI observation, apart from natural astrophysical sources, the signal we
receive at a given frequency may contain three contributors: system noise, RFI
and potential ETI signal. The received signals of X and Y directions are
separate and independent. Assuming that the intensity of noise can be modeled
by complex normally distribution in both X and Y directions and at all
frequencies during the observation, then, the four correlation data should
follows chi-squared distribution after Fourier transform [16]. We generate
linearly polarized ETI and RFI synthetic observation data combining the chi-
squared noise as an example. The intensity of the noise is calculated based on
data of the targeted observation of FAST for Kepler-438. The most notable
event was found in the observation of FAST for Kepler-438 around 1140.6040 MHz
[10].

In order witness the phase difference between $Q^{\prime}$ and $U^{\prime}$,
the observation should be long enough so that the change of parallactic angle
can be greater than 45∘. We set the observation time to $\sim$4 hours for ETI
and terrestrial RFI synthetic observation. 4-hour observation time is
sufficient for the change in $q$ greater than 45∘ (See left panel in Fig 2).
The variations and the phase difference between $Q^{\prime}$ and $U^{\prime}$
are obvious in the waterfall plot, while the Stokes parameters of RFI are
relatively stable. Since the waterfall plot only presents a general
polarization features of the synthetic observation data, we extract the
$Q^{\prime}$ and $U^{\prime}$ data of synthetic ETI signal and synthetic RFI
to further analyze the polarization features quantificationally (See two right
panels in Fig 2). The small fluctuation is caused by the chi-squared
background noise. $Q^{\prime}$ and $U^{\prime}$ of synthetic RFI only
fluctuate around the mean values, while the synthetic ETI signal shows
sinusoidal variations and 45∘ phase difference between $Q^{\prime}$ and
$U^{\prime}$.

Figure 2: Left: The changes of parallactic angle, altitude and azimuth of
Kepler-438 during the $\sim$4-hour observation in local time 2021.06.29 of
FAST. Right top: The variation of $Q^{\prime}$ and $U^{\prime}$ of the
synthetic ETI signal during the $\sim$4-hour observation of Kepler-438. Right
bottom: The variation of $Q^{\prime}$ and $U^{\prime}$ of the synthetic RFI
during the $\sim$4-hour observation of Kepler-438.

##  III Discussions

###  III.1 Stellar Burst in SETI Observation and Polarimetry

Although what we are interested in is the ETI signal emitted from the
exoplanets, our current technology does not yet allow us to focus our
telescopes perfectly on an exoplanet in real observations, what we actually
observe is the planetary system consisting of a star and the planets orbiting
that star. Apart from ETI signal from exoplanets, it is more possible that we
can detect the radiation from the host stars. M-type stars are the most common
type of stars in the Milky Way [17], and a lot of studies on the search for
Earth-like planets and their habitability are more focused on M-type stars
[18, 19, 20]. Due to the strong magnetic fields (usually kilogauss, [21, 22]),
longer rotational braking times [23, 24, 25] and more active emission for a
given rotation period [26], the proportion of active stars among M-type stars
is much larger than among G-type stars, and the flare activities, a generic
result of magnetic reconnection [27], also occurs more frequently and more
energetically than solar flare [28, 29, 30, 31].

The stellar radio flare typically lasts seconds to hours and covers frequency
from megahertz to gigahertz. In past decades, observations reveals that the
stellar radio bursts are commonly characterized by close to 100% circular
polarization [32, 33, 34, 35] with extremely high brightness temperatures up
to excess of $10^{14}$ K [36, 37]. Such circularly polarized bursts can be
visualized by Stokes-$I$ and Stokes-$V$ waterfall plots (usually with time as
the horizontal axis and frequency as the vertical axis) for the full
observation band. Many of these stellar flare also exhibit frequency drift
properties due to the intrinsic motion, and the drift rate can be written as
[38]

$\dot{\nu}\approx-\nu\frac{v_{s}\cos\phi}{2H_{n}},$ (4)

where $\nu$ is frequency, $\phi$ angle between the beam direction and the
vertical, $v_{s}$ is the propagation velocity and $H_{n}$ is the density scale
height. For given observed drift rate, frequency and scale height (which can
be estimated from X-ray observations), we can make constrain on the
propagation velocity, or we can obtain the estimation for scale height
$H_{n}$, which is relevant to the hydrostatic equilibrium of the star, from
the observed drift rate, frequency and the constrain on propagation velocity.

The stellar flare are also relevant to space weather and habitability of
exoplanets. A planet in the habitable zone around an M-type star is likely to
be tidally locked [39], and the habitable zone is located closer to the M-type
star since the luminosity of M-type star is lower. The atmosphere of such a
planet, therefore, is usually exposed to a severer magnetic environment, in
the face of problems such as magnetospheric compression and atmospheric
erosion [40, 41], ultimately leading to problems with the habitability of the
planet. The potential habitable planet: Kepler-438 b, is probably suffering
the energetic stellar flares from its host star occurring every $\sim$200 days
[42]. Since the probability of detecting signals of interest in SETI
observations is very low and the observing resources of telescopes are
limited, the inclusion of stellar radio bursts as scientific goal in SETI
observations can not only improve the efficiency of the use of observational
data, but also make it possible to study the magnetic activity of stars and
space weather through the data obtained from observations, and use them to
assess the habitability of exoplanets. Such assessment can also help us to
further optimize the selection of our future observational targets.

###  III.2 Advantages of Polarization in SETI

Polarization is a new physical criterion in the signal identification, which
can be applied to any telescope with two orthogonal linear polarization
directions, and it also show some other advantages additionally.

First, compared with the commonly used physical criterion: frequency drift,
polarization can reduces the observation time required to identify ETI
signals. For a signal emitted from a transmitter on an exoplanet, the
frequency drift of the signal is caused by the rotations and orbits of the
exoplanet and Earth, where the orbit of exoplanet usually makes greater
contribution on the frequency drift since the orbital periods of habitable
planets around M-type stars are relatively shorter. Such frequency drift can
be characterized by pseudosinusoidal curve of the frequency change[43]. To
witness such pseudosinusoidal curve, we need a relatively long observation
time as the orbital periods of exoplanets are still too long compared with our
observation time. For example, the orbital period of Kepler-438 b is 35.23319
days [44], even if we expect to witness the peak or trough parts of the curve,
at least a quarter of the the orbital period of the exoplanet is required. As
we discuss in the previous sections, several hours observation time is enough
to witness the sinusoidal variations of $Q^{\prime}$ and $U^{\prime}$, since
the change of parallactic angle only depend on the rotation of Earth for the
given observation locations and observation targets.

Second, polarization also allows us to include the stellar burst as scientific
goal into targeted SETI observation, which greatly improve the use of
observation data. Most of the observed radio stellar bursts are highly
circularly polarized, the polarization features of the bursts can be
characterized and visually by Stokes-$I$ and Stokes-$V$ waterfall for
intensity and circular polarization. The studies on stellar bursts can provide
us the information of the magnetic activity of stars, space weather and
habitability of the exoplanets. The assessment of habitability can further
optimize the selection of future target selection.

###### Acknowledgements.

We thank the anonymous referees for the comments that helped us greatly
improve this article. This work was supported by National Science Foundation
of China under grant No. 11929301.

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# Control of bow shock induced three-dimensional separation using bleed
through holes

Hemanth Chandravamsi  Sourabh Bhardwaj  K. Ramachandra  R. Sriram  Department
of Aerospace Engineering, Indian Institute of Technology Madras, India

###### Abstract

The unsteady three-dimensional separated flow on a wall induced by a square
protrusion (approximately twice the local boundary layer thickness in width
and height), and its control by means of passive suction through holes, is
investigated using wind tunnel experiments at Mach $2.87$. The baseline flow
without any control was characterized and compared against the cases with
bleed. A bow-shaped separation line on the wall with a mid-span separation
length of $5.57\delta$ from protrusion face was traced from oil-flow
visualization. The averaged pressure distribution surveyed using static
pressure ports placed on the wall has mapped plateau, high-pressure, and a
low-pressure region in the separated flow, distinctive to three-dimensional
interactions. Ten control configurations were tested with suction holes placed
along mid-span in the different pressure zones. Significant spanwise ‘Mean
Reduction in Separation Length’ of up to $0.93\delta$ was observed from oil-
flow visualization. A comparison of observations from various control
configurations suggested that bleeding the flow from the high-pressure region
could in general delay the separation and reduce the bubble size. Further,
time-resolved schlieren visualizations have confirmed reduction in both ‘mid-
span separation length’ and ‘shock-intermittent-region’ with the introduction
of suction in high-pressure region. Fourier and Proper Orthogonal
Decomposition analysis done on the schlieren data has confirmed the presence
of low-frequency separation-shock oscillations at Strouhal Numbers of order
$10^{-2}$, both with and without control. Furthermore, the amplitudes of
separation-shock oscillations in the spectrum were reduced with the
introduction of suction simultaneously from two holes placed in high and low-
pressure regions.

††preprint: AIP/123-QED

##  I Introduction

Unsteady separated flows induced due to Shock Wave Boundary Layer Interactions
(SBLI) can cause violent thermo-mechanical loads which are capable of causing
catastrophic damage to the components of supersonic/hypersonic systems such as
intakes, isolators and control surfaces. With seventy years of history on SBLI
research and the recent advancements in diagnostics and computational
facilities, various aspects of turbulent SBLI due to nominally two-dimensional
geometric configurations including the “low-frequency unsteadiness” were well
resolved Clemens and Narayanaswamy (2014).

Several studies from the past few decades have established how various flow
parameters such as shock strength, Mach number ($M$), Reynolds number ($Re$),
and boundary layer profile affect SBLI in general, as well as the induced flow
separation Delery (2011). In addition, the unsteady aspects and the arguments
supporting upstream and downstream influences are also studied extensively
using both experimental Ganapathisubramani  _et al._ (2007); Dussauge  _et
al._ (2006), and computational approaches Pirozzoli and Grasso (2006). All
these studies together establish a good understanding of the physics of two-
dimensional SBLI. However, in several practical engineering applications,
three-dimensional interactions are often encountered.

Although there are reports in the literature addressing three-dimensional
SBLI, each of them was configuration specific, lacking any universal picture.
Westkaemper (1968) and Voitenko  _et al._ (1966) in 1966 have independently
reported the features of three-dimensional separated flowfields induced by
cylindrical protrusions placed in supersonic flow. Subsequently in the
following years, extensive experimental Zheltovodov and Knight (2011) and a
few RANS based computational studies Schmisseur  _et al._ (2000); Zheltovodov
_et al._ (2000) were performed on three-dimensional configurations such as
sharp unswept fins Zheltovodov (1982), sharp swept fins Zheltovodov and
Schülein (1986), semi-conesA. A. Zheltovodov (1991) and swept compression
ramps Settles  _et al._ (1980). The nature of shock topology, streamline
patterns in the vicinity of the object, pressure distribution and important
flow transitions under varying geometric and flow parameters were studied in
the above mentioned investigations. In the recent past, Mowatt and Skews
(2011) have investigated the interaction of three-dimensional curved shock and
boundary layer on a variety of curved surfaces using a set of experimental and
RANS based simulations. Pickles et al. Pickles  _et al._ (2019) in their
recent study have investigated three-dimensional SBLI under fin-on-cylinder
configuration. Their experimental and computational studies have revealed the
topological features of the separated flowfield along with the extent of
three-dimensional relief offered by the cylinder’s lateral surface under
various fin-angles, inflow Mach numbers and cylinder diameters.

With only a few reported investigations on such diverse configurations,
possible universal aspects concerning three-dimensional interactions are not
yet understood, even for classes of problems such as the SBLI due to bow
shock. Further, owing to the challenges in terms of computational cost to
perform high fidelity numerical studies using Large Eddy Simulations (LES) and
Direct Numerical Simulations (DNS), and limited three-dimensional
characterization capability of experimental flow diagnostics, researchers in
the past have largely managed to capture only the time averaged picture of the
interactions. Recent studies by Bhardwaj  _et al._ (2022) was the first to
propose a general scaling law (valid for various protuberance shapes and sizes
as well as supersonic freestream Mach numbers) relating separation length in
protuberance induced SBLI with the inviscid bow shock’s radius of curvature
and freestream conditions. However, a comprehensive understanding of the
unsteady aspects of these interactions is not well established in the
literature. With relatively sparse knowledge on unsteadiness, it is also
difficult to evaluate the critical aerodynamic and thermal loads associated
with three-dimensional interactions, which are essential in designing
efficient and safer high-speed flights.

Characterizing the interaction dynamics and underlying mechanisms is also
essential to implement control mechanisms to overcome the adverse effects.
Various active and passive flow control techniques for unsteady SBLI have been
explored and demonstrated successfully for nominally two-dimensional
configurations. Passive control devices, primarily due to their simplicity in
implementation and other essential benefits, have attracted several
researchers and engineers as potential candidates for mitigating separation
and unsteadiness. Among the various passive control methods, boundary layer
suction/bleed has received significant attention Raghunathan (1988); Delery
(1985) in all speed regimes. In this technique, fluid close to the wall,
either upstream of the separation or inside the bubble, is removed using
suction, taking out the local low momentum fluid.

The bleed configurations can be classified into three categories based on the
vent design: slots/perforations/holes. The use of these devices is not only
limited to the control of SBLI but has also been widely applied to control
other flow fields with separation and unsteadiness. For instance, in the
context of supersonic intake flow control, Herrmann  _et al._ (2011) and
Soltani  _et al._ (2015) have used bleed through slots to control the ‘intake
buzz.’ With regard to two-dimensional-SBLI too, slot control has attracted
significant attention, especially since slots can be in nominally two-
dimensional configurations. Krogmann  _et al._ (1985), and Hamed  _et al._
(1995) have used slot(s) oriented normal to the primary flow direction
producing a two-dimensional control effect for interactions in transonic and
supersonic regimes respectively. With the former being experimental and the
latter numerical, both studies report the effect of various bleed parameters
in controlling the two-dimensional SBLI. Contrary to placing slots normal to
the flow, studies were also performed with slots oriented along streamwise
direction, essentially inducing a three-dimensional control effect. For
instance, Smith  _et al._ (2003) and Holden and Babinsky (2005) have used a
series of streamwise bleed slots to delay the onset and reduce the extent of
shock induced separation. Interestingly, the study by Holden and Babinsky
(2005) notes that although the streamwise slot configuration they employed is
three-dimensional, the control effect induced on the interaction stands
relatively two-dimensional (meaning the controlled flowfield remained nearly
uniform along the spanwise direction).

Perforated strips were also employed by researchers to control SBLIs. Thiede
_et al._ (1984) have experimentally shown that perforated strips can delay
shock induced separation and stabilize the shock motion, improving the airfoil
characteristics on which these interactions occur. Jegadheeswaran  _et al._
(2019) have investigated the effect of a perforated wall in scramjet engine
setup through two-dimensional numerical simulations. They report that the
bleed from perforations causes the size of the separation bubble to shrink by
inducing a feedback loop between the upstream and downstream regions of the
separation bubble. However, such nominally two-dimensional control
configurations may not be effective for three-dimensional flowfields.

Alongside slots and perforations, holes were also employed to control SBLI.
However, holes can be placed in different arrangements. Their number, size,
and relative location to the interaction region can play an important role.
One simple arrangement widely used to control two-dimensional interactions is
to have a uniformly spaced array of holes that produces a global two-
dimensional effect. A few applications that employ this kind of arrangement
include Ghosh  _et al._ (2010) and Willis  _et al._ (1995), in which they
control separation due to an impinging shock boundary layer interaction. In
the hypersonic regime, Sriram and Jagadeesh (2014) have performed shock tunnel
experiments to control impinging shock induced separation through an array of
spanwise bleed holes. Considering the small size of the bleed holes and the
spacing between them in their experiments, their bleed set up can be
categorized to fall between perforations and hole based control. However,
unlike most slot and perforation configurations, a bleed hole that is
sufficiently distanced from other bleed holes induces a three-dimensional
effect. Literature on hole control suggests that such three-dimensional bleed
hole configurations can be used to control two-dimensional interactions.
Schoenenberger  _et al._ (1999), and Bodner  _et al._ (1996) have investigated
the effect of a single bleed hole on the development of turbulent boundary
layer downstream without the presence of any shock or SBLI. Rimlinger  _et
al._ (1992) (1992) have numerically studied the effect of a single bleed hole
and its position relative to impinging shock and demonstrated its control
effect. The single bleed hole was noted to result in three-dimensional control
effect in their flow field. The same group, later in 1996, extended their
study Rimlinger  _et al._ (1996) and explored the effect of rows of multiple
holes arranged three-dimensionally in a staggered fashion. They report that
when the holes are placed sufficiently upstream of the incident shock, the
bleed effect can adequately block the shock-induced adverse pressure gradient
from propagating further upstream, thereby controlling separation. Such
studies tell us that even in nominally two-dimensional SBLIs, the three-
dimensional effect from bleed holes can induce a good control effect. While
carrying out the present study, we have hypothesized that the same three-
dimensional control effect from the holes can also be used for controlling
three-dimensional interactions, possibly with better efficiency. Furthermore,
in the context of three-dimensional interactions, the placement of hole(s) can
even be critical. To the best of our knowledge, there is no detailed study in
the literature concerning the bleed control of three-dimensional shock induced
separation due to protrusion.

The research presented in the current article is a part of the broad study
initiated at the Department of Aerospace Engineering, IIT Madras to understand
and address the issues concerning three-dimensional SBLI. The current study
concerns the characterization of three-dimensional shock boundary layer
interaction induced by a square protrusion placed in a supersonic flow (of
nominal Mach number 2.93) and a demonstration of passive suction based
strategic control to mitigate separation and flow unsteadiness. This canonical
three-dimensional configuration of the square protrusion (with its face
oriented perpendicular to incoming flow) was especially chosen for the current
study to explore the implications of the interaction between strong detached
shock (generated due to the bluntness of protrusion) and turbulent boundary
layer. The results from the study are thus qualitatively applicable to any bow
shock induced separation, such as that in blunt fin induced SBLI.

The paper is organized as follows. After the present section, the experimental
setup designed to carry out the current study, i.e., the wind tunnel, flow
diagnostics, and control setup, shall be detailed in section II. Following
that, in section III, the mean and unsteady nature of the baseline111The term
‘baseline’ in this paper is used to refer to the flow configuration without
any control. case i.e., the protrusion induced three-dimensional separated
flowfield, is discussed. Then, in sections IV and V, the effect of bleed holes
placed inside the separated region under various configurations (designed
based on baseline results) is discussed. In particular, tools such as Fourier
analysis and snapshot Proper Orthogonal Decomposition (POD) were employed to
get a deeper understanding of the unsteady dynamics of the baseline flowfield
and the control effect. Finally, the conclusions are laid out.

##  II Experimental set-up and flow diagnostics

###  II.1 Wind tunnel facility and model set-up

Figure 1: (a) Schematic of the wind tunnel designed to study protrusion
induced three-dimensional shock boundary layer interaction. (b) Cross-
sectional view of nozzle and test-section (pen placed as a reference for
scale)

Experiments were performed in the supersonic blowdown wind tunnel at the Gas
Dynamics Laboratory, IIT Madras (for details, refer to Bhardwaj  _et al._
(2022)). The tunnel is being operated at a nominal stagnation pressure
($P_{o}$) and stagnation temperature ($T_{o}$) of $6\pm 1.1\%$ bar (absolute)
and $300\pm 1\%$ K, respectively, with each run lasting nearly $10$ seconds.
The schematic of the nozzle and test-section assembly designed for the current
study is shown in Fig 1. The set up includes a 396 mm long test-section with a
rectangular cross-section of span and height, $100$ mm and $113$ mm,
respectively, with a provision of optical access on the side walls (for visual
inspection and schlieren visualization). This test-section is attached to the
exit of a contoured supersonic nozzle of exit to throat area ratio of $3.96$.
The schematic of the set up with origin (origin is at the spanwise center, on
wall protrusion junction) and the coordinate axis orientation can be
understood from Fig 1a. In order to make alterations as well as to enable
taking it out to capture the photograph of surface streakline pattern after
the experimental run, the base-plate (described by plane ABCD in Fig 1a) was
designed in a modular fashion to be easily removable from the test section
while maintaining the desired geometrical integrity of the whole set up.

Figure 2: (a) Precisely moving pitot probe assembly used to survey the
boundary layer and freestream of the wind tunnel mounted at protrusion
placement spot. (b) Boundary layer profile probed at protrusion mounting
location using the precisely moving pitot-probe arrangement (without any
obstacle/model being placed in it).

The freestream conditions and boundary layer were characterized using a
precisely moving pitot-probe arrangement shown in Fig 2a. The internal and
external diameters of the pitot probe are $0.9$ mm and $1.8$ mm respectively.
The freestream Mach number in the test section was $2.87\pm 4.4\%$. A summary
of the pitot survey results is presented in Fig. 2a and table 1. The pitot
survey in the test section (without mounting the protuberance) was done at a
distance of $170$ mm from the nozzle exit, which is also where the
protrusion’s face was mounted. The time-averaged velocity profile obtained
from the pitot survey is shown in Fig 2b, from which the boundary layer
thickness ($\delta$) was measured to be $7$ mm. Based on boundary layer
thickness, the protrusion’s height ($h$) and width were chosen to be roughly
two times $\delta$ ($15$ mm), with an overall dimension of $15\times 15\times
15$ mm3 (The protrusion can be seen in Fig 1). The dimensions of the cube were
so chosen to facilitate the comparison of the current study’s results with the
corresponding two-dimensional case (forward facing step) study conducted by
Murugan and Govardhan (2016). The step height in their case was taken as two
times their incoming boundary layer thickness. The protrusion was mounted at
the spanwise center, with one of its face perpendicular to the streamwise
direction.

Table 1: Free stream conditions inside the test-section and boundary layer thickness at protrusion placement spot $M_{\infty}$ | $Re_{\infty}$ | $U_{\infty}$ | $P_{\infty}$ | $T_{\infty}$ | $\delta$
---|---|---|---|---|---
$2.87\pm 4.4\%$ |  $40.4\times 10^{6}$ 1/m |  $612\pm 1.2\%$ m/s |  $19200\pm 0.05\%$ Pa |  $114\pm 1\%$ K |  $7$ mm

###  II.2 Flow diagnostics

Oil-flow visualization, schlieren imaging, and surface pressure measurements
(averaged) were the flow diagnostics used in the current study. The details of
the diagnostics are presented in the paper by Bhardwaj  _et al._ (2022). Oil-
flow surface streakline visualization was performed on the base plate where
the protrusion was mounted for both baseline and control cases to highlight
important features such as the separation line and streakline pattern inside
the separation bubble. A mixture of Titanium Dioxide (TiO2) and SAE-$30$ grade
mineral oil with Oleic acid as the emulsifying agent was prepared and
spattered in the region of study to carry out oil-flow experiments Oil (2022).
The steady-state streakline pattern generated at the end of each experiment is
captured using a camera. The geometry of the separation line is also recorded
for comparison and analysis purposes by means of grid lines drawn on the
surface. Multiple experiments were performed to check the repeatability of the
results and found out to be so.

Schlieren visualization with light passing perpendicular to the side walls was
used to observe the separated flow field’s shock patterns and features. The
high-speed ‘Photron FASTCAM SA4, Model 500K-M1’ camera was used to capture the
images. With a time resolution of $30,000$ frames per second, schlieren
visualizations were also performed to characterize the shock motion and
unsteadiness quantitatively.

The time-averaged surface pressure distribution on the bottom wall in the
separated flow field was surveyed by means of pressure ports and an electronic
pressure scanner (Scanivalve DSA-3217). For this purpose, a layout of thirty
holes (static pressure taps) of $0.6$ mm diameter each was made on the base
plate with the help of insights gained from baseline oil-flow experiments. The
layout of the ports is designed in a fashion to understand the pressure
distribution and identify various pressure zones corresponding to the
interaction. In addition, readings obtained from a set of four experiments
were considered to acquire a converged set of average pressure values.

###  II.3 Control set up

Figure 3: Suction chamber and baseplate assembly for control experiments: (a)
Schematic sketch (b) Photograph of the set-up.

Passive suction is implemented in the separated flow region through circular
holes along the centerline on the bottom surface. The control experiments were
designed based on the insights gained from the surface static pressure
measurements obtained on the bottom surface for the baseline case. Based on
separation features and pressure distribution characterized by baseline
experiments, a total of ten bleed-hole configurations were designed to
understand the effect of suction from various pressure zones present in the
separated region. To set up various control configurations, two base-plates
were prepared with three bleed holes of $5$ mm diameter on each of them. Using
two plates enabled the placing of holes at locations that were only slightly
off from locations in the first plate, so as to see if small changes in hole
placements may vary the flow drastically. Different control configurations
were achieved by strategically closing some holes and keeping the others open.
To close the holes, a hard cylindrical rubber of a slightly larger diameter
than the hole was passed through and flush mounted into the holes with a flat
cut on the plate surface. The specific details of the geometric positioning of
the bleed holes for various configurations shall be discussed in section IV
after presenting the data of surface static pressure.

To draw the fluid out from these bleed holes (each of 5 mm diameter), a
rectangular suction-chamber of dimensions $149\times 64\times 60$ mm3 was
placed beneath the holes under the bottom surface of the test-section as shown
in Fig 3. The bleed holes drilled on the test-section’s baseplate were exposed
to the vacuum chamber’s suction from beneath. The chamber is evacuated by an
external vacuum pump which generates sufficient vacuum to choke all the bleed
holes.

It is reasonable to expect that the bleed mass flow rate and the extent of
local flow field manipulation it results in are correlated. Thus it is of
interest to measure the bleed mass flow rate of the different control
configurations. It is worth noting that the bleed rate can vary from one
configuration to another depending on the location of the bleed hole(s) in
different pressure zones. Therefore, in an attempt to understand the
correlation between the bleed mass-flow, hole positions and control achieved,
mass-flow rate measurements were performed for some critical control cases
using a ‘Bronkhorst D-6380-DR’ mass-flow meter. The mass-flow meter was placed
after the vacuum chamber before the bleed air enters into the vacuum pump. The
mass-flow rate readings presented are accurate to within $\pm 0.5\%$ kg/hr.
The experimental results shall be discussed in the following section.

##  III Three dimensional shock induced separated flowfield

The time-averaged features of the flowfield due to a square protrusion under
the same operating conditions as in the present work were detailed in the
recent paper by Bhardwaj  _et al._ (2022). In this section, we revisit some of
the key aspects of time-averaged flowfield to establish the nomenclature,
concepts required for further discussion, and the rationale behind the control
strategy drawn from various pressure regions. Additionally, this section also
presents a characterization of unsteadiness in the flowfield based on time-
resolved schlieren. These aspects of the baseline case are subsequently used
for comparison with the control cases.

###  III.1 Oil flow visualization

Oil flow visualization for the baseline case reveals the streakline pattern on
the base-plate, highlighting the flow separation line and the streakline
pattern inside the separation bubble, as shown in the Fig 4a. The separation
line can be identified as a bow shaped thin region of convergence of the
streak lines between the incoming flow and the separation bubble. The
separation length, which is defined as the distance between the separation
line and the protrusion surface at the spanwise centre, was measured to be
$5.57\delta$ for the square-faced protrusion as indicated in the figure. With
the help of grid markings made on the bottom surface, the separation line is
traced and plotted as shown in Fig 4b on the right (in solid-black line). The
bow-shock position on bottom wall corresponding to inviscid flow over this
protrusion was also represented in the Fig 4b for comparison (this bow-shock’s
position was found through numerical Euler simulation, as discussed in
Bhardwaj  _et al._ (2022)). It is worth noting that, due to the interaction,
the (separation) shock has moved upstream by an amount of $5.3$ times the
stand-off distance of the inviscid bow-shock along midspan. Also, the current
midspan separation length can be noted to be roughly $1.6$ times lower than
the separation length ($8.2\delta$) reported for the corresponding two-
dimensional case (forward facing step) by Murugan and Govardhan (2016).

Figure 4: (a) Oil-flow surface streakline pattern for the baseline case with
separation line highlighted, (b) Separation line position for the baseline
case (solid black line) traced from oil-flow visualization, and Inviscid bow-
shock position from numerical Euler simulations

###  III.2 Surface static pressure measurements

With the reference of the separation line traced from oil-flow visualization,
the layout for static pressure ports was designed as shown in Fig 5a (each
circle represents a sensor there, with its radius and colour indicative of the
magnitude of local pressure). It can be seen that the line along spanwise
centre was particularly resolved to have a total of 9 ports. In addition to
that, at a few streamwise locations, taps were placed symmetrically on either
side of the centreline to ensure symmetry of the flow. Otherwise, most of the
taps were concentrated on the left side of the centreline as seen in Fig 5a
(bottom portion in the figure), expecting symmetry in the field (as it turned
out to be from measurements). It can also be seen that along the separation
line there are a total of six pressure taps. Along the spanwise centreline, it
can be observed that the first two taps measure freestream pressure. In the
third sensor, which is a little upstream of the mean separation line, the
pressure starts to rise. This is attributed to the upstream influence, due to
which the pressure starts to rise before the separation. Following this, there
is a region of plateau pressure with a value close to twice the freestream
pressure inside the separation bubble up to a distance of around $4\delta$
from the protrusion. Downstream of this plateau region, the surface pressure
drops, reaching values as low as $1.5$ times freestream pressure between the
distances $\delta$ and $2\delta$ away from the protrusion. This local region
of low pressure in the protrusion induced separation bubble is due to the
three-dimensional relieving effect, and such a region is not observed in two-
dimensional SBLI in which the plateau pressure inside the bubble is directly
followed by the pressure rise at reattachment. The relieving in protrusion
induced SBLI is due to the spanwise turning of streamlines inside the
separation bubble which is evident from the surface oil-flow streakline
patterns presented in Fig 4a. It is in this region where the relieving begins
that the pressures are low. However, very close to the protrusion, the
pressures are very high (5 times the freestream pressure) due to reattachment
and stagnation of flow at the protrusion front face. The trend is observed
even along lines that are close to the spanwise centre line, though the values
of the peak pressure are observed to be lower. A similar pressure profile with
pressure dip was also reported by Schülein and Zheltovodov (2001) in 2001 in
their study on three-dimensional separation due to fins (see figures 5.17a and
5.22 in the book by Babinsky and Harvey (2011)). In the neighbourhood of the
spanwise centreline, based on the surface pressure distribution, three
distinct regions can be demarcated, viz., plateau region, low-pressure zone
and high-pressure zone. Far away from the centreline, the plateau pressure
after the separation is followed by a pressure drop, and no high pressure
zones are observed. The high-pressure is thus confined to a width comparable
to the width of the protrusion. Based on the pressure distribution along the
spanwise centreline, control experiments were designed to explore the effect
of suction from various pressure regions on the separation length and
unsteadiness.

Figure 5: (a) Surface pressures at various locations on bottom surface
measured through the surface static pressure taps. The centres of the circle
are the positions of the static pressure taps. Colour and radius of each solid
circle represents the magnitude of pressure. (b) Surface pressure distribution
along midspan ($z/\delta=0$). Various pressure regions are indicated.

###  III.3 Schlieren flow visualization

The Schlieren visualization (Fig. 6a) reveals various features present in the
flowfield such as: incoming boundary layer, separation shock, reattachment
shock, re-circulation bubble and shear layer as labelled in Fig 6b. As shown
in the Fig. 6, in the spanwise central plane, type-6 Edney shock interaction
Edney (1968) was observed between separation shock (labelled with number 1)
and reattachment shock (labelled with number 2). The interaction of these two
shocks resulted in two reflected shocks labelled with numbers 3 and 4 Fig. 6.

Figure 6: (a) An instantaneous schlieren snapshot taken for the baseline case
(raw image), (b) Various features of the three-dimensional shock boundary
layer interaction flow field labelled.

Combining the oil-flow data and the schlieren visualizations, a schematic of
the three-dimensional separated flow upstream of the square protrusion is
shown in Fig 7. The schematic shows important features such as the three-
dimensional separation shock topology (represented with red dashed lines), the
separation line (SF’), the surface streakline pattern, shock interaction in
the midspan plane, and the ‘recirculation’ bubble inside the separation zone
in the midspan plane. The bubble is not shown to have closed streamlines in
the schematic (in contrast to the two-dimensional bubbles) since the
streamlines spiral towards a core and subsequently turn spanwise, resulting in
a horseshoe vortex (detailed topology is discussed in Bhardwaj  _et al._
(2022), 2022).

Figure 7: Schematic of three-dimensional separation in front of square
protrusion.

With the time-resolved schlieren imaging, the positions of separation shock
and reattachment shock were seen to vary temporally. Some instantaneous
schlieren snapshots taken during the run are shown in the Fig. 8a. The shock
position and separation bubble size can be clearly seen to vary over time in
these images. For instance, this difference can be clearly appreciated if two
snapshots corresponding to $0\>\mu s$ and $600\>\mu s$ are compared. Also
presented in the Fig. 8 are the mean and standard deviation of pixel intensity
values considering $8000$ instantaneous snapshots. The thick and blurred
nature of the separation shock in the mean picture also illustrates the
unsteady nature of the flowfield. The non-zero values present in the standard
deviation picture shown in Fig. 8c also demonstrate the same.

Figure 8: (a) Instantaneous schlieren visualization snapshots (without
control). (b) Mean and (c) Standard deviation of pixel intensity of schlieren
images

To analyze the dynamics of separation shock, the instantaneous separation
shock position in the midspan plane, as apparent in the instantaneous
schlieren images, was extracted from each snapshot through image processing
and a program written to find the pixel locations corresponding to the
separation shock. First of all, the image processing was done by adjusting the
brightness, contrast and sharpness of the raw image (shown in Fig. 9a) so as
to display the separation shock with high contrast. This is followed by
detecting pixels on the surface of separation shock (in midspan plane), by
spotting the locations with a sudden jump in pixel intensity. Subsequently,
the shock position was reconstructed as a single straight line in each
snapshot through the identified pixel locations. The steps involved in the
process are also pictorially shown in Fig. 9.

Figure 9: (a) Raw sample snapshot from time resolved schlieren (b) Processed
schlieren image with separation shock looking more distinct (c) Scanning
region for extraction of pixels corresponding to the separation shock (points
on the separation shock indicated). (d) Reconstructed separation shock, using
best fit line passing through the first dark pixels from top-down, in the
scanning region.

The best fit line for the shock is projected onto the wall to obtain the
instantaneous shock foot position. After being able to extract shock foot
position from each snapshot captured over a time period of $0.27$ sec ($8000$
snapshots), a shock-foot location time series has been plotted (Fig 10a). The
shock foot can be seen to sweep between distances $5\delta$ and $7\delta$ from
the front of protrusion (in the spanwise centred plane). The mean position of
the shock foot was found to be at $6.1\delta$ from the protrusion. The
standard deviation of the shock foot (which represents its degree of presence
away from the mean position) was found to be $0.5\delta$. The effect of
control on these two quantities will be discussed in the next section with a
few important control configurations.

In order to gain an understanding of the degree of shock’s presence at various
spatial locations on the bottom surface, the Probability Density Function
(PDF) of the shock foot data is obtained. Since the shock-foot data obtained
is discrete, kernel density estimation method is used to obtain the PDF (with
Gaussian kernel function). The PDF has been plotted about the mean shock foot
position. The PDF can be seen to be mostly symmetrical about the mean, with
maximum amount probability density in the range of $-\delta$ to $\delta$. This
corresponds to a sweeping length of approximately $2\delta$.

Figure 10: (a) Time series of shock foot collected using image processing. (b)
Probability Density Function of shock foot on the spanwise central plane.

##  IV Effect of suction on the separation and unsteadiness

A total of ten suction hole configurations were designed and experimented
with, by placing the holes only along the midspan in all the layouts. The
placement of suction holes were strategically chosen to explore the effect of
suction from the three pressure regions identified from surface pressure
measurements. The names of the configurations are given in the format
C_no.1_no.2_no.3 (first letter C stands for ‘Configuration’). The numbers:
no.1, no.2, etc- denote bleed-hole positions, in terms of their distance (in
mm) from the protrusion front-face. For example, C_2.5_12.5 represents the
configuration with two holes, one of them with its centre at $2.5$ mm from the
protrusion, the other with its centre at $12.5$ mm from the protrusion.

It was hypothesized that the bleed from different locations on the plate
(zones) may result in different bleed flow rates, which in turn can be among
the factors determining the control effectiveness of particular
configurations. Configuration C_2.5 employs a bleed hole at 2.5 mm (or
$0.36\delta$) from the protrusion, which is in the region of high pressure
from where the highest bleed mass-flow is expected. The Other configurations
C_2.5_12.5, C_2.5_12.5_27.5, and C_2.5_27.5 were designed to explore the bleed
effect from the hole placed at $x=-2.5$ mm in combination with holes placed in
low and plateau pressure regions which correspond to locations $x=-12.5$ mm
and $x=-27.5$ mm respectively. The rest of the configurations were designed to
explore the control effect due to holes placed solely in the plateau and low
pressure regions.

###  IV.1 Oil flow visualization

The oil-flow streakline patterns generated for various control cases were
shown in Fig. 11 along with their corresponding midspan separation lengths.
The first four configurations (a,b,c and d) are those with a hole in the high
pressure region (at $x=-2.5$ mm), and the rest has suction only from upstream
locations. Separation line geometries from these images were extracted with
the help of grid markings made on the bottom surface, and plotted together, as
shown in the Fig. 12, to compare the profiles for various cases against the
baseline case; inviscid bow-shock profile for the baseline case is also
plotted for reference.

Figure 11: Oil-flow surface streak line patterns for various control cases.
The separation line corresponding to the separation shock is indicated with
the red dashed line. The configuration name and the Mid Span Separation Length
(MSSL) is indicated above each of the corresponding images. Grid lines on the
surface of the baseplate can be also seen in the images which are drawn for
measurement purposes. Each grid box measures 5 mm ($0.71\delta$) in both its
width and height. Figure 12: Geometry of separation lines for control
configurations (a) C_2.5, C_2.5_12.5, C_2.5_12.5_27.5 and C_2.5_27.5 (b)
C_12.5_27.5, C_40, C_40_30, C_30, C_15 and C_15_30 in comparison to the
geometry of baseline separation line (solid black line).

At a first glance of the geometry of streakline patterns shown in Fig. 11,
some important features and differences can be noted in comparison with the
baseline case. For configurations C_2.5_12.5, C_2.5_12.5_27.5 and C_2.5 a
reduction in separation length along the entire span can be clearly observed.
Further, for the configurations having a hole in high pressure region
($x=-2.5$ mm), the separation line and the surface streakline pattern were
symmetric about midspan. However, for some other configurations the separation
line can be seen to be asymmetric (C_40_30, C_30 and C_15). In some
configurations hinging/pulling effect towards the bleed holes can be observed;
this hinging of the separation line in the mid-span is followed by the quick
catching up of the separation line with that of the baseline case, or even an
increase in separation length at some spanwise locations (local separation
length being the x-distance from protrusion front face to the separation
length at the considered spanwise location). The separation line corresponding
to the configuration C_12.5_27.5 can be seen to have its position hinged
towards the bleed hole at $x=-27.5$ mm ($-3.9\delta$). A similar pulling
effect was seen for most configurations (C_12.5_27.5, C_40, C_40_30, C_30)
with a bleed hole in the plateau pressure region. It was also noted that the
separation line for C_15_30 experiencing the pulling effect of suction from
the bleed hole at $x=-30$ mm (or $-4.3\delta$), has indeed increased
separation length at some locations away from the midspan, although the
separation length in the midspan was reduced (with the separation line hinged
on the front of the hole).

To quantify the performance of the control configurations, the Mean Reduction
in Separation Length (MRSL) throughout the span was calculated for each
configuration. MRSL essentially quantifies the overall amount of
reduction/increase in separation length due to the flow control. The values of
MRSL for the different control configurations are reported in the Table 2,
along with the midspan separation length and bleed mass-flow rate (bleed mass
flow was measured only for few critical configurations). The observations
presented in the table, in general, suggest that the introduction of suction
at the high pressure region will lead to a delay in overall separation. With
MRSL of $0.86\delta$, the best performance can be seen for the configuration
C_2.5_12.5 (combination of two suction holes one from high pressure and
another from low). This configuration also had a reduction in midspan
separation length of $0.92\delta$ which, however, is not the highest reduction
in midspan separation length. Configurations with a hole in the plateau but
without a hole in the high-pressure region generally had the highest reduction
in midspan separation length, which was $1.29\delta$. However, these
configurations had considerably lesser MRSL, one of them (C_15_30, with a
reduction in midspan separation length of $1.15\delta$) even having a negative
MRSL (due to larger separation lengths than baseline case for a wider range of
span).

In fact, all the configurations in which the bleed hole was placed at
$x=-2.5$mm ($-0.36\delta$) (which is in high pressure zone), can be seen to
show good performance in reducing the separation length throughout the span,
and also preserving the symmetry about midspan. Further, these configurations
have also reported higher bleed mass-flow rates compared to any other
configuration without bleed hole at $x=-2.5$ mm. This clearly demonstrates
that it is advantageous, in terms of reduction in separation length, to have
suction from high pressure region of the separation bubble rather than
employing bleed from other places in the midspan.

In configurations C_15 and C_15_30 suction was employed from relatively low
pressure regions of the separation region (between $x=-\delta$ to
$-2.5\delta$). The performance from these configurations was observed to be
poor with an overall increase in the size of the separation bubble. Thus,
suction from low pressure regions of the separation bubble can be said to be
undesirable in general.

Although the bleed mass-flow that was achieved through configuration C_2.5 is
higher than the configuration C_2.5_12.5, control performance was seen to be
better for the configuration C_2.5_12.5. It can be inferred from this
observation that, the bleed placement is of greater importance for flow
control than the bleed mass flow rate. In fact, the configuration of bleed
placement dictates the bleed mass flow rate.

Table 2: Comparison of performance and bleed mass-flow rates gained from various control configurations. Configuration | Midspan separation distance | Mean reduction in separation length (MRSL) | Bleed massflow rate (kg/hr)
---|---|---|---
C_2.5_12.5 | $4.65\delta$ | $0.86\delta$ | 13.1
C_2.5_12.5_27.5 | $4.78\delta$ | $0.77\delta$ | -
C_30_40 | $4.28\delta$ | $0.71\delta$ | 4.35
C_30 | $4.28\delta$ | $0.68\delta$ | -
C_2.5 | $4.86\delta$ | $0.54\delta$ | $16.6$
C_2.5_27.5 | $5.08\delta$ | $0.54\delta$ | $11.9$
C_40 | $5.72\delta$ | $0.51\delta$ | 3.52
C_12.5_27.5 | $4.28\delta$ | $0.02\delta$ | -
No control | $5.57\delta$ | $0$ | $0$
C_15_30 | $4.41\delta$ | $-0.03\delta$ | -
C_15 | $5.35\delta$ | $-0.06\delta$ | -

###  IV.2 Time resolved schlieren visualization

Time resolved Schlieren visualization data with the control configurations
reveals some important details concerning the effect of control on
unsteadiness. Experiments were performed on three control cases: C_2.5_12.5,
C_2.5 and C_12.5_27.5. Configuration C_2.5_12.5 was chosen as it is the best
performing case of all, and configuration C_12.5_27.5 was chosen in order to
see the effect of one bleed hole placed close to the separation location but
without a hole in high pressure region. Configuration C_2.5 was chosen to see
the solo effect of bleed from the hole in the high pressure region.

Figure 13: Time series of shock foot for (a) C_2.5_12.5, (b) C_2.5 and (c)
C_12.5_27.5. (d) Mean separation distance versus standard deviation for
baseline and control configurations.

The separation shock foot location at the midspan was extracted from each
snapshot using the same procedure described in the previous section in Fig. 9
for the baseline case. The shock foot location time series for control cases
C_2.5_12.5, C_2.5 and C_12.5_27.5 are shown in Fig. 13(a,b, and c). The
corresponding mean and standard deviation of shock foot location for each case
was computed.

The mean separation shock location for the control case C_2.5_12.5 calculated
by averaging the shock foot data is found to be at $4.87\delta$ from the
protrusion, which is approximately $20\%$ ($1.23\delta$) lesser (downstream)
than the baseline case’s mean separation length ($6.1\delta$). In a similar
fashion, the mean shock foot distances for the cases C_2.5 and C_12.5_27.5 are
also lower by $13\%$ and $8\%$ each with their mean positions at distances
$5.29\delta$ and $5.59\delta$ from protrusion foot respectively. One can note
that for the C_12.5_27.5 case, although the separation point at midspan is
much downstream ($4.28\delta$), the reduction in midspan separation shock is
not as much as the reduction in midspan separation length. It must also be
noted that this configuration almost has zero MRSL, and at midspan the
separation line turns sharply and reaches the position of the baseline case in
a short spanwise distance, which could be the reason for the observed mean
shock foot position at midspan.

A plot of Mean ($\mu/\delta$) versus Standard deviation ($\sigma/\delta$) of
shock foot position obtained for each of these cases is shown in Fig. 13d to
understand the relation between the effect of control in reducing ‘mean
separation length’ and in ‘shock intermittent region (sweeping region)’. The
arrow in the image indicates the direction towards a desirable flowfield
condition. Broadly, suppression in the intermittent region seems to be
associated with a reduction in overall separation length. It can be noted that
despite having only one bleed hole in the case of C_2.5 it has performed
significantly well, especially in suppressing the shock intermittent region.
In addition to the bleed hole in the high pressure region, when a bleed hole
is also introduced in the low pressure region (C_2.5_12.5), the combined
performance of suction from these two holes results in nearly comparable
suppression of shock intermittent region. This is also the configuration with
the best MRSL. This clearly demonstrates the superior effectiveness of bleed
from high-pressure region in controlling both separation and shock
oscillations. However, with suction from one hole in low pressure region and
one in the plateau region (i.e., for C_12.5_27.5 case) the performance does
not look the best in terms of reduction in both separation length and standard
deviation.

Probability Density Function (PDF) plots were obtained for the control cases
too based on the shock foot data presented in Fig 13(a,b, and c), using the
same procedure employed for the baseline case as discussed in section III.3.
Fig 14 shows probability density of control cases C_2.5_12.5, C_2.5 and
C_12.5_27.5 along with the baseline case. All the curves can be seen to be
symmetric with Gaussian like distribution. The curves corresponding to the
configuration C_2.5_12.5 and C_2.5 are narrower and taller relatively, which
illustrates that the standard deviation corresponding to these two cases is
considerably low when compared with the other two cases. This also implies
that the intermittent region corresponding to these two configurations is
comparatively smaller. However, the distributions for C_12.5_27.5 and baseline
cases are more spread out and have relatively lower peak values. This
corresponds to the fact that their separation shock foot oscillates in a wider
region.

Figure 14: Probability Density Functions (PDFs) of shock foot time series
about mean for baseline, C_2.5, C_2.5_12.5 and C_12.5_27.5 control
configurations

##  V Fourier and Proper Orthogonal Decomposition (POD) analysis of shock
unsteadiness

The reduction in mean separation lengths and shock foot sweeping range, with
the introduction of suction control in configurations C_2.5_12.5 and C_2.5 was
clearly recognized from the statistical analysis presented in the previous
subsection. In this subsection, the frequency components present in the shock
oscillations for baseline, C_2.5_12.5 and C_12.5_27.5 cases are analysed by
performing Fourier analysis and Proper Orthogonal Decomposition (POD) of
temporal schlieren snapshot data.

###  V.1 Fourier analysis of shock oscillations

Time series of shock foot position was obtained from the best line fit for the
separation shock from the instantaneous snapshots as discussed in the previous
section. However, since the best line fit could filter out some frequencies
which could otherwise be observed at individual shock locations, shock foot
time series is not subject to Fourier analysis. Instead, a scan of intensity
along a horizontal scanning line passing through the shock, at the level of
the edge of the (undisturbed) boundary layer, is used to identify the
instantaneous shock position at this particular height, whose time series was
analysed.

The schlieren snapshots were first processed before extracting the shock
position information to increase the precision of the analysis. Fig. 15 shows
a processed instantaneous schlieren image acquired using the procedure
described in section III.3 (with reference to Fig. 9). The shock was detected
by scanning pixel intensity values from left to right along a scanning line at
$y=\delta$ shown with blue dotted lines in Fig 15. The corresponding intensity
variation along the scan line has also been shown above it with a solid red
line. The pixel location corresponding to the first sudden change in intensity
value was taken as the position of the separation shock surface in the midspan
passing through the scan line in each snapshot. The process was automated
through a python program to locate shock position on the scan line in all
other snapshots for baseline, C_2.5_12.5, and C_12.5_127.5 cases. The time
series containing locations of pixels corresponding to separation shock on the
scan line for each snapshot for various cases is obtained and used for the
Fourier analysis, whose results are presented subsequently.

Figure 15: (a) Sample instantaneous raw schlieren image, (b) Processed
schlieren image with scanning line and intensity along the line indicated for
shock position detection.

To estimate the Power Spectral Density (PSD) - $G(f)$ for the Fourier analysis
presented in this section, MATLAB’s pwelch function was used. A total of
$8000$ snapshots captured at $30,000$ Hz was used for the analysis, which
suffices to resolve the expected low-frequency unsteadiness (around $1000$ Hz)
characteristic of two-dimensional SBLIs. Fig. 16 shows the pre-multiplied
Power Spectral Density ($f\times G(f)$) versus the Strouhal number for time-
series data constructed for baseline, C_2.5_12.5 and C_12.5_27.5 cases using
the procedure mentioned above. While it is observed that the most energetic
Strouhal numbers for all three cases are in the order of $10^{-2}$, a
difference in the spread of the dominant frequencies is noted among them.
These frequencies, which correspond to three-dimensional interactions, can be
noted to be the same as the widely reported low frequencies found in two-
dimensional interactionsClemens and Narayanaswamy (2014). In comparison to the
baseline case, the spectrum of dominant oscillation frequencies corresponding
to C_2.5_12.5 is spread widely around $St=10^{-2}$ with lesser power. This
suggests that the effect of suction has resulted in wider distribution of
energy among various frequencies. In contrast to this, configuration
C_12.5_27.5 has exhibited a distinct peak at a Strouhal number of $1.8\times
10^{-2}$ ($\approx 1600$ Hz). This interesting observation suggests that the
presence of a bleed hole at the separation zone causes separation shock to
oscillate almost periodically. Further, it may be recalled that asymmetry
about mid-span was observed in separation lines for some cases with upstream
bleed holes (such as separation/plateau pressure zones). It leads to a
speculation that these ‘oscillations’ could also be characterised by
alternating forward and backward motion of the separation line on either side
of the mid-span, although a confirmation of this speculation requires
sophisticated diagnostics and visualisation.

Figure 16: Power spectral density (PSD) of shock position extracted from
schlieren images for: (a) Baseline, (b) C_2.5_12.5 and (c) C_12.5_27.5 cases.

###  V.2 Snapshot POD analysis

Proper Orthogonal Decomposition (POD) is a dimensionality reduction technique
that can be used to decompose temporal flow-field data into a series of
respective “orthogonal spatial basis functions” or also called as “POD modes”.
Each POD mode presents a low-dimensional description of the entire unsteady
flow-field with certain energy. This technique has been widely used by fluid
dynamicists since 1967 Lumley (1967) especially on the experimental PIV data-
sets Fujiwara  _et al._ (2020) and data from computational simulations
Sengupta and Dey (2004) to uncover important spatial and temporal scales
associated with the flow-field. However, in a few recent studies, the
technique has also been applied to the field of view schlieren snapshots Berry
_et al._ (2017). In addition, the temporal characteristics associated with
each mode can also be studied by performing Fourier analysis of ‘temporal mode
coefficients’ which are obtained by projecting the snapshot vector data onto
the intended orthogonal basis vector (POD mode). These ‘temporal mode
coefficients’ essentially carry dynamical information corresponding to the
mode. Details of the various steps and the algorithm used to perform snapshot
POD in the current work are being omitted here for brevity. A good summary of
snapshot POD and the method used in the current study can be found in Meyer
_et al._ (2008).

The current subsection is dedicated to extract frequencies corresponding to
separation shock dynamics as observed in dominant POD modes and compare them
with the frequencies obtained in the previous sub-section V.1 where the
separation shock is tracked using image processing. To do so, various mode
shapes were extracted from the time resolved schlieren data, and Fourier
analysis was done on the mode coefficients to extract the frequencies present
in each corresponding mode. Before proceeding into the POD analysis of shock
oscillations, firstly, the outlier frequencies that do not contribute to the
flow physics will be identified.

####  V.2.1 Identification of the outlier frequencies

Figure 17: A processed schlieren snapshot and windows of cropped regions
considered for POD analysis.

To avoid any frequencies corresponding to freestream turbulence to appear in
the spectrum, POD analysis was done on the processed images which were
obtained through the method detailed in section III.3 (with reference to Fig.
9). Further, the analysis of the SBLI was only restricted to a cropped portion
of the entire image, marked with a red box in Fig. 17. Before carrying out the
main POD analysis, the outlier frequencies such as the low frequencies
corresponding to the tunnel/set-up vibrations were characterized by performing
a separate POD analysis. This POD analysis was done on a window containing
only the wall and freestream which is highlighted with a blue dashed rectangle
in Fig. 17. The mode-1 of the analysis, shown in Fig. 18a, which carries
$70.91\%$ of the total energy, clearly corresponds to tunnel wall motion; this
can be inferred by noting the highlighted portions in the mode shape. The low
frequencies corresponding to this mode are of Strouhal numbers $5.8\times
10^{-4}$ and $7.9\times 10^{-4}$ which needs to be dis-regarded if appeared in
the POD analysis done in the main window. The second mode shown in Fig. 18b
carries much lesser energy ($\approx 9\%$) in comparison with the first mode,
and can be neglected.

Figure 18: (a) Various POD mode structures (corresponding mode and energy is
indicated on top) and (b) PSD spectrum of POD mode coefficients.

####  V.2.2 POD modes and the spectrum of temporal coefficients

The first two decomposed POD modes of schlieren data (density gradient
fluctuation fields) and their corresponding spectrum are presented in Fig. 19
for baseline, C_2.5_12.5, and C_12.5_27.5 cases respectively. The
corresponding mode energy is also presented at the top of each mode shape. The
color range for the intensity of pixels ($I$) in the mode images presented
here is normalized with the maximum intensity value ($I_{max}$) in the image.
It was observed that, in all three cases, a significant amount of energy
($>30\%$) is present in the first two modes. In addition, a drastic drop in
energy content can be observed between the first and second modes in all three
cases. It can be observed that the region corresponding to the separation
shock motion is clearly highlighted in the first mode structure in all three
cases. This means that physically the first mode carries the dynamics
corresponding to separation shock oscillations in all three cases (Fig 19a,b,
and c). The contribution from other parts of the flow is noted to be minimal
in mode-1 as the magnitude of mode intensity values at locations apart from
the separation shock region are close to zero. In comparison to mode-1 of the
baseline case, the high and low-intensity regions (red and blue) of C_2.5_12.5
mode-1 shown in Fig. 19b can be noted to be much thinner. It can be re-
confirmed here again that for case C_2.5_12.5, the intermittent region
corresponding to the separation shock is thinner and oscillation amplitudes
are much less.

The spectrum of mode coefficients for the first two modes of the baseline case
and C_2.5_12.5 case can be noted to have frequencies that are distributed in a
broadband range around $St=10^{-2}$. Whereas in the case of C_12.5_27.5, the
mode-1 spectrum has a relatively narrow band distribution with high amplitudes
peaks close to $St=1.9\times 10^{-2}$ or $f=1660$ Hz (indicated in the figure
with a red circle). These distinct peaks in the narrow band suggest a possible
periodic oscillation of the shock for this bleed configuration, as also noted
in the previous subsection on shock oscillations. The observations noted here
in the Strouhal number values for both baseline and control cases are in great
consistency with the spectra shown in Fig. 16.

It can be noted from the PSD plots that while the frequencies have slightly
increased due to the introduction of control, the amplitudes have reduced
almost by a factor of two. For instance, while the peak amplitudes
corresponding to mode-1 of the baseline case is close to $2\times 10^{7}$, the
peak amplitudes for the mode-1 spectrum of control cases are only close to
$1\times 10^{7}$. The lower amplitudes may suggest lesser shock sweeping
lengths, which is an indication of the amount of control achieved. In the
mode-2 spectrum of C_2.5_12.5 and C_12.5_27.5 cases, the tunnel oscillation
frequencies are clearly visible. These Strouhal number values (marked in the
mode-2 spectra of Fig. 19) are in very close agreement with the frequencies
identified in Fig. 18a. Considering the amplitudes of tunnel oscillations as a
reference, the energy of oscillations corresponding to SBLI in C_2.5_12.5 case
is much lower and comparable with those of tunnel oscillations, unlike the
baseline case, which has much higher oscillation energy with reference to
tunnel oscillations.

To summarize the observations presented in this section, firstly, it was noted
that the introduction of suction at high and low-pressure regions together
caused the shock oscillation amplitudes to reduce by almost two times.
Consequently, this has caused the separation shock to oscillate in a smaller
intermittent region with approximately two times higher frequency compared to
the baseline case. However, when the suction was introduced at the low and
plateau pressure regions, the oscillation amplitudes (in mode-1 Fig. 19c) and
the intermittent region’s size remained almost unchanged. Nevertheless, the
frequency distribution of the shock oscillations has turned to a narrowband,
with shock oscillating at approximately two times the peak frequency noted in
the baseline case. These observations, in general, suggest that suction from
the deeper part of the separation bubble, i.e., from the high-pressure zone
offers a better control strategy.

Figure 19: Mode shapes and PSD of mode coefficents for (a) Baseline (b)
C_2.5_12.5 (c) C_12.5_27.5. The mode number and its energy are indicated on
top of each mode shape.

##  VI Conclusions

Experiments with supersonic flow ($M=2.87$) over square protrusion, roughly
twice the local boundary layer thickness ($\delta$) in width and height,
resolved some critical aspects associated with the three-dimensional shock
structure and separation bubble in-front of the protrusion. Separation line
was identified using surface oil flow visualization; the separation at
spanwise centre was identified to be $5.57\delta$ from the protrusion face.
From the time-resolved schlieren observations, the separation shock-foot was
found to be at $6.1\delta$ from protrusion on an average, and had a shock
sweeping-length of approximately $2\delta$. It was noted from the surface
pressure survey that the three-dimensional relieving effect, observable from
the streaklines from oil flow visualization, has resulted in three pressure
zones, namely: plateau region downstream of separation, followed by a low-
pressure region, and then a high-pressure region in front of the protrusion,
unlike only two pressure zones that are typically found in two-dimensional
interactions. The measurements and insights obtained from these experiments
provided the basis for the design of bleed configurations (in positioning the
suction holes at various pressure zones along the spanwise centreline), and
served as the baseline case with which the experimental results with bleed
control were compared.

Experiments with bleed control were performed with a total of ten different
suction hole configurations. Oil-flow visualizations have revealed that the
effect of suction from various pressure regions has different implications on
the separation line geometry, and the unsteady dynamics. A few configurations
such as C_2.5_12.5 and C_2.5_12.5_27.5 with a bleed hole in the high pressure
region have showed reduction in separation length throughout the span, along
with midspan reduction of $0.92\delta$ (16.5% less) and $0.80\delta$ (14.2%
less) respectively. On the other hand, configurations without any hole in high
pressure region and with a hole in plateau region, such as C_12.5_27.5 and
C_15_30 had only shown a reduction in midspan separation length (as high as
$1.29\delta$ in some cases), due to the presence of bleed hole close to the
separation line. Despite smaller separation lengths along the midspan in these
two cases, the overall performance in reducing separation length along span
was observed to be poor, due to the larger separation length when compared
with baseline case in some range of spanwise distances away from spanwise
center. To quantify the overall reduction in separation length across span,
Mean Reduction in Separation length (MRSL) was computed as the spanwise
averaged distance of the separation line from the protrusion face. Out of all
the configurations employed, configuration C_2.5_12.5, having two bleed holes,
one in the high-pressure zone and other in low-pressure zone, showed good
overall performance, with MRSL of $0.86\delta$. However, in few configurations
without any hole in the high pressure zone, the MRSL was even observed to be
negative, that is, having an increase in separation length overall despite
higher reduction in separation length in midspan. Comparison of observations
from various control configurations suggests that bleeding the flow from the
high pressure region in the separation bubble (near the foot of the
protrusion) will produce high reduction in the separation lengths in general.
The bleed mass-flow rate measurements suggest that the control performance has
a stronger dependency on the hole placement rather than the net amount of
bleed rate through the holes.

The Fourier analysis and snapshot POD has illustrated the low frequency
separation shock unsteadiness (similar to two-dimensional interactions) in the
order of $St=10^{-2}$ for baseline and control configurations C_2.5_12.5 and
C_12.5_27.5. Specifically, broadband frequencies in the range of approximately
$St=3\times 10^{-3}$ to $4\times 10^{-2}$ ($250$ to $3500$ Hz) were observed
in baseline and C_2.5_12.5 cases. Along with the reduction in mean separation
length and shock sweeping length discussed above, the introduction of suction
in control configuration C_2.5_12.5 has also attenuated the amplitude of
frequency peaks observed in the spectrum meaning the oscillations are
comparatively less energetic. In contrast to the baseline and C_2.5_12.5
cases, a narrow band, almost looking like a single peak, around a
$St=1.9\times 10^{-2}$ ($\approx 1600$ Hz) was noted for C_12.5_27.5 case,
with the bleed hole placed near the separation line, making the oscillations
periodic.

###### Acknowledgements.

The research reported in this article is partly supported by the Science and
Engineering Research Board (SERB), New Delhi, under Grant No. SRG/2019/001793.
The authors would like to thank Prof. G Rajesh and his group at the Department
of Aerospace Engineering, IIT Madras, for kindly letting us use some of their
experimental equipment for this study. Additionally, the authors would like to
thank Siva Vayala for the discussions on this topic.

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