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Spiral instability-modes on rotating cones in high-Reynolds-number axial
flow
Sumit Tambe,1Ferry Schrijer,1Leo Veldhuis,1and Arvind Gangoli Rao1
AWEP, Aerospace Engineering, Delft University of Technology, Klyuverweg-1, 2629HS Delft,
The Netherlands
(*Electronic mail: A.GangoliRao@tudelft.nl)
(*Electronic mail: L.L.M.Veldhuis@tudelft.nl)
(*Electronic mail: F.F.J.Schrijer@tudelft.nl)
(*Electronic mail: S.S.Tambe@tudelft.nl)
(Dated: 17 February 2022)
This work shows the behaviour of an unstable boundary-layer on rotating cones in high-speed flow conditions: high
Reynolds number Rel>106, low rotational speed ratio S<1–1.5, and inflow Mach number M=0.5. These conditions
are most-commonly encountered on rotating aero-engine-nose-cones of transonic cruise aircraft. Although it has been
addressed in several past studies, the boundary-layer instability on rotating cones remained to be explored in high-speed
inflow regime. This work uses infrared-thermography with POD approach to detect instability-induced flow structures
by measuring their thermal footprints on rotating cones in high-speed inflow. Observed surface temperature patterns
show that the boundary-layer instability induces spiral modes on rotating cones, which closely resemble the thermal
footprints of the spiral vortices observed in the past studies at low-speed flow conditions: Rel<105,S>1, and M≈0.
Three cones with half-cone angles ψ=15◦,30◦, and 40◦are tested. For a given cone, the Reynolds number relating
to the maximum amplification of the spiral vortices is found to follow an exponential relation with the rotational speed
ratio S, extending from low- to high-speed regime. At a given rotational speed ratio S, the spiral vortex angle appears
to be as expected from the low-speed studies, irrespective of the half-cone angle.
I. INTRODUCTION
Instability and transition of the rotating boundary-layers
has classically been a subject of great interest due to its abun-
dance in nature (rotating planets, stars, etc.) and in indus-
try (turbo-machinery, wheels, rotating projectiles, etc.). De-
pending on the geometry, centrifugal or cross-flow instabil-
ity gives rise to coherent spiral vortices in an unstable lami-
nar boundary-layer on a rotating body, e.g. disk, cone, and
sphere. Upon their spatial growth, the vortices alter the basic
boundary-layer profile by enhancing the mixing of high- and
low-momentum flow, and subsequently, lead the boundary-
layer towards a turbulent state1,2.
Investigating this transition process is important for effi-
ciency improvements in engineering applications. For exam-
ple, in compressor cascades, the state (laminar/turbulent) of
the hub end-wall boundary-layer influences the aerodynamic
losses3. In aero-engines, the hub-end-wall boundary-layer be-
gins right from the tip of a rotating nose-cone. Here, ac-
curately assessing the aerodynamic losses in engine compo-
nents (fan and compressors) requires knowing the instability
behaviour of the rotating nose-cone boundary-layer.
In the past, instability mechanisms in rotating boundary-
layer flows were first discovered through studies on a disk ro-
tating in still fluid. One of the first experiments by Smith 4
showed that the laminar boundary layer over a rotating disk
exhibits sinusoidal velocity fluctuations, before transitioning
into a fully turbulent state. Later on, the experiments of Gre-
gory, Stuart, and Walker 5revealed that these velocity fluctu-
ations are due to the spiral vortices formed near the rotating
disk surface. Here, the radially increasing tangential velocity
of the disk surface creates a pressure gradient, leading to an
inflectional radial velocity profile. This system exhibits an in-
viscid instability, named cross-flow instability, which allows
perturbations to grow and form the spiral vortices. These vor-
tices appear co-rotating in their cross sections6, and subtend
the wave angle εof around 14◦with the outward radial vector.
Since a disk is a cone with half angle ψ=90◦, the bound-
ary layer instability mechanism on a rotating broad cone
(30◦.ψ<90◦) is similar to that over a rotating disk. In still
fluid, the cross-flow instability on rotating broad cones leads
to the formation of co-rotating spiral vortices2,7–14. When the
half-cone angle ψis decreased from 90◦to 15◦, the wave an-
gle εof a vortex, subtended with the meridional line over
a rotating cone, decreases from 14◦to 0◦1,7. Interestingly,
around ψ=40◦to 30◦, the instability mechanism changes
from the cross-flow to the centrifugal type, due to the dom-
inant effect of centripetal acceleration on the boundary layer
for low half cone angles7. This results in toroidal vortices over
a rotating slender cone (ψ<30◦) which are counter-rotating
in the cross-section, similar to those observed in concentric
cylinders15,16, concave walls17, and rotating cylinders in cross
flow18–20. The exact range of half-cone angles where this
change in instability mechanism occurs has not been identi-
fied yet. For further discussions, the cones with ψ&30◦are
considered as broad cones, which are closer to the rotating
disk, and the cones with ψ<30◦are considered as slender
cones, which are closer to the rotating cylinder case.
When an axial inflow is enforced on rotating cones, both
cross-flow and centrifugal instabilities induce spiral vortices.
Their onset and growth strongly depend on two parameters:
the local Reynolds number Rel=ρuel/µand local rotational
speed ratio S=rω/ue21,22. Here, lis the meridional dis-
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PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0083564
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FIG. 1. Schematic of a rotating cone under axial inflow.
tance from the cone apex, ρis the fluid density, µis the dy-
namic viscosity, ris the local radius, ωis the angular velocity
of a cone, and ueis the velocity just outside the boundary
layer21–25. Figure 1 shows a schematic of spiral waves over a
rotating cone in axial inflow, along with the geometry param-
eters.
Although the past literature has explored the boundary-
layer instability on rotating cones, the investigations were lim-
ited to low-speed inflow conditions, i.e. low inflow Reynolds
numbers ReL=ρUL/µ<105, high base rotational speed ra-
tio Sb=rbω/U>1, and incompressible flow. Here, Lis the
total meridional length of a cone, rbis the base radius, ωis
the angular velocity and Uis the inflow velocity. However,
rotating nose-cones of transonic aircraft engines typically face
high-speed inflow conditions: high inflow Reynolds number
ReL>106, low base rotational speed ratio Sb<1.5 and Mach
number M≈0.5–0.6 (although the transonic aircraft cruises
at Mach number around 0.8, the engine intakes reduce the in-
flow Mach number for the efficient fan operation26,27). But,
the boundary-layer instability on rotating cones remained to
be explored in high-speed inflow conditions. This hinders in
accurately assessing the aerodynamic performance of an aero-
engine, and therefore, restricts the design space explorations
for any additional efficiency improvements.
Past experiments were limited to the low-speed flow
regime because they relied on particle-based flow visualisa-
tion to detect the instability-induced spiral vortices on rotat-
ing cones21,28. They deposited particles (Titanium tetrachlo-
ride) on the cone surface before starting an experiment and
observed their transport during the operation. Due to sev-
eral practical challenges, this type of methods are not suitable
for high-speed experiments, which most-often use blow-down
tunnels. For example, due to the short durations (around 20s)
of each wind-tunnel run, the cones have to be kept rotating
before the wind tunnel starts. For the base rotational speed ra-
tios of interest (Sb≈0.7–1.2), the angular velocity of a cone
is high in a high-speed inflow, causing high centrifugal forces
on the deposited particles on the rotating cone surface. This
will cause the particle transfer before the operating condi-
tions are achieved, making the flow visualisation of the insta-
bility modes challenging. Recent experimental studies often
use hot-wire anemometry to reconstruct the spiral instability-
modes on rotating cones in still fluid14,29,30. However, due to
the short operating times of high-speed wind-tunnels, scan-
ning the velocity field on rotating cones with a hot-wire probe
requires several wind-tunnel runs for each operating point,
making it practically infeasible for exploring the parameter
space. Therefore, detecting boundary-layer instability on ro-
tating cones remained challenging at flow conditions that are
typical for a realistic flight.
In the present work, instead of particle-based visualisation
methods and hot-wire anemometry, infrared thermography is
used to detect the spiral instability-modes by measuring their
thermal footprints on rotating cones in high-speed inflow con-
ditions: ReL>106,Sb<1 and M=0.5. The experimental
method described in Tambe et al. 25 is applied here to detect
the instability-induced flow features on rotating cones. This
method is relatively easy to implement in high-speed con-
ditions. It overcomes the limitations of the previously used
measurement techniques because the method is non-intrusive
and provides instantaneous surface temperature distributions,
requiring only a single wind-tunnel run for each operating
point. Three different cones with ψ=15◦,30◦and 40◦are
tested. Section II describes the experiments. Section III de-
scribes flow fields surrounding the cones in the wind-tunnel
test-section. Section IV presents visualisations of the spiral
vortices. Section V shows the growth and spatial character-
istics of the spiral vortices in the parameter space Relvs S.
Section VI presents observed spiral vortex angles. Section
VII concludes the chapter.
II. DESCRIPTION OF EXPERIMENTS
Experiments are performed in TST-27, a transonic-
supersonic wind-tunnel at the faculty of Aerospace Engineer-
ing, Delft University of Technology (TU Delft). Figure 2
shows a schematic of TST-27. This blow-down wind-tunnel
uses pressurised air stored in a separate reservoir at a total
pressure between 20 and 40 bar. The pressurised air is ex-
panded through a variable area channel to achieve prescribed
flow conditions. A variable choke downstream of the test sec-
tion is used to achieve a desired subsonic Mach number. The
desired Reynolds number is determined by varying the total
pressure in the settling chamber. The test-section is rectan-
gular with a constant width of 0.28 m and variable height,
which for the present case is 0.253 m. To simulate flow con-
ditions within aero-engines, the wind tunnel is operated at
settling chamber pressures between 1.4 and 1.5 bar resulting
in ReL>106and at the inflow Mach number M=0.5. The
undisturbed inflow in an empty test section is uniform except
the tunnel-wall boundary-layers, as described in appendix A.
Figure 3 shows the measurement setup. As contemporary
aero-engines are found to use variety of nose-cone shapes,
three different cones are chosen for this investigation: a slen-
der cone with half angle ψ=15◦, and two broad cones with
ψ=30◦and ψ=40◦. Note that previous studies7,31 have in-
vestigated cones of ψ=15◦and 30◦in low-speed and there-
fore, these cases are retained in these high-speed investiga-
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FIG. 2. A schematic of the transonic-supersonic wind-tunnel TST27.
FIG. 3. Measurement setup.
tions. Broader cones with ψ&45◦are excluded from the test
because they experience high drag which is beyond the limit
of the present rotating setup. All the models have a constant
base diameter D=100 mm. In the test section, area blockage
due to the cone increases from 0% at the tip to 11.1% at the
base (excluding the wall boundary-layers), which is compara-
ble to that of a typical aero-engine. The models are rotated by
a brushless motor at various rotational speeds (8000 to 36500
RPM) to achieve different combinations of local Reynolds
number Reland local rotational speed ratio S. The models
are made of Polyoxymethylene (POM) which has favourable
thermal properties for the infrared measurements25. The sur-
face is smoothed up to the r.m.s. roughness lower than 1 µm.
The models are statically balanced around the rotation axis.
The tip eccentricity is around 5 µm.
Generally in an unstable system, small environmental dis-
turbances can undergo growth to form coherent flow struc-
tures. In the present setup, these disturbances come primar-
ily from three sources: (a) free-stream turbulence of around
3.5−4% of the mean velocity as measured in an empty test
section using particle image velocimetry, (b) surface rough-
ness of the cone ≈1µm and (c) remnant dust particles in the
air below the filter size <10µm.
Infrared thermography has been shown to be a useful tool
for measuring the spiral instability modes over a rotating
disk32 and a rotating cone25. This technique is applied in the
present study to detect these instability modes from their sur-
face temperature footprints, see figure 3 for the measurement
setup. For the half-cone angles ψ=40◦,30◦and 15◦, the in-
frared camera has viewing angles β=71.4◦,68.7◦and 64.9◦,
and the object distance do=0.33m,0.4m and 0.47m, respec-
tively. The surface temperature is recorded as the digital pixel
intensity I. The integration times of infrared acquisitions are
varied between 250µs to 25µs such that the angle swept by
the rotating cones during each acquisition is minimized, while
retaining a sufficient signal to noise ratio to measure the sur-
face temperature fluctuations (represented by I′). The results
show that at the same operating conditions, different integra-
tion times do not significantly alter the observations of insta-
bility modes; see appendix B for an example. At high rotation
rates (RPM >30000) of broad cones, the integration times are
lowered (50µs to 25µs), and therefore, the signal contrast is
increased by irradiating the cones with a theatre lamp (575W).
Table I details the technical specifications of the setup.
Before the wind-tunnel test, both the model and the pres-
surised stagnant air are at ambient temperature. When the
tunnel starts, the air expands into the test-section and the
static temperature of the air drops, which cools the model
surface. During the wind-tunnel operation time (around 20s),
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FIG. 4. Photographs showing (a) rotating cone in the test section of TST-27, a transonic-supersonic wind-tunnel and (b) infrared camera
viewing through the germanium window.
Camera FLIR (CEDIP) SC7300 Titanium
Noise equivalent temperature difference (NETD) 25 mK
Spatial resolution 0.43 −0.6 mm/px
Integration time 25 −250 µs
Acquisition frequency 200 Hz
Number of images per dataset 2000
Heat source Theatre lamp (575 W)
TABLE I. Specifications of the Infrared Thermography setup.
the model cools down and its temperature drops continuously.
This trend is removed by subtracting a moving average with
the kernel size of 20 instances (corresponding to 0.1s) from
the dataset. Since this operation is only a precursor to the
subsequent modal analysis, the data after moving mean sub-
traction is referred to as raw data.
To obtain the local flow properties along the cones, static
pressure is measured on the non-rotating cones using a six-
teen channel Scanivalve DSA3217 pressure acquisition sys-
tem. The pressures are measured at different circumferential
positions of a cone in a rectangular test-section to obtain a cir-
cumferential mean static pressure at a given radius. The total
pressure and temperature in the settling chamber, and static
pressures at the wind-tunnel walls are also recorded.
III. FLOW FIELD OVERVIEW
The present wind-tunnel configuration features rotating
cones in an internal flow, unlike the majority of past studies in
low-speed-open-jet facilities. Such internal flow closely rep-
resents the most-encountered flow conditions inside an aero-
engine. In a symmetry plane, inviscid flow develops between
two bounding streamlines near the test-section wall and cone
U
M
Wall
Wall
Mavg
Ml
Cone
FIG. 5. Schematic of the internal flow within the test section.
surface, see figure 5. Near the cone, the undisturbed oncoming
flow slows down upon an encounter with the cone-tip region
and turns to follow the cone surface.
Here, estimating flow properties at the edge of a cone
boundary-layer is crucial to compute the local flow param-
eters Reland Sthat govern the stability characteristics over
the rotating cones21,22. For this purpose, the local Mach
number Ml, just outside the boundary-layer, is obtained us-
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FIG. 6. Mach number variations over the cone meridians: (a) local Mach number of the potential flow near the cone surface obtained from
surface pressure measurements and (b) area averaged Mach number of the flow between a cone and test section walls obtained from the local
area ratio.
ing the isentropic relations and measured flow properties:
circumferential-mean static pressure along the non-rotating
cones and total pressure in the settling chamber. This, along
with the measured total temperature, gives the local static tem-
perature at the cone surface under the isentropic flow assump-
tion. This gives the local sound speed, which together with
the local Mach number Mlis used to calculate the meridional
velocity uof an inviscid flow along the cone, with the uncer-
tainty of ±0.038u. This velocity is used as an approximate es-
timate of the edge velocity uejust outside the cone boundary-
layer. In low-speed studies, this type of assumption is often
used where the boundary-layer edge velocity ueis approxi-
mately estimated as the meridional velocity of the potential
flow over a cone21,23. Furthermore, local air density is calcu-
lated using the ideal-gas relation, and local dynamic viscosity
is obtained from the Sutherland’s law.
The local Mach number Mlis low near the cone-tip
and increases downstream along the cone beyond the free-
stream Mach number, see figure 6a. The area-averaged flow
Mach number (resulting from one-dimensional-isentropic
area-Mach relation) also shows a gradual increase along the
cone due to the area contraction, see figure 6b. With in-
creasing half-cone-angle ψ, flow accelerates steeper along the
cone, as evident from figure 6.
The local Mach number distributions from figure 6a are
used to obtain the flow parameters: local Reynolds number
Reland rotational speed ratio S. With variable total conditions
and rotational speed, different distributions of these parame-
ters are obtained along the cone length to investigate a wider
region of the parameter space Relvs S.
IV. VISUALISATIONS OF INSTABILITY MODES
The instantaneous surface temperature footprints allow vi-
sualising the instability modes over the rotating cones. Proper
orthogonal decomposition (POD) approach is used to iden-
tify the modes of surface temperature fluctuations in the mea-
surement dataset, each consisting of 2000 images. The POD
modes corresponding to the measurement noise (wavelength
λθ<4 pixels) are discarded. Remaining POD modes are used
to selectively reconstruct the instability modes using criteria
based on the azimuthal wavelength λθ(see Tambe et al. 25 for
further details).
Figure 7 shows (a) the instantaneous surface temperature
fluctuations in the raw data and associated (b and c) POD re-
constructions of a rotating broad cone with ψ=40◦. The top
row images are as observed in the camera sensor plane; and
corresponding images in the bottom row are unwrapped cone-
surfaces (figures 7d, e and f). The raw data reveal that a wave
pattern (λθ≈πr/8) appears on a rotating cone, overlaid with
a long wavelength modulation (λθ>πr/4). These two types
of wave patterns are separated using POD modes. The long
wavelength modulation is reconstructed using the POD modes
having an azimuthal wavelength λθ>πr/4, i.e. an azimuthal
number of waves n<8, see figures 7b and e. The short-
wavelength pattern is reconstructed from the POD modes hav-
ing an azimuthal wavelength between πr/4>λθ>4 pixels,
see figures 7c and f; this pattern shows nearly constant az-
imuthal spacing, which indicates azimuthal coherence.
Similar short and long wave temperature patterns have been
observed on rotating cones in low-speed conditions (e.g., see
Tambe et al. 25 ). The low-speed investigations have clarified
that the short-wave temperature pattern corresponds to the spi-
ral vortices, and the long-wave pattern is aligned with the spi-
ral vortices. This can be viewed as the azimuthal modulation
of the spiral vortex strength. The striking similarity of the
patterns observed in low-speed and present high-speed case
suggests that the short-wave pattern in figure 7c corresponds
to the footprint of the spiral vortices. Further investigation is
required to identify the physical reasoning behind the appear-
ance of the long-wave pattern.
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Similar wave patterns are observed also for a rotating broad
cone with ψ=30◦, as shown in figure 8. Here, a coherent
spiral vortex footprints with λθ≈πr/5 (figures 8c and f) has
a major contribution to the raw image (figures 8a and d), in
addition to the long wave modulation (figures 8b and e).
The stability analyses of rotating broad cones in compress-
ible still fluid12 and with incompressible axial inflow8have
shown that, with decreasing half-cone angle ψ, the range of
unstable wavelengths gets broadened, and increasingly longer
wavelengths can destabilise the flow. Results of Kobayashi
and Izumi 7agree with this trend as the azimuthal number de-
creases (wavelength increases) with decreasing half-cone an-
gle ψ. A similar trend is observed in the present study, where,
for a lower half cone angle ψ=30◦(figure 8), longer wave-
lengths of λθ≈πr/5 are observed as compared to λθ≈πr/8
observed for ψ=40◦(figure 7).
Figure 9 shows a raw image and the corresponding POD re-
constructions for a slender cone with ψ=15◦. Here, a coher-
ent spiral vortex pattern of λθ≈πr/4 (figure 9c) is observed
along with the long wave modulation (figure 9b). However,
on the slender cone, other instances with coherent patterns
of λθ.πr/3 are also observed, see figure 17 for example.
Therefore, the upper bound for the low-order reconstructions
in figures 9c and f is set at πr/3 (relating to the azimuthal
number n=6) instead of πr/4 (relating to the azimuthal num-
ber n=8) used for broad cones. This highlights the need of a
separate investigation to identify these vortices by measuring
the velocity field.
Overall, the visualizations show that, in high-speed condi-
tions of M=0.5 and ReL>106, the instability modes over
rotating cones appear to be spiral vortices, similar to those
observed in low-speed conditions. The spiral vortex patterns
on broad cones (figures 7 and 8) are similar to the spiral vor-
tices observed in the low-speed cases over rotating disks and
broad cones; compare figures 7 and 8 with visualisations of
Kobayashi 1, Kohama 2, 6 . On a rotating slender cone, the spi-
ral vortex fronts are oriented more towards the axial direc-
tion as compared to the past low-speed studies21,25 . This is
expected as the present observations on slender cones are at
low local rotational speed ratios S<0.5, whereas past low-
speed studies encountered high rotational speed ratio S>1.
This agrees with the previously identified trend where with
decreasing rotational speed ratio spiral vortex-fronts become
more axial.
Flow investigations in a meridional plane are necessary
to confirm whether these spiral vortices are co- or counter-
rotating. Due to the limitations of the present experiments,
i.e. limited optical access for a PIV light-sheet, the cross-
sections of the spiral vortices in the meridional plane remains
unknown.
V. GROWTH OF SPIRAL VORTICES
The spiral vortices observed in the present study are con-
vective instability modes. The effect of their spatial growth on
the surface temperature fluctuations is observed from the sta-
tistical RMS of the thermal footprints I′
RMS on a rotating cone,
computed over a dataset (2000 images, acquired at 200Hz).
Figure 10a shows I′
RMS on a rotating broad cone (ψ=30◦),
computed over a raw dataset containing the instance shown in
figure 8a. The region where the spiral vortices appear shows
high levels of temperature fluctuations. The RMS fluctua-
tions are axisymmetric because of the symmetry of axial in-
flow. Therefore, the spatial growth of the spiral vortices is
characterised using a circumferential mean of I′
RMS along the
meridional lines (see figure 10b). Downstream from the cone
apex (l/D=0), the I′
RMS stays low until a point (l/D≈0.57)
where it suddenly starts growing. This location relates to a
critical point suggesting the onset of the spiral vortex growth.
At their origin, the spiral vortices are expected to have weak
thermal footprints which is usually below the measurement
sensitivity31. Therefore, a critical point in the present case
relates to the onset of a rapid growth of the spiral vortices
rather than the point of their origin. This point is objectively
defined as an intersection of the linear parts of I′
RMS curve (ap-
proximated with least square linear fits shown as gray lines in
10b). The growth of the spiral vortices continues until a max-
imum amplification (l/D≈0.72), after which I′
RMS starts to
decrease.
In figure 11, the curves of Relvs Sshow how these flow
parameters vary over a cone at different operating condi-
tions. The critical and maximum amplification points on
these curves mark the region of the spiral vortex growth.
The Reynolds numbers corresponding to these points decrease
with the rotational speed ratio S. With increasing half-cone
angle ψand for a fixed Reynolds number Rel, critical and
maximum amplification points of the spiral vortex growth ap-
pear to shift towards higher rotational speed ratios S. This
suggests that with increasing half-cone angle the boundary-
layer becomes more stable such that a stronger relative ro-
tation effect (represented by higher rotational speed ratio S)
is required to cause the spiral vortex growth. Previously,
this trend has been observed for rotating cones in still fluid7
and predicted for rotating broad cones in incompressible axial
inflow22. However, understanding the mechanism behind this
trend requires a separate theoretical analysis, which is beyond
the scope of the present work.
Figure 12 compares the spiral vortex growth region at the
present high-speed conditions with the low-speed investiga-
tions from the literature. Kobayashi et al.23 have tested ro-
tating cones with varying free-stream turbulence level in low-
speed conditions. Their observations show that the transition
Reynolds number, where the velocity fluctuations start to re-
semble a typical turbulent spectrum, remains unaltered by the
free-stream turbulence level. However, the critical Reynolds
number (corresponding to the onset of instability modes) is
lowered by a higher intensity of the free-stream turbulence
(see figure 12). Consequently, the region between the criti-
cal and transition points (transition region) becomes broader.
The free-stream turbulence level in the present experiments
is around 3.5−4% of the free-stream velocity U. Therefore,
the present study can be compared to the low-speed results of
Kobayashi et al. 23 with the similar turbulence intensity (3.6%
of U). The comparison suggests that the transition region (as
schematically highlighted in figure 12) reported in past low-
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FIG. 7. Thermal footprints of the spiral instability modes and their POD reconstructions over a rotating broad cone with half angle ψ=40◦
shown in image planes (top row) as well as unwrapped cone surfaces (bottom row). (a) and (d) raw instance, (b) and (e) POD reconstruction
of a long-wave pattern (λθ>πr/4), (c) and (f) POD reconstruction of a short-wave pattern (πr/4>λθ>4px). ReL=1.3×106and Sb=1.1.
FIG. 8. Thermal footprints of the spiral instability modes and their POD reconstructions over a rotating broad cone with half angle ψ=30◦
shown in image planes (top row) as well as unwrapped cone surfaces (bottom row). (a) and (d) raw instance, (b) and (e) POD reconstruction
of a long-wave pattern (λθ>πr/4), (c) and (f) POD reconstruction of a short-wave pattern (πr/4>λθ>4px). ReL=1.4×106and Sb=1.
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FIG. 9. Thermal footprints of the spiral instability modes and their POD reconstructions over a rotating slender cone with half angle ψ=15◦
shown in image planes (top rows) as well as unwrapped cone surfaces (bottom row). (a) and (d) raw instance, (b) and (e) POD reconstruction of
a long-wave pattern (λθ>πr/3), (c) and (f) POD reconstruction of a short-wave pattern (πr/3>λθ>4px). ReL=3.2×106and Sb=0.72.
speed studies, can be extrapolated to the high-speed condi-
tions of the present study. This shows that the spiral instability
modes similar to those studied in the past can be expected on
real aero-engine nose-cones.
Figure 13 shows the maximum amplification points on ro-
tating cones in low- and high-speed inflows; obtained from the
literature (available only for ψ=15◦)31 and present investi-
gation, respectively. For the investigated cones of ψ=15◦,
the instability behaviour remains the same in low- and high-
speed cases. Here, maximum amplification Reynolds number
follows an exponential relation with the rotational speed ratio
Rel,m=CSa1, extending from low-speed to high-speed condi-
tions (shown using dashed line in figure 13). Here, constants
Cand a1are expected to depend on half-cone angle ψ. For
ψ=15◦,a1=−2.62 and C=2.3×105. The present high-
speed investigation is performed in a close-test section where
the area contraction accelerates the flow along the cone, unlike
in the low-speed investigations in an open-jet. However, this
does not seem to affect the maximum amplification of the spi-
ral vortices in Relvs Sspace (figure 13). Moreover, increasing
inflow Mach number from M<0.02 to M=0.5 has insignif-
icant effect on the trend of maximum amplification points in
Relvs Sspace (for ψ=15◦).
VI. SPIRAL VORTEX ANGLE
The spiral vortex angle εis obtained from the POD recon-
structions as shown in figure 14. Here, the spiral vortices ap-
pear linear in l,θcoordinate system. First, a trace of tem-
perature fluctuations (I′) over a 60◦sector is obtained along
θ, at a fixed l/D. This trace is cross-correlated with tem-
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FIG. 10. RMS of surface temperature fluctuations in a dataset over a rotating broad cone (ψ=30◦) shown in an (a) image plane and (b) as a
circumferential average along the meridional length for raw data. ReL=1.4×106and Sb=1.
FIG. 11. Growth characteristics of the spiral vortices.
perature fluctuation patterns at an incremented l/D, sliding
along the whole range of θand a cross-correlation peak is
found. This procedure is repeated on the new peak location
until the whole transition region is scanned. A least-square
linear curve (shown as red in figure 14a) is fitted to the loci
of cross-correlation peaks (shown as circles in figure 14a) :
θ=ml +c(where mand care fit parameters). The spiral
vortex angle is obained as ε=sin−1(1/q1+l2m2sin4(ψ)).
As known from low-speed studies, the spiral vortex angle
depends on the local rotational speed ratio S21. Generally, the
spiral vortex angle increases with decreasing rotational speed
ratio S. Physically, this means that as the effect of rotation
is reduced, the spiral vortex fronts turn towards the oncoming
flow direction. A similar trend is observed at high-speed con-
ditions, as evident from comparing present spiral vortex an-
gles with the low-speed data from Kobayashi et al. 23 in figure
15. In this case, the measured spiral vortex angles in high-
speed cases agree well with those from the low-speed studies.
This suggests that the Reynolds number does not influence the
spiral vortex angles. Figure 15b is a zoomed in view of figure
15, showing the measured spiral vortex angle ε.
VII. CONCLUSION
The instability of the boundary-layers over rotating cones
is studied in an enforced high-speed flow at ReL>106and
M=0.5. Two broad cones with half angles ψ=30◦and 40◦
and a slender cone with ψ=15◦are tested in a transonic-
supersonic wind tunnel. The instability modes are identified
from their surface temperature footprints, measured using in-
frared thermography. Following are the important conclu-
sions:
1. Visualisations show that instability induced spiral vor-
tices appear on rotating cones facing high-speed inflow
at ReL>106and M=0.5, similar to the inflows in
aero-engines. Their appearance is similar to the spi-
ral vortices observed over rotating cones in low-speed
conditions.
2. In high-speed conditions, the surface temperature on ro-
tating cones fluctuates in two distinct patterns: a short
wavelength footprints of spiral vortices and long wave-
length modulations of the spiral vortex strength. These
patterns show similarities with those observed in the
low-speed cases.
3. Local Reynolds number Reland rotational speed ratio
Sgovern the growth of spiral vortices as expected from
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FIG. 12. Comparisons of the growth characteristics of the spiral vortices at high-speed conditions to those reported in the literature for
low-speed conditions21,23,24,31. (a) ψ=15◦and (b) ψ=30◦.
low-speed studies.
4. At a fixed local Reynolds number Rel, increasing half-
cone angle ψshifts the growth of the spiral vortices at
higher rotational speed ratios S. This trend was also
observed in past low-speed studies.
5. The maximum amplification Reynolds number follows
an exponential relation with the rotational speed ratio
Rel,m=CSa1, extending from low- to high-speed con-
ditions. Constants Cand a1are expected to depend on
the half-cone angle ψ. For ψ=15◦,a1=−2.62 and
C=2.3×105.
6. The trend of spiral vortex angle εvariation w.r.t. the ro-
tational speed ratio Sis common in both low- and high-
speed conditions.
This study has experimentally explored the parameter space
of boundary-layer instability on rotating cones to the flow con-
ditions relevant to aerospace applications, e.g. aero-engine-
nose-cones during a transonic flight. The study has shown
that the instability-induced spiral vortices can be expected to
appear on nose-cones of civil transport aircraft. This is ev-
idently observed as the boundary-layer transition region on
rotating cones extends to the parameter space of Relvs Sat
Rel>106and S.1 in a similar fashion as expected from
the past low-speed studies. Comaparison of low- and high-
speed investigations shows that the local flow Mach number
Ml=0.02-0.6 does not influence the maximum amplifications
points in the Relvs Sspace. However, the cross-sectional
view of the spiral vortices remains to be investigated. This
information can help in identifying the underlying instability
mechanism, as centrifugal instability induces counter-rotating
vortices and cross-flow instability usually induces co-rotating
vortices1. Furthermore, the effect of surface roughness on
the spiral vortex growth needs to be investigated, as in real-
ity, aero-engine nose cones may experience surface roughness
as isolated elements from foreign object impacts, or as dis-
tributed roughness arising from, fasteners, manufacturing and
coating techniques.
ACKNOWLEDGEMENT
The authors thank technical staff–Peter Duyndam, ir. Frits
Donker Duyvis, ir. Eric de Keizer, Dennis Bruikman, and
Henk-Jan Siemer–of High-speed lab, AWEP, Aerospace En-
gineering, TU Delft for their technical support during the ex-
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FIG. 13. Maximum amplification points of the spiral vortex growth on rotating cones in present high-speed cases compared to the low-speed
case31.
periments. This work was funded by the European Union
Horizon 2020 program: Clean Sky 2 Large Passenger Aircraft
(CS2-LPA-GAM-2018-2019-01), and CENTRELINE (Grant
Agreement No. 723242).
CONFLICT OF INTEREST
Authors have no conflict of interest to report.
Appendix A: Undisturbed inflow
The inflow velocity is measured using two-component pla-
nar particle image velocimetry (PIV) in a symmetry plane of
the empty test section (without a model). The image pairs
are captured using LaVision imager sCMOS camera equipped
with 105mm Nikkor objective lens. The flow is seeded with
DEHS droplets of size ≈1µm. The particles are illuminated
with double-pulsed laser Evergreen 200. Commercial soft-
ware DaVis 8.4.0 is used for image acquisition and vector cal-
culations. A multipass approach with decreasing interrogation
window size (from 128 ×128 pixels to 32 ×32pixels) is used
to compute the vectors resulting in a vector pitch of 0.23mm.
The total of 450 image pairs are acquired at the rate of 10Hz
(using three windtunnel runs of 150 image pairs each).
The velocity measurements show that the undisturbed in-
flow in an empty test section of TST27 windtunnel is uniform,
as seen from the time-averaged stream-wise velocity (u) in fig-
ure 16a. It is known from the previous boundary-layer stud-
ies performed in this windtunnel (although supersonic, but at
the order of Reynolds number comparable with the present
study), that the order of displacement thickness of the tunnel
wall boundary-layers is usually below 1% of the test section
width (also, δ99 <7% of the test section width)33. Therefore,
this does not pose any non-uniformity to the cones placed at
the centre of the test-section because their base diameter is
only around 35% of the test section width. Figure 16b shows
the typical turbulence intensity distribution of the inflow in
terms of RMS of streamwise velocity fluctuations u′
RMS.
Appendix B: Effect of integration time
In the present study, the cones are rotated at high r.p.m.
(8000 to 33000) to achieve the desired rotational speed ratios
in high-speed inflow. Spiral vortices are observed with an in-
frared camera at finite integration times, ranging from 25µs
to 205µs. During the integration time, a cone surface rotates
with respect to a stationary camera sensor plane. A sensor
records the temperature of this moving surface as it passes
through its field of view. For a coherent spiral vortex pattern
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FIG. 14. Traced wavefront of the spiral modes in land θspace used
to obtain the wave angle ε.
this does not alter the observed pattern. Figure 17 shows the
rotating cone at same operating conditions but observed with
three integration times tint =205µs, 105µs and 50µs. The
long wave modulations do not show any significant difference,
see figures 17b, e and h. The short wavelength patterns get
slightly sharper with decreasing integration time, see figures
17c, f and i.
The locations where the spiral vortex growth is observed
does not show significant changes with changing integration
time, see figure 18a. As expected, the lower integration time
reduces the signal strength as evident from the reduced I′
RMS
at tint =50µs as compared to that at tint =205µs in figure 18.
The effect of integration time on the spiral vortex angle εis
also minimal, as evident from figure 18b.
AIP PUBLISHING DATA SHARING POLICY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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FIG. 15. (a) Comparisons of the wave angle εobtained in the present study to those reported in the literature21 against the local rotational
speed ratio S. (b) A zoomed-in view of the vortex angles obtained in the present study.
FIG. 16. Undisturbed inflow profiles in an empty test section (a) time-averaged stream-wise velocity and (b) turbulence intensity.
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FIG. 17. Effect of integration time tint on the thermal footprint measurements of the spiral vortices. (a),(d),(g) raw instance, (b),(e),(h) POD
reconstruction of a long-wave pattern (λθ>πr/3), and (c),(f),(i) POD reconstruction of a short-wave pattern (πr/3>λθ>4px). (a),(b),(c)
tint =205µs, (d),(e),(f) tint =105µs, (g),(h),(i) tint =50µs.
FIG. 18. Effect of integration time tint on (a) surface temperature fluctuations I′
RMS and (b) spiral vortex angle ε.
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