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Experiments in Fluids (2020) 61:151
https://doi.org/10.1007/s0034802002975x
RESEARCH ARTICLE
Experimental characterization ofthehypersonic ow aroundacuboid
ThomasW.Rees1 · TomB.Fisher2· PaulJ.K.Bruce1· JimA.Merrield3· MarkK.Quinn2
Received: 20 December 2019 / Revised: 9 April 2020 / Accepted: 9 May 2020 / Published online: 12 June 2020
© The Author(s) 2020
Abstract
Understanding the hypersonic ﬂow around faceted shapes is important in the context of the fragmentation and demise of
satellites undergoing uncontrolled atmospheric entry. To better understand the physics of such ﬂows, as well as the satellite
demise process, we perform an experimental study of the Mach 5 ﬂow around a cuboid geometry in the University of Man
chester High SuperSonic Tunnel. Heat ﬂuxes are measured using infrared thermography and a 3D inverse heat conduction
solution, and ﬂow features are imaged using schlieren photography. Measurements are taken at a range of Reynolds numbers
from
40.0 ×103
to
549 ×103
. The schlieren results suggest the presence of a separation bubble at the windward edge of the
cube at high Reynolds numbers. High heat ﬂuxes are observed near corners and edges, which are caused by boundarylayer
thinning. Additionally, on the side (oﬀstagnation) faces of the cube, we observe wedgeshaped regions of high heat ﬂux
emanating from the windward corners of the cube. We attribute these to vortical structures being generated by the strong
expansion around the cube’s corners. We also observe that the stagnation point of the cube is oﬀcentre of the windward
face, which we propose is due to sting ﬂex under aerodynamic loading. Finally, we propose a simple method of calculating
the stagnation point heat ﬂux to a cube, as well as relations which can be used to predict hypersonic heat ﬂuxes to cuboid
geometries such as satellites during atmospheric reentry.
Graphic abstract
Extended author information available on the last page of the article
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151 Page 2 of 22
1 Introduction
A half century of human space ﬂight and exploitation has
resulted in approximately 6000 satellites being placed in
orbit around the Earth. Due to their limited operational
lifespan, fewer than one in six of those satellites are still
operational, leaving a large number of decommissioned
satellites in orbit around the Earth (ESA’s Annual Space
Environment 2019). Due to the limited availability of
Earth orbits, the presence of decommissioned satellites
in space increases the risk of inorbit collisions and the
associated risks of space debris. To address these prob
lems, satellites must be disposed of at their endoflife.
The method of disposal generally depends on the orbit
the satellite is placed in. Due to their high altitudes, satel
lites in Geostationary Earth Orbit (GEO) are often raised
to a graveyard orbit well away from the other common
orbits. On the other hand, decommissioned satellites in
Low Earth Orbit (LEO) are often left alone at the end
of their life, as their low orbits will gradually decay due
to the atmospheric drag experienced in LEO. Eventually
this results in an uncontrolled atmospheric reentry. For
smaller satellites, a reentry event will induce temperatures
and forces large enough to destroy the satellite, with with
only limited parts making groundfall. As the satellite size
increases however it becomes more and more likely that
signiﬁcant satellite mass will hit the ground.
Satellite debris impacting the Earth carries with it a
casualty risk which, although small for any given reentry
event, cannot be neglected due to the sheer number of sat
ellites in orbit above the Earth. As a result, it is generally
accepted that space users have a duty to minimize the risks
associated with reentry events (Merriﬁeld etal. 2014). To
this end, the European Space Agency issued an instruction
in 2014 that the casualty risk for any reentry event should
be no greater than 1 in
104
(Dordain 2014). A number
of other national space agencies, including NASA, also
adhere to this ﬁgure (IADC 2007). Estimates of the ground
casualty risk associated with reentry are calculated using
dedicated tools (Koppenwallner etal. 2005; Martin etal.
2005) which must take into account the number of objects
involved, their fragmentation and demise mechanisms, the
eﬀective crosssections of the surviving components, their
most likely locations as they hit the ground, as well as an
accurate population density map of the Earth.
In particular, there remains considerable uncertainty
in predictions of the aerothermodynamic heating rates
induced by the hypersonic flow around satellites dur
ing reentry. This is largely due to the fact that satellite
geometries are signiﬁcantly diﬀerent from most other re
entry bodies—they are typiﬁed by sharp corners, facets,
and multiscale structures. These features cause strong
expansions and compressions in the ﬂow around the sat
ellite, signiﬁcantly thinning or thickening the boundary
layer and therefore increasing or decreasing local heating
rates. Beyond the obvious importance of understanding
what the maximum heat ﬂux to a body is, some recent
studies have suggested that satellite fragmentation mecha
nisms are driven by failure of fasteners and glues rather
than melting of body panels (Soares and Merriﬁeld 2018).
As these components are often located near corners and
edges, fully understanding the heating rates at these loca
tions is particularly important.
The fundamental roadblock to a better understanding of
the reentry heating rates to satellites is that there is very
little freely available highﬁdelity data, either experimental
or numerical, of hypersonic aerothermal heating to faceted
shapes such as cuboids, plates, or cylinders. Heating rates
to ﬂatended cylinders were analysed experimentally and
theoretically in Eaves (1968), Inouye etal. (1968), Klett
(1964), Kuehn (1963), andMatthews and Eaves (1967). In
particular, the work of Matthews and Eaves (1967) iden
tiﬁed that, at certain conditions, a separation bubble can
form immediately downstream of a cylinder’s expansion
edge. Unfortunately, there are no data available for heating
rates under the separation bubble. Nevertheless, the authors
suggested that these heat ﬂuxes, and the separation bubble
formation, are highly dependent on the Reynolds number. A
2D CFD study investigating the eﬀect of Reynolds number
on the hypersonic ﬂow around faceted shapes conﬁrmed that
the formation of such a separation bubble was dependent on
Reynolds number (Rees etal. 2018), and that the presence of
a separation could signiﬁcantly decrease local heating rates.
This reduction in local heating is especially signiﬁcant in the
context of satellite demise as it will result in an increased
casualty risk. In recent freeﬂight experiments of a cube in
a hypersonic ﬂow (Seltner etal. 2019), the authors claimed
the presence of a separation at the leading edge of the cube,
but the resolution of the schlieren was not high enough to
capture it in detail. In addition to cylinders, heating rates
to cuboids have also been studied in the reports of Crosby
and Knox (1980), and Laganelli (1980), who experimentally
measured heat ﬂuxes to a cube in a Mach 8 ﬂow at discrete
locations using thinfoil calorimeters. However, these stud
ies only report results at one ﬂow condition and at limited
discrete locations on the model surface.
This scope of this work is the experimental study of
the hypersonic ﬂow around a cuboid shape, with empha
sis placed on leveraging modern measurement techniques
to obtain accurate heat ﬂux measurements, especially near
the corners and edges of the geometry. To achieve these
highﬁdelity heat ﬂux measurements, the temperature ﬁeld
is measured over the entire surface of the wind tunnel model
using InfraRed Thermography (IRT), and the heat ﬂux is
calculated by the solution of a threedimensional inverse
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Experiments in Fluids (2020) 61:151
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heat conduction problem (3DIHCP). In this way, the Stan
ton number,
CH
is calculated at every point on the surface
of the cube, including in the regions closest to the corners
and edges. The use ofIRT is advantageous due to the fact
that it is a nonintrusive measurement technique, with a high
spatial resolution, low response time, and high sensitivity. In
addition to the IRT measurements, the ﬂow structure around
the cube is imaged with schlieren photography to further
study the separation formation described in Matthews and
Eaves (1967), Rees etal. (2018), andSeltner etal. (2019).
2 Flow facility, models, andtest conditions
2.1 High SuperSonic tunnel (HSST)
Experiments have been performed in the University of Man
chester’s High SuperSonic Tunnel (HSST), based in the
department of Mechanical, Aerospace and Civil Engineer
ing. HSST is a longduration blowdown facility with an
electric resistive heater and a swappable nozzle. A schematic
diagram of the wind tunnel is presented in Fig.1, and a table
of the achievable ﬂow conditions with a Mach 5 nozzle are
presented in Table1. Optical access to the working section is
provided by two parallel, rectangular quartz windows which
span the full length of the useful test jet. Infrared access is
aﬀorded via a
75 mm
diameter uncoated germanium win
dow. Fixed model mounting positions are provided by an
arcbalance sting, which allows the model to be mounted
at angles of attack of
±20◦
. Model orientation on the sting
is aﬀorded by a keyway and grubscrew arrangement. A
detailed description of the tunnel and its operation can be
found in Erdem (2011), andFisher (2019).
2.2 Schlieren
Schlieren images were acquired through Töpler’s Ztype
schlieren method. Two 12 in.diameter f/7.9 mirrors pass the
light from a Newport optics model 66921 Xenon arc lamp,
typically at 450 W, onto the knifeedge in the cutoﬀ plane
which is then focused though a
500 mm
focal length ach
romatic doublet lens onto the camera sensor. The images
are captured with a commercial Nikon D5200 24megapixel
DSLR camera.
2.3 Models, materials, andmounting
Two model geometries are tested: a
30 mm
length cube
and a
30 mm
diameter hemisphere, with the hemisphere
model being used to validate the IRT and heat ﬂux calcula
tion techniques (see Sect.6.1). The models were mounted
to a sting adaptor, which allows the model to ﬁt to the
arcbalance sting (Fig.2). By swapping the sting adap
tor, the models can be mounted in diﬀerent roll orienta
tions, allowing diﬀerent facets of the cube to be imaged
by the IR camera and schlieren. For IRT measurements,
the cube model is oriented in a rolled
45◦
orientation such
that three surfaces of the cube are imaged simultaneously
(see Fig.5a for a sample IR image of a cube model). In this
way, temperature data at a corner of the cube are obtained.
Table 1 HSST characteristic ﬂow conditions with a Mach 5 nozzle
Parameter Min Max
T0
range [K] 320 950
P0
range [kPa] 200 850
Run time [s] 0.5 7.5
Enthalpy [kJ/kg] 19.8 654
Re ×105
[m−1]9.69 226
Test jet diameter [m] 0.152
Test gas Air
Fig. 1 Schematic diagram of the HSST facility
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For schlieren measurements, the cube is mounted in an
unrolled orientation so that all relevant ﬂow structures can
be imaged.
The models are manufactured from MACOR®, a machin
able glassceramic. It was chosen due to its favourable ther
mal properties, high emissivity, success in previous experi
mental hypersonic IRT applications (Cardone etal. 2012),
and ease of machining. Measurements of the temperature
variation of MACOR’s thermal properties are known from
Imbriale (2013), and are plotted in Fig 3. The directional
emissivity variation of MACOR has been reported in Car
done etal. (2012). Imbriale (2013) correlated this data to
ﬁnd a correlation of the form
with coefficients
𝜀0=0.934
,
a=0.0098
, and
b=2.4
.
The temperature variation of the emissivity of MACOR is
unknown, and is assumed to be negligible in the current
tests. However, measurements of the emissivity temperature
variation ofsimilar ceramic materials such as fused silica
glass (Clayton 1962) suggest that the emissivity variation
of such materials is very small up to temperatures of the
order of
530 K
.
The sting adaptors are manufactured from
RigurTM
, a 3D
printed simulated polypropylene with a high thermal defor
mation temperature (Rigur Polyjet 2016).
2.4 Test conditions
The conditions achievable in HSST (Table1) are much less
energetic than real reentry ﬂows, which generally have
enthalpies on the order of tens of MJ/kg and Mach num
bers on the order of 25–30 (for reentry from LEO). Despite
the divergence in total energy between the present tests and
ﬂight conditions, the higher density and lower velocity in
HSST means that the Reynolds numbers achieved in the
wind tunnel are still representative of reentry Reynolds
numbers, which are typically on the order of
104
at 80 km.
Furthermore, due to the exponential increase in the density
of the Earth’s atmosphere during the initial stages of re
entry, the Reynolds number of reentry ﬂows can increase
rapidly at altitudes around 80 km while the Mach number
only varies weakly.
For these two reasons, the experimental ﬂow conditions
are speciﬁcally chosen to investigate the eﬀect of Reynolds
(1)
𝜀
(𝜃)=
(
𝜀
0
cos (𝜃)
)
a
cosb(𝜃)
,
Fig. 2 Experimental setup in the HSST working section
300 350 400 450 500
1.34
1.36
1.38
1.4
1.42
1.44
1.46
700
750
800
850
900
950
1000
Fig. 3 Temperature variation of MACOR thermal properties given by
Corning Inc. and as reported in Imbriale (2013)
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Experiments in Fluids (2020) 61:151
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number on the ﬂow ﬁeld and surface heat ﬂuxes, while main
taining a constant Mach number. The tested ﬂow conditions
are presented in Table2. IRT measurements are taken at four
conditions at Mach 5 with nominal
T0=800 K
and nominal
P0
varying from 200 to
800 kPa
. Further schlieren images
are taken at a ﬁfth condition with nominal
T0=350 K
. Due
to the fact that hypersonic ﬂow ﬁeld behaviour is relatively
weakly dependent on Mach number (known as Mach num
ber independence), these experimental conditions can still
provide valuable insight into reentry ﬂow behaviour.
3 Infrared measurements anddata
processing
3.1 Infrared camera andcalibration
The calibration of the infrared camera follows the basic
principles described in Carlomagno and Cardone (2010).
The infrared camera used is a FLIR A655SC ﬁtted with a
25◦
FOV lens. The detector resolution is
640 ×480
pixels,
and the frame rate is
50 Hz
. The IR calibration is performed
using a Fluke 9132 portable infrared calibrator. The cali
brator consists of a quasiblackbody target with
𝜀=0.95
which can be heated up to 500 °C in 0.1 °C increments. The
calibration is performed insitu, that is with the calibrator
placed in the tunnel working section with the camera view
ing the calibrator through the germanium window. In this
case, the total radiant intensity detected by the camera
ID
,
can be written as:
where
𝜏atm
and
𝜏opt
are the transmissivities of the atmosphere
and germanium window,
Iobj
bb
,
Iamb
bb
, and
Iopt
bb
are the radiant
intensities corresponding to a black body at the tempera
tures of the target, the atmosphere and the optical window
respectively. The emissivities of the target and the window
are denoted
𝜀
and
𝜀opt
respectively. By substituting Planck’s
law into Eq.2, and assuming that the absorptivity of the
(2)
I
D=𝜏opt𝜏atm 𝜀I
obj
bb +𝜏opt𝜏atm (1−𝜀)I
amb
bb
+(1−𝜏
atm
)Iatm
bb
+𝜏
atm
𝜀
opt
Iopt
bb
,
atmosphere is negligible, that is that
𝜏atm =1
, the following
expression is obtained:
where
Tobj
,
Tamb
, and
Topt
refer to the temperatures of the
object, the ambient environment, and the window, respec
tively, and R, B, and F are coeﬃcients of radiation. Assum
ing that
𝜏opt
is constant with temperature, then this coeﬃcient
multiplies into the calibration coeﬃcient R. Furthermore, if
𝜀opt
and
Topt
are constant (which is reasonable for the test
facility in question) then the last term of Eq.3 becomes a
constant C which must be found during the calibration. The
calibration equation, therefore, becomes:
The addition of a constant C to the calibration equation was
proposed by Zaccara etal. (2019) as a way of taking the
camera NonUniformity Correction (NUC) into account and
regulate the diﬀerent gains and zero oﬀsets of each pixelof
the Focal Plane Array. In our case, it simply represents and
corrects for any emission of the germanium window. We are
able to take the camera NUC into account by calibrating the
camera to a NUCcorrected intensity value called the Object
Signal, which is calculated by the FLIR A655SC’s ﬁrmware.
During the calibration, the signal of the calibration tar
get is recorded at 55 evenly spaced data points between
293.6
and
676.5 K
. The LevenbergMarquardt nonlinear
least squares algorithm is used to calculate the calibra
tion coeﬃcients R, B, F, and C in Eq.4. The resulting
(3)
I
D=𝜏opt𝜀
R
eB∕Tobj −F+𝜏opt(1−𝜀)
R
eB∕Tamb −
F
+𝜀opt
R
e
B∕Topt
−F
,
(4)
I
D=𝜀
R
e
B∕Tobj
−F
+(1−𝜀)
R
e
B∕Tamb
−F
+C
.
Table 2 Experimental ﬂow conditions
Reynolds numbers are calculated using the cube length
L=30 mm
Case no. M
Re ×
10
3
P0
, kPa
T0
, K IRT
1 5 40.0 208 782 Y
2 5 79.5 424 796 Y
3 5 109 620 831 Y
4 5 148 835 825 Y
5 5 549 810 348 N
300 400 500600 700
200
400
600
800
1000 Calibration Values
Curve Fit
Fig. 4 Camera calibration curve
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calibration curve is shown in Fig.4, while the calibration
parameters, including the coeﬃcient of determination and
the RMS error are presented in Table3.
The camera is mounted to a Minitec frame ﬁxed to the
ﬂoor of the laboratory. It is positioned at a
48.5◦
angle to
the horizontal axis of the model (Fig.2), which allows it
to image three sides of the cube model.
3.2 Image processing
This section describes the image processing algorithm used
to convert the raw IR data acquired by the camera (Fig.5a)
to temperature values suitable for input to the heat ﬂux
calculation. First, the raw data are ﬁltered using a three
dimensional SavitskyGolay ﬁlter in both space and time.
Following ﬁltering, the IR video is stabilised to remove the
eﬀect of model and sting vibration during tunnel startup and
shutdown. The image registration algorithm used to stabi
lise the IR video is the singlestep discrete Fourier transform
approach proposed by GuizarSicairos etal. (2008), which
has already been used successfully on IR videos (Avallone
etal. 2015). This algorithm calculates the displacement
Table 3 Summary of the
infrared camera calibration Parameter Value
R7160
B1420
F1.3
C74.0
Coeﬃcient of determi
nation
R2
0.997
RMS error 0.1
Fig. 5 Image processing steps for the cube IR data
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Experiments in Fluids (2020) 61:151
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between two images to a subpixel accuracy by computing
the upsampled crosscorrelation between an image and a
reference image by the fast Fourier transform.
Following image registration, the locations of the diﬀer
ent surfaces of the cube in the IR image must be identiﬁed,
and an aﬃne transformation calculated to transform the
perspective view of each of the faces to the square arrays
corresponding to the mesh used in the heat ﬂux calcula
tion method described in Sect.4. Previously, Cardone etal.
(2012) produced a mapping between a surface mesh and
an IR image by means of an optical calibration of the IR
camera. Due to the simplicity of the geometry considered
in this case, we take a much simpler approach. The edges
of the cube in the IR image (Fig.5b) are identiﬁed with a
fuzzylogicbased edge detection algorithm. Following this,
the most likely location of the cube edges are extracted, and
their intersections are used to deﬁne the corners of the cube.
These corner locations are used to deﬁne the moving points
of an aﬃne transformation to a square (Fig.5c). During
the calculation of the aﬃne transformation, the IR image is
downsampled to give a ﬁnal spatial resolution of the cube
temperature maps of
2 pix/mm
(Fig.5d). The sensitivity of
the heat ﬂux calculation to errors in both the edge detection
algorithm as well as the downsampling procedure are dis
cussed in Sect.6.5.
Once the aﬃne transformation has been calculated for
every IR frame, the temperature map on each surface of the
cube can be calculated by applying the infrared calibration
(Eq.4) to each surface. By applying the calibration indepen
dently to each surface, the variation in emissivity of each
surface (due to the directional emissivity eﬀect) can be taken
into account. To calculate the directional emissivity for each
cube face, the viewing angle
𝜃
to each surface is calculated
using knowledge of the camera viewing direction
𝐖
(this is
known from the camera’s orientation in its mounting posi
tion), and thenormal vector to each of the cube’s faces
𝐧
(which is known from the cube’s orientation on the sting):
The sensitivity of the computed heat ﬂux values to errors in
the model emissivity, both due to surface ﬁnish as well as
directional eﬀects, is discussed in Sect.5.
4 Hypersonic heat ux calculation
Once the temperature history of an object has been meas
ured, the heat ﬂux to the surface of the object can be cal
culated by a physical model—the heat conduction equa
tion—of the heat transfer in the measurement area. Walker
(5)
𝜃
=arccos
𝐖⋅𝐧
𝐖
.
and Scott (1998) identiﬁed three diﬀerent classes of such
solutions:
1. Analytical techniques
2. Direct numerical techniques
3. Inverse techniques.
The ﬁrst of Walker and Scott’s three solution classes uses a
theoretical closed form solution to the 1D form of the heat
equation, such as that proposed in Cook and Felderman
(1966) and Kendall and Dixon (1996). If the boundary ﬂux
is piecewise constant and the body material properties are
constant, this is a very simple and quick way of calculating
the heat ﬂux to a body. However, the restrictions on the form
of the boundary conditions, as well as the requirement for
the material properties to be constant are both strong limita
tions. The second of Walker and Scott’s techniques addresses
the ﬁrst’s drawbacks by numerically solving the heat conduc
tion equation using numerical techniques. These could be
a ﬁnite diﬀerence, ﬁnite volume, or ﬁnite element method
with implicit or explicit timestepping. The experimental
temperature measurements are given as Dirichlet boundary
conditions. This allows more ﬂexibility in the solution, as
it permits 2D and 3D conduction to be taken into account,
as well as variable material thermal properties, such as
described in Häberle and Gülhan (2007) and Henckels and
Gruhn (2004). The primary drawback of this technique is
that it involves diﬀerentiating the experimental data, thus
magnifying any experimental error or noise.
The third, most sophisticated and robust method is to
solve an inverse heat condution problem (IHCP). Typically,
inverse problems involve the calculation of an object’s
boundary conditions using knowledge of some internal
conditions (Ozisik and Orlande 2000). In the context of
measuring heat ﬂuxes using IRT, the heat ﬂux to the sur
face is estimated by considering the evolution of the surface
temperature. IHCPs, while oﬀering maximum ﬂexibility
in heat ﬂux calculations (Avallone etal. 2015) also have
their disadvantages, namely their signiﬁcant complexity and
computational cost. In addition to this, they are illposed
problems as their solutions are not unique, meaning their
solutions are extremely sensitive to small changes in input
data (Ozisik and Orlande 2000).
Due to the fact that hypersonic heat fluxes tend to be
extremely high, a common assumption made when calculating
heat ﬂuxes with any of the above techniques is that any trans
verse heat transfer within the body is small compared to the
convective heat ﬂux to the body. In this case, it is reasonable to
neglect any transverse conduction and only consider the heat
ﬂux normal to the body surface. This assumption gains further
justiﬁcation when heat ﬂux is being measured to a model with
a low thermal diﬀusivity. However, in cases where signiﬁcant
transverse conduction is present, the dimensionality of the heat
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Experiments in Fluids (2020) 61:151
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conduction equation must be increased in order to achieve an
accurate solution. Previous applications of a 2DIHCP in the
context of hypersonic aerothermodynamics include Avallone
etal. (2015) and Zaccara etal. (2019), where a 2DIHCP solu
tion was used to calculate the heat transfer caused by turbu
lent transition on a wedge and cone respectively. Both results
showed very limited dependence on transverse conduction.
Other authors (Nortershauser and Millan 2000; Sousa etal.
2012) have performed 3DIHCP solutions on problems con
sidering the heating of small test articles by ﬂames and electric
heaters, but they only considered relatively small computa
tional domains (approximately 300 cells). In the case of the
present work, any assumptions of 1D or 2D conduction cannot
be made—the conduction in the model near the corners and
edges will be strongly two or threedimensional and therefore
the 3DIHCP solution must be solved.
In the remainder of this section, we brieﬂy describe a
method of solving the IHCP on the cuboidal domain by the
conjugate gradient method with adjoint and sensitivity prob
lems, a commonly used IHCP solution methodology (Huang
and Wang 1999; Imbriale 2013). In the summary below, we
follow the derivation of Ozisik and Orlande (2000).
4.1 Denition ofthedirect problem
We start by introducing the direct problem. Consider a
cuboid domain
𝛺
with surfaces
S=S1, ..., S6
. The heat equa
tion on this domain, with Neumann boundary conditions can
be written as:
where
𝜌
,
cp
, and k are the material density, speciﬁc heat
capacity, and thermal conductivity respectively. The material
temperature is T, q is the conductive heat ﬂux, and
𝐱
and t
are the space and time variables.
4.2 Denition oftheinverse problem
The inverse problem can be described as follows: ﬁnd the
value of q(S,t) which gives the known temperature evolu
tion at each measurement point
Ym
on S (the details of the
computational mesh are given in Sect.4.6). In practise, this
is every pixel of the IR image. Start by deﬁning the cost
functional for this problem:
where
Tm
is the solution to 6 for q(S,t) at each measurement
point, and M is the total number of measurement points. The
(6)
⎧
⎪
⎨
⎪
⎩
𝜌cp(T)𝜕T
𝜕t=∇
(k(T)∇T),∀𝐱∈𝛺
T(𝐱,t)=Ti(𝐱,t),∀𝐱∈𝛺,t=0
k(T)𝜕T
𝜕n=q(𝐱,t),∀𝐱∈S,0 <t<tf
(7)
f(q(S,t)) = ∫tf
0
M
∑
m=1
(
Tm(q(S,t)) − Ym
)
2dt
conjugate gradient method (CGM) attempts to iteratively
construct a value of q(S,t) of the form:
where the subscript n denotes the iteration count.
4.3 Calculation ofthesearch step size
andthesensitivity problem
The step size
𝛽n
is taken to be the step size by which the cost
functional
fn
reaches a minimum in the direction
pn
. The
expression for
𝛽n
can therefore be found by minimisingEq.7,
giving:
where
𝛥T
is deﬁned as the directional derivative of the tem
perature T in the direction of q. To ﬁnd an expression for the
evolution of
𝛥T
, it is assumed that a perturbation of
q+𝛥q
in Eq.6 causes a perturbation
T+𝛥T
in the solution. Sub
stituting these in the direct problem, subtracting the original
direct problem from the resulting equations and ignoring 2nd
order terms yields the sensitivity problem:
4.4 Calculation oftheconjugate direction
andtheadjoint problem
The conjugate direction,
pn
can be calculated by the equation
where
∇f
is the gradient of the cost functional and
𝛾n
is
called the conjugation coefficient, calculatedusing the
FletcherReeves formula:
The only unknown quantity in Eqs.11 and 12 is the gradient
of the cost functional,
∇f
. The expression for the evolution
of
∇f
is known as the adjoint equation:
(8)
qn+1(S,t)=qn(S,t)+𝛽npn(S,t)
(9)
𝛽
n=
∫t
f
0
M
m=1
Tm−Ym
𝛥Tdt
∫tf
0
M
m=1
(𝛥T)2dt
.
(10)
⎧
⎪
⎨
⎪
⎩
𝜌cp(T)𝜕𝛥T
𝜕t=∇
(k(T)∇𝛥T),∀𝐱∈𝛺
𝛥T(𝐱,t)=0, ∀𝐱∈𝛺,t=0
k(T)𝜕𝛥T
𝜕n=𝛥q(𝐱,t),∀𝐱∈S,0 <t<tf
(11)
pn(qn)=−∇fn+
𝛾
npn−1,
(12)
𝛾
n=
∫t
f
0
∫
S(∇fn)2dSdt
∫tf
0
∫
S
(∇f
n−1
)2dSdt
.
(13)
⎧
⎪
⎨
⎪
⎩
𝜌cp(T)𝜕𝜆
𝜕t= −∇(k(T)∇𝜆),∀𝐱∈𝛺
𝜆(𝐱,t)=0, ∀𝐱∈𝛺,t=tf
k(T)𝜕𝛥T
𝜕n=2(T−Y),∀𝐱∈S,0 <t<tf
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Experiments in Fluids (2020) 61:151
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Page 9 of 22 151
where
𝜆=∇f
. The derivation of the adjoint problem follows
a process similar to the one used to derive the sensitivity
problem. The details can be found in Ozisik and Orlande
(2000).
Of note is the fact that, due to the nature of the ﬁnal value
problem (Eq.13), the gradient of the cost functional at the
ﬁnal time is zero, that is,
∇f(tf)=𝜆(tf)=0
and, therefore,
the conjugate direction will always be zero at the ﬁnal time.
To overcome this singularity, the gradient at the ﬁnal time
is modiﬁed as follows:
where
𝛥t
is the time step used to solve the heat equation. In
this way, the eﬀect of the singularity is reduced. To further
reduce the eﬀect of the singularity, the IHCP solution cal
culation is extended beyond the end of the tunnel run by 50
time steps (during this time the heat ﬂux to the cube is zero).
4.5 Stopping criterion
The stopping criterion of the CGM can be deﬁned either by
a tolerance criterion, or when the algorithm reaches a mini
mum, that is, when there is negligible change in the solu
tion after a direction reset. In other words, if there are no
measurement errors, the stopping criterion can be deﬁned as
where
𝜀 << 1
. In practice, the temperature measurement
error will place a constraint on how small
fn
can become.
Following Huang and Wang (1999)and Ozisik and Orlande
(2000), the temperature measurement residuals will be
approximately equal to the standard deviation of the tem
perature measurement errors, that is:
Therefore,
𝜀
in Eq.15 can be expressed by:
which gives the appropriate value of
𝜀
for the current
problem.
4.6 Algorithm
In summary, the CGM for solving the IHCP can be described
in the following steps at each iteration n:
1. Solve Eq.6 with
qn
as the boundary conditions.
2. Check the stopping criterion (Eq.15). If satisﬁed, exit
the solution.
3. Solve the adjoint problem (Eq.13) to obtain
𝜆=∇f
.
(14)
∇f(tf)=𝜆(t−𝛥t),
(15)
fn<𝜀,
(16)
Tm−Ym≈𝜎.
(17)
𝜀
=M𝜎
2
t
f,
4. Calculate the conjugation coeﬃcient
𝛾
by the Fletcher
Reeves formula (Eq.12).
5. Calculate the conjugate direction
pn
(Eq.11).
6. Solve the sensitivity problem (Eq.10) with
𝛥q=pn
.
7. Calculate the step length
𝛽n
(Eq.9).
8. Update the solution to obtain
qn+1
(Eq.8).
Previous applications of the CGM (such as Imbriale 2013)
to hypersonic problems did not take into account the tem
perature variation of the material thermal properties k(T)
and
cp(T)
. To do this, we calculate the values of these
properties at every spatial point and time step during the
solution of the direct problem (step 1 in Sect.4.6). These
values are then used at the corresponding locations and
times in the solution of the adjoint and sensitivity prob
lem on the same domain. In this sense, k and
cp
in Eqs.10
and 13 could be written as functions of space
𝐱
and time t
rather than temperature T. These values are then updated
at each iteration during the solution of the direct problem.
For this work, the algorithm has been implemented
in Matlab. The direct, sensitivity, and adjoint problems
are solved with the same forwardtime centralspace
(FTCS) ﬁnite diﬀerencing scheme on an equally spaced,
structured grid with
dx =0.5 mm
. To reduce the compu
tational expense of the IHCP, the ﬂow around the cube
is considered symmetrical and the IHCP is only solved
over one quarter of the cube, that is
−0.5 ≤z∕L≤0.5
and
0≤s∕L≤1.5
(see Fig.5d for coordinates), giving a ﬁnal
grid size of
54 ×103
points. The boundary conditions at
the three surfaces of the domain where temperatures are
unknown are considered to be adiabatic. This is justiﬁed
as the transverse conduction normal to the boundaries in
these regions is likely to be small.
Once the conductive heat ﬂux q has been evaluated,
the modiﬁed Stanton number
CH
is calculated, deﬁned as
where
qrad
is the total radiative heat ﬂux away from a surface,
given by the StefanBoltzmann equation,
qconv
is the convec
tive heat ﬂux to the surface, and
H0
is the total enthalpy of
the freestream ﬂow, given by:
In the remainder of this work, any reference to Stanton num
ber refers to the modiﬁed Stanton number as deﬁned above.
The wall and freestream enthalpy values
hw
and
h∞
are cal
culated using the HOT thermal database package for Matlab
and Octave (Martin 2019).
(18)
C
H=
q+q
rad
𝜌
∞
u
∞(
H
0
−h
w)
=
q
conv
𝜌
∞
u
∞(
H
0
−h
w),
(19)
H0
=h
∞
+u
2
∞
∕
2.
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Experiments in Fluids (2020) 61:151
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151 Page 10 of 22
5 Error sensitivity analysis
To validate the IHCP approach to calculating heat ﬂuxes
outlined in Sect.4, as well as to estimate the errors asso
ciated with the calculations, it is important to investigate
both the uncertainty inherent to the IHCP solution, as well
as the sensitivity of the solution methodology to errors in
the input data. To conduct such an analysis, it is necessary
to use a synthetic (rather than experimental) dataset. In
this way the precise diﬀerence between a true
CH
value
(which is known for the synthetic dataset) and the IHCP
calculated value can be found.
For the sensitivity analysis in this study, the synthetic
heat ﬂuxes applied to the cube geometry are chosen such
that they approximate the expected experimental values.
On the stagnation surface of the cube, the heat ﬂuxes are
given by a modiﬁed version of the expression given by
Klett (1964) for the heat ﬂux to a ﬂatended cylinder:
where L is the cube length, s and z are the coordinates shown
in Fig.5d, and
r2=s2+z2
. The conductive heat ﬂux value at
the stagnation point is chosen as
q0
=7×104W/m
2
.
On the streamwise surfaces of the cube, the heat ﬂux
distribution is given by the Eckert reference temperature
method for a ﬂat plate hypersonic boundary layer. To
include temporal variation in the sample data, the heat
ﬂux values are multiplied by a factor of
(1−e−t∕𝜏)
where
𝜏=8.1667
.
Once the temperature maps on each surface have been
calculated by a solution of the direct problem (Eq.6), the
temperatures are passed backwards through the IR cali
bration to obtain equivalent Object Signal values. This
makes it possible to investigate how various errors (such
as in emissivity and ambient temperature measurement)
propagate through the calibration procedure.
When calculating the Stanton number values, the syn
thetic values of
H0
,
𝜌∞
, and
u∞
are found from typical
HSST total temperature and pressure data for a run at
nominal conditions of
T0=800 K
and
P0=800 kPa
.
Errors are assumed to enter the Stanton number calcula
tion from the following sources:
1. Error inherent in the IHCP solver.
2. Error in the ambient temperature measurement.
3. Error in the camera calibration.
4. Error in assumed emissivity values.
5. Error in the initial temperature distribution over the
model.
6. Error in the value of thermal conductivity.
7. Error in the value of speciﬁc heat capacity.
(20)
q
(s,z)=q0
1+0.6
2r
L
3.3
8. Error in measurement of the wind tunnel total tempera
ture.
9. Error in measurement of the wind tunnel total pressure.
To investigate the sensitivity of the data processing proce
dure to each of these error sources, the inputs to the IHCP are
perturbed by an amount approximately equal to the estimated
measurement error
𝛥
. The resulting error in the Stanton num
ber
𝛥CH
is then characterised using the normalized root mean
squared error:
where
̂
CH
is the true value of the Stanton number, given
by the synthetic data, and
tstart
and
tend
are the time indices
where a steadystate Stanton number is achieved. A sum
mary of the error sources, their assumed error (and justi
ﬁcation), and the resulting values of
𝛥CH
can be found in
Table4.
Traditionally, the global accuracy of the calculated Stanton
numbers could be estimated as the root sum of the squares of
the
𝛥CH
values. However, by this method, the error associ
ated with the IHCP methodology will also be present in each
approximation of the partial derivatives, and the total error will
contain incidences of the IHCP error. To correct for this, we
write the sum of squares error equationas
(21)
𝛥
CH=
�
M(tend −tstart )
∑
tend
t=tstart
∑
M
m=1(CH(𝐱m,t)− ̂
CH(𝐱m,t))2
∑
tend
t=tstart ∑
M
m=1
̂
CH(𝐱m,t)
,
(22)
𝛥
CHtot
2=
(
𝛥CH
𝛥Tamb
𝛥Tamb
)2
+
(
𝛥CH
𝛥TIR
𝛥TIR
)2
+(𝛥CH
𝛥𝜀
𝛥𝜀)2
+(𝛥CH
𝛥Ti
𝛥Ti)2
+(𝛥CH
𝛥k
𝛥k)2
+(𝛥CH
𝛥cp
𝛥cp)2
+(𝛥CH
𝛥T0
𝛥T0)2
+(𝛥CH
𝛥P0
𝛥P0)
2
−7𝛥C2
HIHCP
Table 4 Error sources and sensitivity
Error source Variation
𝛥
𝛥C
H
%
(%)
IHCP solution – 1.5
Ambient temperature measurement
Tamb [K]
1.7% 1.2
Infrared calibration
TIR [K]
𝜎=0.1
K 1.2
Model initial temperature distribution
Ti[K]
2.7% 1.2
Material emissivity
𝜀
8.0% 8.3
Material thermal conductivity
k[J/kg K]
6.0% 1.2
Material speciﬁc heat capacity
cp[J/kg K]
2.0% 1.2
Freestream total temperature
T0[K]
2.5% 7.2
Freestream total pressure
P0[Pa]
4.2% 4.4
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Experiments in Fluids (2020) 61:151
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Page 11 of 22 151
so that only one incidence of
𝛥CHIHCP
is included in the ﬁnal
error value. This gives a global error in
CH
of 12%, which is
dominated by the errors due to the material emissivity (see
Table4). It is important to note, however, that this value is
not constant across the entire computational domain. Fig
ure6 shows how the error value changes across the com
putational domain. In this case, we deﬁne a time average
error,
𝛿CH
as:
For the ﬂat faces of the model, the error is approximately
9%, below the global error of 12%, but in regions of strong
multidimensional conduction the error can increase up to
15%. It is notable that the error is much higher near the
edges and corners of the cube, where threedimensional con
duction is strongest.
The eﬀect of the internal conduction and the magnitude
of the error in the regions where the conduction is strongest
could bedirectly experimentally assessed by placing ther
mocouples internally in the model. Unfortunately, the eﬀect
of installing such instrumentation would signiﬁcantly com
plicate the infrared measurements and the IHCP solution.
Due to the extremely low thermal diﬀusivity of MACOR,
the thermocouples would have to be placed very near the
surface of the cube. The placement of these instruments
(23)
𝛿
CH=
�
(tend −tstart )
∑
tend
t=tstart
(CH(𝐱m,t)− ̂
CH(𝐱m,t))2
∑
tend
t=t
start
̂
CH(𝐱m,t)
.
would aﬀect the distribution of the cube material as well
as the surface temperature of the cube. These factors would
have to be corrected for in the solution of the IHCP which
would require a much more complex mesh.
6 Results
6.1 Heat ux measurement validation
withahemisphere model
Taking IRT measurements of temperature histories and cal
culating heat ﬂuxes with an IHCP solution are both complex
processes with many possible sources of error in both exper
imental setup (including IR calibration) and data reduction
algorithms. Therefore, to validate the combined experimen
tal setup and the IHCP solver, they are used to calculate
the experimental stagnation point heat ﬂux to a hemisphere.
This is a particularly useful geometry with which to validate
the experimental setup as the stagnation point heat ﬂux on
a hemisphere is well characterised, and can be accurately
calculated using a number of diﬀerent methods.
For this application, we obtain a theoretical value of the
heat ﬂux at the stagnation point of a sphere by solving the
self similar form of the boundary layer equations (for their
derivation see Anderson (2006)):
where
f�=
𝜕
f∕
𝜕𝜂
=u∕ue
,
g=g(
𝜉
,
𝜂
)=h∕he
, and
C=𝜌𝜇∕𝜌e𝜇e
. In these relations,
𝜂
and
𝜉
are the LeesDor
onitsyn variables, subscripts
e
refer to ﬂow variables at
the edge of the boundarylayer, and Pr is the ﬂow Prandtl
number. The reader is referred to Anderson (2006) for more
details about these equations as well their derivations. The
equations are solved using the tridiagonal solution method
described in Blottner (1979) and implemented in Adams
(2002). This is preferred to using an existing correlation
(such as the Fay and Riddell (1958) or Sutton and Graves
(1971) correlations) due to the fact that the freestream con
ditions used for these experiments lie outside of the range of
validity of these correlations.
These theoretical heat ﬂux values are compared to experi
mental values at the stagnation point of a
30 mm
diameter
hemisphere in a hypersonic ﬂow with
M=5
,
T0=805 K
,
and
P0=200 kPa
(which corresponds to a freestream
Reynolds number of
Re∕m=1.23 ×106
). The experimen
tal heat ﬂux values are calculated using both the IHCP solu
tion described above, as well as the Cook and Felderman
(1966) equation, a Class 1 method using the classiﬁcation
(24)
Cf ��
�
+ﬀ�� =
f�
2−
g
C
Pr g��
+fg�=0,
Fig. 6 Contours of the error in Stanton number
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Experiments in Fluids (2020) 61:151
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151 Page 12 of 22
system described in Sect.4. The IHCP is solved assuming
1D conduction only (i.e. neglecting any transverse con
duction in the hemisphere model) at the stagnation point.
A plot of experimental Stanton numbers at the stagnation
point of the hemisphere during the run of the HSST is pre
sented in Fig.7, alongside the theoretical value. The time
averaged 1DIHCP value during the windtunnel steady
state
2<t<6
is
CH0
=0.0275
, while the Cook and Fel
derman value is
CH0
=0.0281
, slightly higher likely due
to its assumption of constant material thermal properties.
The boundarylayer self similar solution gives a value of
CH0
=0.0259
, 6% smaller than the IHCP solution, and 8%
smaller than the Cook and Felderman value. How much the
diﬀerence between the theoretical and experimental heat ﬂux
values is due to the IRT measurement technique and heat
ﬂux calculation method, or due to errors in the freestream
ﬂow parameter estimation could be ascertained by perform
ing an additional test with a conventional heat ﬂux probe,
instrumented with a thinﬁlm heat ﬂux gauge. This was not
performed in the present experimental campaign.
The RMS error of the IHCP solution is 1.7%, while
for the Cook and Felderman solution it is 2.2%. These are
slightly lower than the values of 3–4% calculated by Aval
lone etal. (2013), likely due to the slightly higher noise
(
𝜎=0.8 K
) in the temperature histories gathered by Aval
lone etal.
The results described in this section, in combination with
the error sensitivity analysis described in Sect.5 gives fur
ther conﬁdence in the cube results discussed below.
6.2 Cuboid schlieren results
Schlieren photographs of the cube model at high (
549 ×103
)
and low (
40.0 ×103
) Reynolds numbers are presented in
Fig.8. The schlieren appears to conﬁrm the appearance of
diﬀerent ﬂow structures with increasing Reynolds number as
predicted by Rees etal. (2018). Most notable is the appear
ance of an apparent separation shock at the windward edge
of the cube at high Reynolds numbers (Fig.9) which is not
present at lower Reynolds numbers. This pattern is similar to
that imaged by Matthews and Eaves (1967) around a cylin
der, and supports the conclusion made in Rees etal. (2018)
that a separation bubble can form on the sides of a cube at
hypersonic speeds, as Reynolds number is increased, even if
it was not present at lower Reynolds values. The freestream
total temperature was lowered signiﬁcantly to achieve the
high Reynolds number condition, and therefore it was not
possible to take any highquality IRT data at this condition.
0246
0
0.02
0.04
0.06
0.08
0.1
Cook and Felderman (1966)
Current 1D IHCP
Theoretical value
Fig. 7 Hemisphere stagnation point Stanton number evolution during
a tunnel run
Fig. 8 Schlieren images for a cuboid at
M=5
at high and low Reyn
olds numbers
Fig. 9 Labelled schlieren at
M=5
and
Re =549 ×103
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Experiments in Fluids (2020) 61:151
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Page 13 of 22 151
As a result the separation bubble’s eﬀect on the heat ﬂux
could not be quantiﬁed.
6.3 Heat uxes tothecube
The solutions to the 3DIHCP for the cube model at the
fourdiﬀerent freestream conditions are plotted in Fig.10.
These are timeaveraged Stanton numbers calculated by
averaging overthe steadystate portionof the tunnel run time
(
2<t<6
s). The trends in the contours are as expected,
with heat ﬂux increasing due to the thinning boundary layer
as the edges and corners are approached. Notably, in addi
tion to the increases of heat ﬂux at the edges of the cube,
there are also wedgeshaped regions of high
CH
along the
streamwise edges of the cube. These appear to show some
Reynolds number dependence, getting wider as the Reyn
olds number is increased. Figures11 and 12 show plots of
Stanton number normalised by the stagnation point Stan
ton number in both the streamwise (at the centreline of the
cube,
z∕L=0.50
) and spanwise (at
s∕L=1.0
) directions.
The identiﬁcation of the stagnation point Stanton number
is discussed in Sect.7.1. Figure11 shows that the wedges
of increased Stanton number along the streamwise edges
of the cube are bounded either side by a region of slightly
decreased Stanton number. Furthermore, Fig.12 shows that
on the side face of the cube, the Stanton number tends to
reach a maximum at
s∕L=1.0
.
The solutions to the 1DIHCP (neglecting any transverse
conduction) for the cube model at the four freestream con
ditions are presented in Fig.13. Due to the nature of the
1D solution, the 1DIHCP can be solved across the whole
measurement surface of the cube (no symmetry assumption
is necessary) and so results for the entire cube (rather than
a quartercube as in Fig.10) are shown. These results show
broadly similar behaviour to the 3DIHCP solutions, albeit
with lessdeﬁned changes in heat ﬂux near the corners and
edges of the model.
One thing which is notable in the 1DIHCP contours
that is not immediately obvious in the 3DIHCP contours
is the presence of regions of lower Stanton number on the
stagnation surface of the cube. These regions correspond
to the lowtemperature regions visible in the raw IR image
(Fig.5a). These regions are oﬀcentre of the stagnation sur
face, and the relative strength of the reduction in heat ﬂux
appears to increase with Reynolds number. These regions
are also present in the 3DIHCP solution, however, due to
the noisier nature of the 3D solution, theyare not as clear.
6.4 Eects ofthree‑dimensional conduction
The transverse (conductive) heat transfer within the model
is timedependent. Asdiﬀerent parts of the model heat up
at diﬀerent rates the temperature gradients (and therefore
internal conduction rates) willvary with time. A compre
hensive analysis of the eﬀect of internal conductionon the
accuracy of the 1D versus the 3DIHCP solution would take
this variation into account. However, for the purposes of this
analysis, we will simply compare the timeaveraged diﬀer
ence between the
CH
values obtained with the 3D and 1D
conduction assumptions. We deﬁne contours of the diﬀer
ence due to dimensionality as
which are plotted in Fig.14. The results show that, although
the eﬀect of highdimensional conduction is unimportant
on the stagnation surface and large parts of the sidefaces of
the cube, failing to account for 3D conduction in the regions
near the corners and edges of the cube can result in errors
up to 500%. These errors are likely lower at the start of the
tunnel run time and higher towards the end. Furthermore,
there are regions of signiﬁcant highdimensional conduction
around the edges of the hot wedges described above. Despite
these errors associated with the 1DIHCP, the 3DIHCP
solutions also have some limitations. First, the 3D results
are noisier than the 1D results. Thisnoise is captured by the
error contoursshown in Fig.6. More importantly, the 3D
solutions rely on an accurate imagetoIHCP input perspec
tive transformation. If the location of the edge of a cube is
assumed to be even slightly wrong the
CH
calculation can be
signiﬁcantly aﬀected.
6.5 Sensitivity toimage processing andquality
To investigate the sensitivity of the 3DIHCP solution to
errors in the identiﬁcation of theedge locations of the cube,
seven analyses of solution’s sensitivity to the edge location
were performed. For this analysis, the four corners deﬁning
the cube’s stagnation surface (see Fig.5c) were moved up
and down by 3 pixels (in the raw IR image), to give a total
of 6 diﬀerent edge locations, in addition to the baseline loca
tion used in the main results. Furthermore, to investigate
the eﬀect of the downsampling of the IR image, another
analysis was performed wherethe image was processed with
minimal downsampling, resulting in a solution with a spa
tial resolution of 3.7 pix/mm rather than 2.0 pix/mm.
The results of these sensitivity analyses on the Case
4 (high Re) results can be seen by examining the cube
centreline Stanton numbers (Fig.15). As the corners get
moved up the IR image (that is, away from the stagnation
surface), there a region of nonphysical negative Stanton
number that appears after moving the edge location only
2 pixels. These negative Stanton numbers were reported in
Rees etal. (2019), but were not identiﬁed asbeing caused
by errors in the image processing method. As the corners
(25)
𝜀
D=
C
H1D
−C
H3D

C
H3D
,
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get moved down the IR image, that is towards the stagnation
surface, the increase in Stanton number across the cube edge
becomes nonmonotonic after a movement of 3 pixels. The
nonphysicality of this behaviour suggests that the current
image processing algorithm can locate the edge of the cube
in the IR image to within
±2.5
pixels. However, Fig.15c
suggests that the spatial resolution of the image, as well
as the downsampling during the image processing could
alsoaﬀect the accuracy of the edge location. Although the 2
pix/mm and 3.7 pix/mm lines in Fig.15c look very similar,
Fig. 10 Stanton number contours calculated using a 3DIHCP solution for a cube at Mach 5 and at four diﬀerent Reynolds numbers
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we note that the extra noise present in the highresolution
solution may be causing a nonmonotonic change in heat
ﬂux through the cube edge at
s∕L=0.5
. Finally, we note that
the downsampling does not otherwise aﬀect the 3DIHCP
solution, suggesting that downsampling process does not
signiﬁcantly aﬀect the results.
Obviously, accurate identiﬁcation of the cube’s edge
locations in the infrared image is a crucial aspect of obtain
ing accurate heat ﬂuxes using the 3DIHCP solution. Edge
detection algorithms generally rely on the assumption that
edges are locations of high gradient (even if they may
not place the edge at the region of highest gradient). This
assumption is not true in the case of the windward edge
(
s∕L=0.5
) of the cube. Therefore, the localisation of this
edge of the cube, as well as the streamwise edge at
z∕L=0.0
relies on the assumption that the edge is located at the region
of highest temperature, which is not justiﬁed a priori. This
problem could be mitigated by performing an optical cali
bration of the camera and then ﬁtting the IHCP cube mesh
mapped to its surface (as discussed in Sect.3.2 and by Car
done etal. (2012)).
7 Discussion
7.1 Stagnation surface behaviour
The most striking result of the stagnation surface contours in
Figs.10 and 13 is the presence of the regions of lower Stan
ton number. The variation in the Stanton number of the pre
sent experimental data due to these ‘cold spots’ is up to 30%.
The cold spot location and strength also appears to show a
Reynolds number dependence, getting stronger and further
oﬀcentre as the tunnel total pressure (and therefore Reyn
olds number) increases. Although this behaviour perhaps
indicates that the cause of these regions is nonuniformity of
the freestream ﬂow, previous studies of the ﬂow uniformity
of HSST (Erdem 2011; Fisher 2019) suggest that any non
uniformity across the test jet is negligible. Alternatively, an
imperfection in the model surface conditions could cause
such a behaviour. However, the tests were performed with
two diﬀerent cube models, ruling out this explanation.
Instead, we propose a diﬀerent explanation: that these
regions are in fact the stagnation points of the cube, which
have been moved oﬀcentre by the ﬂex of the tunnel mount
ing sting. The stagnation point on the windward surface of
the cube should manifest itself in the Stanton number con
tours as such a cold spot. This is due to the fact that as the
ﬂow accelerates away from the stagnation point on the wind
ward surface, the boundarylayer will thin, increasing the
heat ﬂux, making the stagnation point the region of lowest
heating on the stagnation surface. If the model is perfectly
aligned in the freestream ﬂow direction, the stagnation
point would be at the geometric centre of the stagnation
surface. However, due to the planar nature of the stagnation
surface of a cube (in other words, the radius of curvature
is inﬁnite), the location of the stagnation point is likely to
be highly sensitive to the cube attitude—a small change in
the cube attitude may result in a signiﬁcant change in the
stagnation point location. The ﬂexure of the sting during a
run will change the cube attitude slightly, causing the stag
nation point to move. As the Reynolds number of the ﬂow,
and therefore the aerodynamic force on the cube increases
with the increase in ﬂow total pressure, the sting ﬂex will
get larger, making the change in stagnation point location
and strength stronger. This oﬀset stagnation point would
explain why the plots in Fig.12 do not collapse exactly, and
5.005.0
0
0.2
0.4
0.6
0.8
1
1.2
Re = 40.0 10 3
Re = 79.5 10 3
Re = 109 10 3
Re = 148 10 3
Fig. 11 Spanwise Stanton number proﬁles at
s∕L=1.0
0 0.5 1 1.5
0
0.5
1
1.5 Re = 40.0 103
Re = 79.5 103
Re = 109 103
Re = 148 103
Fig. 12 Streamwise Stanton number proﬁles at
z∕
L
=0.5
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why the Stanton number at the geometric centre of the cube
does not correspond to the stagnation point Stanton number.
To investigate the stagnation point Stanton numbers meas
ured in these experiments, they are compared to correlations
for the stagnation point heating to a ﬂat surface at similar ﬂow
conditions. Previous studies (Matthews and Eaves 1967; Trim
mer 1968) have measured the stagnation point heat ﬂux to a
ﬂatended cylinder and compared it to the stagnation point
heating to a hemisphere. These studies related the relative
heating rates between these two geometries to the eﬀective
stagnation point velocity gradient at the edge of the boundary
layer:
(26)
𝛽
=
du
e
dx
.
We know from many stagnation point correlations that the
Stanton number at a stagnation point is directly proportional
to the square root of
𝛽
:
Assuming that the only diﬀerence between an equivalent
stagnation point ﬂow on a round surface and a ﬂat face is
the velocity gradient, then:
where
CHFF
is the stagnation point Stanton number to a ﬂat
face and
CHSS
is the stagnation point Stanton number to an
equivalent spherical geometry. It should therefore be pos
sible, using knowledge of the diﬀerence in velocity gradi
ent between the two geometries, to ﬁnd a correspondence
(27)
CH∝√𝛽.
(28)
CHFF
∝CHSS ,
Fig. 13 Stanton number contours calculated using a 1DIHCP solution for a cube at Mach 5 and at four diﬀerent Reynolds numbers
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Experiments in Fluids (2020) 61:151
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between the stagnation point heat ﬂux to a ﬂat surface and
an equivalent sphere.
Trimmer (1968) and Matthews and Eaves (1967) per
formed experiments to measure the velocity gradient to
an endon cylinder in hypersonic ﬂow. Cylinders of vari
ous bluntness were tested, from a hemisphericallycapped
cylinder to a ﬂatended cylinder. By comparing the non
dimensionalised velocity gradients measured in these two
studies, the constant of proportionality in Eq.28 is found
to be 0.54 (from the Matthews and Eaves data) and 0.57
(from the Trimmer data). Klett (1964) found similar results,
suggesting a proportionality constant of 0.5, although did
not report what data this value was based on. The Reynolds
number variation of these results (Klett, Trimmer, and Mat
thews and Eaves) are presented in Fig.16, as well as the
current experimental data for a cube. The combined data
show signiﬁcant scatter, although the current experimental
data are generally lower than the earlier data. The average
ratio of the stagnation point Stanton numbers measured
on the cube geometries to an equivalent hemisphere value
(where
Rn=L∕2
) (found using the same selfsimilar solu
tion used in Sect.6.1) is found to be 0.44 on average across
conditions. This is not surprising as the cube will naturally
have an eﬀective nose radius slightly larger than L/2 due
to the presence of the corners. In this sense, a better way
of deﬁning the eﬀective nose radius of a cube would be to
use the diagonal distance across the corners:
Rn=√2L∕2
.
Using this deﬁnition, the average constant of proportionality
for the current experimental data becomes 0.52, closer to the
values reported in the earlier references.
It should be remarked that the previous experimental data
were collected using thinﬁlm heat ﬂux gauges at discrete
locations on the model stagnation surface. Therefore, it
would not have been possible for the authors to capture the
variation in heat ﬂux occurring in the current experimental
data. It is therefore possible that the stagnation point heat
ﬂuxes measured in the earlier data may be larger than the
true stagnation values.
7.2 O‑stagnation behaviour
In the oﬀstagnation regions, that is the side faces of the
cube, the most notable ﬂow feature are the regions of
high heat ﬂux, or wedges, which appear to emanate from
the windward corners of the cube. Such ﬂow phenom
ena have already been observed in CFD simulations of
hypersonic ﬂow around faceted shapes, such as in Gülhan
etal. (2016), but have not been studied experimentally.
The angles of these wedges appear to show dependence
on Reynolds number, suggesting that they are a viscous,
rather than inviscid ﬂow eﬀect. This is in contrast to simi
lar results presented in Rees etal. (2019), which indicated
Fig. 14 Percent errors due to inappropriate conduction assumption
Fig. 15 Centreline Stanton number proﬁles for the Edge Sensitivity analyses for Case 4
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no Reynolds number dependence, leading to the oppo
site conclusion. The likely reason for this discrepancy is
that the results in Rees etal. (2019) only considered a
2DIHCP in the spanwise direction, which resulted in less
spatially accurate wedge shapes. The shapes of the wedges
in the high and low Reynolds number cases are compared
in Fig.17. These proﬁles were obtained by thresholding
the Stanton number contours on the side faces so that only
regions where the heat ﬂux was higher than
0.1%
of the
average heat ﬂux on the surface were visible. The edges of
these thresholded images were then extracted to produce
the proﬁle. The wedge spreading angles for all the tested
conditions are reported in Table5. The heat ﬂuxes under
the wedges can be signiﬁcant, with regions of peak heat
ing reaching up to
100%
of the stagnation point heat ﬂux
value. The average values of the increased Stanton number
caused by these wedges,
C
H
w
(deﬁned as the average value
of the Stanton number bound by the contours in Fig.17)
are reported in Table5.
Notably, the regions of high heat flux wedges are
bounded on either side by a similar region of lower heat
ﬂux (see Fig.11). Such a pattern is often seen in the pres
ence of vortical structures, suggesting that the wedges are
generated by vortices being shed by the corner of the cube.
The Reynolds number dependence of the wedge angle is
further evidence that these wedges are a viscous ﬂow
eﬀect.
Away from these wedges, the heat ﬂux on the side faces
of the cube is much lower than anywhere else. The aver
age values of the Stanton number along the centreline of
the cube,
C
H
c
(that is along
z∕L=0.5
for
s∕L
>
0.5
) are
reported in Table5, and are between 15 and 18% of the
stagnation point value.
7.3 Comparison tosatellite demise heating models
Objectoriented satellite demise tools such as DRAMA
(Martin etal. 2005) prescribe heat ﬂuxes to simple shapes
such as cuboids and cylinders through the use of a heat
ingshape factor
Fsh
, deﬁned as:
where
̂q
is the spaceaveraged heat ﬂux to the object:
and
qss
is the stagnation point heat ﬂux to a sphere with
radius equivalent to the eﬀective nose radius of the object
being considered. By convention, the equivalent nose radius
is taken to be
Rn=L∕2
, rather than
Rn
=
√
2L∕
2
as sug
gested above. In this way, the heat load to an object can be
calculated by ﬁnding
qss
, which is trivial to calculate using
any number of correlations [such as Sutton and Graves
(1971) or Fay and Riddell (1958)], and multiplying by the
shape factor (which is stored in a library containing fac
tors for multiple diﬀerent primitive shapes at many diﬀerent
orientations and attitudes). The shape factors for the experi
mental data are presented in Table5. Although the shape
factors show very little dependence on Reynolds number,
they hide the strong spatial variations in heat ﬂux which
exist in reality. The fact that the highest heat ﬂuxes occur
near the edges of the cube means that these are the regions
(29)
F
sh =
̂q
q
ss
(30)
̂q
=
∫
Sqd
S
S
0.4 0.6 0.8 11.2 1.4 1.6
10
5
0.005
0.01
0.015
Current experimental data
Klett (1964)
Matthews and Eaves (1967)
Trimmer (1968)
Fig. 16 Comparison of ﬂatsurface stagnation point values for diﬀer
ent datasets
Fig. 17 Experimental Wedge shapes at high and low Reynolds num
bers
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Experiments in Fluids (2020) 61:151
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which are likely to fail or melt ﬁrst during destructive re
entry. As a result, the fragmentation and demise of a satellite
is most likely to be driven by the heat ﬂuxes in these regions.
To take the important spatial variations of heat ﬂux into
account, we propose that shape factors
Fsh
should instead
be presented as edgespeciﬁc shape factors, where the aver
age in Eq.30 only includes the cube regions near the edges.
There are two ways of doing this. Either all of the edges of
the cube can be included in the average, or the edges can
be separated into three categories: windward (
s∕L=0.5
),
streamwise (
z∕L=0.5
), and leeward (
s∕L=1.5
, the data
for which are not available from the current experiments),
and then three diﬀerent heat ﬂux averages
̂q
and shape fac
tors can be calculated. To illustrate this, shape factors for
all these diﬀerent deﬁnitions have been plotted in Fig.18.
In this case, the deﬁnition of the ‘edge’ is the region of the
cube surface within 0.1L of the spatial discontinuity. Only
this region is used in the calculation of Eq.30.This deﬁ
nition is slightly arbitrary and the factor of 0.1 could be
increased or decreased depending on the context. For exam
ple, if it is known that the epoxy joints holding the satellite
panel together have a length of
𝛿
then
𝛿∕L
would be the most
appropriate factor to use for edge deﬁnition.
The above results are presented with an implicit assump
tion that the current methodologies used for modelling
satellite demise (broadly outlined in the Introduction) are
the most appropriate way of modelling demise. We implic
itly assume that fragmentation is driven by melting or other
catastrophic failure in regions of maximum heat ﬂux. While
this is a seemingly intuitive model, any model of satellite
demise is extremely challenging to validate experimentally.
Due to the complexity of the demise process, it is possible
that demise is driven by other failure modes, for example
delamination of the aluminium honeycomb sandwich pan
els. Alternatively, studies of Thermal Protection System
(TPS) materials have looked at possible TPS failure mecha
nisms which are initiated by damage to the TPS caused by
micrometeoroid impacts (Agrawal etal. 2013). Such micro
damages could very well be a nucleation point for satellite
demise and failure. If it can be shown that satellite frag
mentation is driven by a phenomenon other than melting at
corners and edges then the results presented in the current
study must be reinterpreted with that in mind.
Finally, as brieﬂy discussed in Sect.2.4, these results are
based on a freestream ﬂow with a fraction of the enthalpy
of a real reentry ﬂow. To further conﬁrm the applicability of
these results to real reentry ﬂows, the current results should
be compared against and supplemented with additional data
from CFD, high energy (shock tube) experiments, or even
ﬂight data. The higher freestream enthalpies considered in
these datasets will cause much higher absolute values of
heat ﬂux, and possibly diﬀerent heating patterns which could
aﬀect the values of the diﬀerent shape factors calculated
above.
8 Conclusions
The Mach 5 ﬂowﬁeld around a cube has been studied exper
imentally, using both schlieren photography to visualize the
ﬂowﬁeld as well as infrared thermography and an IHCP
data reduction method to measure the surface heat ﬂuxes to
the cube. The schlieren images revealed the presence of a
separation bubble on the side surface of the cube at certain
Reynolds numbers. This structure was imaged at a Reynolds
number of
549 ×103
. Unfortunately, due to the low free
stream total temperature required to achieve this Reynolds
number, it was not possible to take any highquality IRT
data, and therefore the separation bubble’s eﬀect on the heat
ﬂux could not be quantiﬁed.
Table 5 Summary of Stanton
number patterns Case no. Re
×10−3
C
H
0
×10
−3
C
H
c
×10
−3
C
H
w
×10
−3
Wedge angle
Fsh
1 40.0 11 1.7 4.4
9.8◦
0.1583
2 79.5 8.1 1.4 3.5
12.0◦
0.1679
3 109 6.4 1.2 2.9
13.0◦
0.1611
4 148 5.7 1.0 2.5
15.1◦
0.1612
0.4 0.6 0.8 1 1.2 1.4 1.6
10 5
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5 All Edges
Windward Edge (s/L = 0.5)
Streamwise Edge (z/L = 0.0)
Whole Cube
Fig. 18 The evolution of diﬀerent shape factors with Reynolds num
ber
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Experiments in Fluids (2020) 61:151
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A detailed error sensitivity analysis of the inverse heat
conduction method for heat ﬂux calculation used showed
that the Stanton numbers calculated by the data reduction
method have an error of 12%, which is dominated by errors
in the material emissivity. These errors appear to be highest
near the regions of the cube where internal transverse con
duction in the cube is strongest, such as corners and edges.
On the stagnation surface of the cube, the heat ﬂux meas
urement results show broad agreement with existing data
for stagnation point heating to ﬂat axisymmetric surfaces.
However, experimental heat ﬂux contours to the stagna
tion surface of the cube shows the presence of distinct oﬀ
centred regions of lower heat ﬂux, which we have termed
‘cold spots’. The Stanton number in these regions can be as
much as 30% lower than the Stanton number to the geomet
ric centre of the cube. Furthermore, their strength appears
to show some dependence on Reynolds number. We deduce
that the cause of these cold spots is due to sting ﬂex during
the tunnel run, causing a misalignment of the stagnation
point. To conﬁrm this hypothesis, a valuable future study
would be to perform an oil ﬂow visualisation of the ﬂow
around the cube. Such an experiment would also conﬁrm the
presence of a separation bubble on the side face of the cube
at higher Reynolds numbers. We propose that the stagna
tion point heat ﬂux to a cube can be estimated by calculat
ing the stagnation point heat ﬂux to a sphere with a nose
radius of
Rn
=
√
2L∕
2
, and multiplying the resulting value
by 0.52, a coeﬃcient very similar to those used to estimate
the heat ﬂux to a ﬂatended cylinder. However, the stagna
tion point heat ﬂux may be as much as 30% lower compared
to other regions on the stagnation surface. The most notable
ﬂow feature on the oﬀstagnation (side) faces of the cube
are wedgeshaped regions of increased heat ﬂux emanating
from the windward corners of the cube. The heat ﬂux under
these wedges can be very high, with regions of peak heating
reaching stagnation point values. The spreading angle of
these wedges show Reynolds number dependence and we
therefore attribute them to the presence of vortical struc
tures that are shed from the corners of the cube. Using the
experimentally measured heat ﬂuxes to the surface of the
cube, we calculated diﬀerent shapefactors describing the
average heating to the cube as a whole as well as edges and
corners, where heating is highest.
Finally, we draw attention to what we believe to be
the two most important weaknesses of the current study.
Firstly, as discussed extensively in Sect.2.4 the ﬂow con
ditions considered in this study are very lowenergy when
compared to true flight conditions. Taking equivalent
experimental infrared measurements at ﬂows correspond
ing to ﬂight conditions would be challenging, as the infra
red data capturing would need to take into the account
the transmissivity of the reacting shock layer, the strongly
varying material properties (including emissivity), and the
very short test times. Preliminary arcjet and plasmatron
studies of the reentry ﬂows around CubeSats (Masutti
etal. 2018) have identiﬁed regions of high heating and
temperature using uncalibrated IR measurements. Rep
etitions of these highenthalpy studies using carefully
calibrated IR measurements would provide valuable data
with which the current measurements could be extrapo
lated to ﬂight conditions. The second weakness of the cur
rent study is that it only considers one orientation of the
model with respect to the freestream. In reality, a satellite
during reentry will be tumbling rather than maintaining
one attitude, constantly changing the heat ﬂux distribution
over the geometry. The time scale of the satellite tumbling
motion is much larger than the timescale of the hyper
sonic ﬂow. As a result, when calculating heat ﬂuxes for a
tumbling geometry, only steady state ﬂowﬁelds need to be
considered. Even with this simpliﬁcation, it is impractical
to gather experimental data at a suﬃciently ﬁnegrained
range of attitude orientations. Therefore, future research
whichconsiders the eﬀect of tumbling on the heat ﬂuxes
to a satellite would have to be largely CFD based, with
judiciously chosen conditions and orientations at which
validation experiments can be performed.
Acknowledgements TWR would like to acknowledge the ﬁnancial
and technical support of the European Space Agency (under NPI 480
2015), Fluid Gravity Engineering, and the Engineering and Physical
Sciences Research Council through the Imperial College Centre for
Doctoral Training in Fluid Dynamics Across Scales(EP/L016230/1).
This work was further supported ﬁnancially by the award of a UK
Fluids Network Short Research Visit grant. The authors would also
like to thank the technicians at the Imperial College Department of
Aeronautics for manufacturing the models used in these experiments.
Compliance with ethical standards
Conflict of interest The authors declare that they have no conﬂicts of
interest.
Open Access This article is licensed under a Creative Commons Attri
bution 4.0 International License, which permits use, sharing, adapta
tion, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source,
provide a link to the Creative Commons licence, and indicate if changes
were made. The images or other third party material in this article are
included in the article’s Creative Commons licence, unless indicated
otherwise in a credit line to the material. If material is not included in
the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will
need to obtain permission directly from the copyright holder. To view a
copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.
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Aliations
ThomasW.Rees1 · TomB.Fisher2· PaulJ.K.Bruce1· JimA.Merrield3· MarkK.Quinn2
* Thomas W. Rees
thomas.rees10@imperial.ac.uk
1 Imperial College London, Exhibition Road,
LondonSW72AZ, UK
2 University ofManchester, Oxford Road,
ManchesterM139PL, UK
3 Fluid Gravity Engineering Ltd., 1 West Street,
EmsworthPO107DX, UK
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