This paper is declared a work of the US Government and is not subject to copyright protection in the United States.
HYPERSONIC BOUNDARY LAYER STABILITY OVER A FLARED CONE IN A QUIET TUNNEL
Jason T. Lachowicz
*
and Ndaona Chokani
†
North Carolina State University, Raleigh NC 27695
and
Stephen P. Wilkinson
‡
NASA Langley Research Center, Hampton VA 23681
Abstract
Hypersonic boundary layer measurements were
conducted over a flared cone in a quiet wind tunnel. The
flared cone was tested at a freestream unit Reynolds
number of 2.82x10
6
/ft in a Mach 6 flow. This Reynolds
number provided laminar-to-transitional flow over the
model in a low-disturbance environment. Point
measurements with a single hot wire using a novel
constant voltage anemometry system were used to
measure the boundary layer disturbances. Surface
temperature and schlieren measurements were also
conducted to characterize the laminar-to-transitional state
of the boundary layer and to identify instability modes.
Results suggest that the second mode disturbances were
the most unstable and scaled with the boundary layer
thickness. The integrated growth rates of the second mode
compared well with linear stability theory in the linear
stability regime. The second mode is responsible for
transition onset despite the existence of a second mode
sub-harmonic. The sub-harmonic wavelength also scales
with the boundary layer thickness. Furthermore, the
existence of higher harmonics of the fundamental suggests
that non-linear disturbances are not associated with “high”
free stream disturbance levels.
Nomenclature
A Disturbance rms amplitude (square of power
spectral density, arbitrary units).
f Frequency (kHz)
R (Re
s
)
1/2
Re
s
Reynolds number based upon freestream
conditions and surface distance from the apex of
the cone, S
r
b
Base radius of cone
S Distance along the surface of the model, measured
from the apex of the cone model
To Total temperature
Tw Surface static temperature
ρU Mass flux
Vs Constant voltage anemometer output voltage
X Coordinate along the cone model axis of
symmetry, measured from the apex of the cone
Y Coordinate perpendicular to the cone axis of
symmetry, measured from the cone axis of
symmetry
Yw Y-location of the wall surface
Greek
-α
i
Non-dimensional amplification rate, -α
i
=
1
2A
dA
dR
δ Boundary layer thickness
λ Disturbance wavelength
η Non-dimensional Y-distance , η =
(Y-Y
w
)R
S
Subscripts
∞ Conditions in freestream
rms Root mean square of fluctuating component
Superscripts
' Fluctuating component of a time dependent
quantity
( )
Time average of a particular quantity
Introduction
The change from laminar to turbulent flow in the
hypersonic boundary layer is accompanied by large
changes in both heat transfer and skin-friction drag. These
changes are important to the aerodynamic design of
hypersonic vehicles since the aerodynamic coefficients are
very sensitive to the large changes in heat transfer and
skin-friction that accompany transition.
1
Furthermore, the
stability, control, and structural design of the vehicle are
affected due to the increased thermal and aerodynamic
loading.
The conical geometry is prevalent in many hypersonic
aerodynamic applications, but only a few stability
experiments
2
of hypersonic cone boundary layers have
been conducted. These studies have provided a
fundamental understanding of the hypersonic boundary
layer stability problem. However, these few stability
experiments have been conducted in conventional
hypersonic wind tunnels where relatively large freestream
disturbances occur. The primary source of the free stream
disturbances is acoustic radiation from convecting eddies
generated by the turbulent boundary layer on the nozzle
wall.
3,4
The frequency content of this incident noise field
provides a stimulus to excite disturbances in the
hypersonic boundary layer which may lead to transition.
Thus, some of the observed anomalies between
experiment and theory
5,6
may be due to effects of the wind
tunnel noise.
*
Graduate Research Assistant, Department of
Mechanical and Aerospace Engineering, Student Member
AIAA. Presently, NRC Research Associate, NASA
Langley.
†
Associate Professor, Department of Mechanical and
Aerospace Engineering, Member AIAA.
‡
Group Leader, Quiet Tunnel & Transition Group, Flow
Modeling and Control Branch, Fluid Mechanics and
Acoustics Division, Senior Member AIAA.
2
In order to provide a more reliable test environment for
the experimentalist, the NASA Langley Research Center
has developed a series of supersonic/hypersonic quiet
tunnels.
7
In these facilities, the free stream noise is
maintained at low levels by treating the settling chamber
flow and maintaining the nozzle wall boundary layer in a
laminar state. Hence, transition on the nozzle wall is
delayed providing lower free stream disturbance levels in
the quiet tunnel relative to conventional tunnels.
The primary objective of the present study is to obtain
experimental hypersonic boundary layer stability data over
a conical body in a quiet tunnel. In a parallel paper, Ref. 8,
the disturbance environment of the quiet tunnel used in the
present work is documented. In this paper, the first
hypersonic boundary layer stability measurements
obtained in a quiet tunnel are presented.
Experimental Apparatus
Test Facility
All tests were conducted in the NASA Langley Nozzle
Test Chamber Facility. This is an open-jet blow-down
facility, and was equipped for the present tests with a
slow-expansion, axisymmetric, quiet Mach 6 nozzle. The
nozzle, which is more fully described in Ref. 9, had a
throat diameter of 1.00", exit diameter of 7.49", and length
from throat to exit of 39.76". The nozzle is equipped with
an annular throat slot to remove the boundary layer that
develops upstream of the throat. As a result, the boundary
layer that forms on the nozzle wall remains in a laminar
state until far downstream. This provides low disturbance
levels in the nozzle test section. The nozzle may be
operated over a range of stagnation pressures from 80 to
200 psia, and stagnation temperatures up to 400 °F. Run
times from minutes to several hours are possible.
Test Model
The model, used in this study, was a 20" long cone with
a curved-flare afterbody, shown in Fig. 1. For sake of
brevity, this model is referred to as the flared cone. The
straight cone surface extended from X=0" to X=10", with
a semivertex angle of 5°. The flare surface had a radius of
curvature of 93.07", and extended from X=10" to X=20".
The sharp model tip had a nominal radius of 0.0001". The
model surface was instrumented with 29 pressure orifices
and 51 thermocouple gages placed along diametrically
opposite rays as shown in the side view of Fig. 1. Hot-wire
boundary layer surveys were conducted along a ray
located 90° from the surface measurement rays as shown
in the top view of Fig. 1.
The flared cone was used instead of a straight cone in
order to induce transition on the model within the quiet
flow capability of the tunnel. The model surface was of
high fidelity, and had a maximum rms radius error of less
that 2.8% of the model boundary layer thickness.
Hot-Wire Probe
s
The hot-wire probes were constructed of 10% platinum
plated tungsten wire of 100 µin. diameter. The wire was
soldered onto 0.005" diameter stainless steel broaches
which were attached to the main probe body. The nominal
length-to-diameter ratio of the wire was 210. The wire was
slack to minimize the “strain-gage effect”. An electrical
contact probe was located about 0.005" below the
broaches in order to determine the location of the model
surface.
Hot-Wire Anemometer
The hot-wire anemometer system used in the present
work was a new, proprietary constant voltage anemometer
(CVA). In this system, a steady DC voltage was
maintained across the hot wire through the use of a
composite-amplifier-compensation circuit. The operating
principles of the CVA are described in detail in Ref. 11.
Only the CVA, in contrast to attempts with constant
current and constant temperature anemometers, provided
the ability to obtain measurable signals in the freestream
of the quiet nozzle flow. The reason for the better
performance of the CVA has not yet been investigated.
The present CVA system had a bandwidth of about 350
kHz with a 40 dB/decade roll-off.
The CVA operation was computer controlled for all
tests conducted. At each boundary layer measurement
point, the wire voltage was automatically changed through
7 levels; the voltage magnitudes were optimized for the
individual wires used. The constant wire voltage was
monitored throughout testing using a 5
1
/2
digit digital
multimeter (DMM). The DMM was also used to measure
the mean CVA output signal. The rms and fluctuating
components were measured from the AC-coupled CVA
output signal. An analog true RMS voltmeter was used to
determine the CVA rms output voltage. Prior to
measurement, the CVA output was high-pass filtered at 1
kHz and low-pass filtered at 1 MHz. An 8-Bit digital
oscilloscope was used to obtain time traces of the
fluctuating CVA output voltage. The high- and low-pass
filter settings were 1 kHz and 630 kHz, respectively, and
the sampling rate was 2 MHz. Standard FFT procedures
employing a Hanning window, data length of 512 points,
and 78 averages were used to obtain the spectra.
A calibration procedure for the CVA was developed,
and is outlined in Ref. 12. This procedure enabled mean
and rms mass flux and total temperatures in the boundary
layer to be obtained. The comparisons of the present data
with computational predictions verified the validity of the
procedure for mean flow quantities. However, the
accuracy of the quantitative rms data could not be verified
for the fixed-time-compensation CVA system.
Nevertheless, the absence of quantitative fluctuation data
did not prevent an analysis of the spatial amplification
rates in the linear and weakly non-linear stability regimes.
In the linear stability region, fluctuations are expected to
grow (or decay) exponentially. This exponential growth is
described by normal mode decomposition
5
which specifies
that the amplification rate is the same for separate
fluctuation components (pressure, velocity, etc.). Thus, an
uncalibrated approach is valid in the linear stability region
as verified from both controlled
13
and uncontrolled
6
stability experiments. In the weakly non-linear stability
regime, analysis of the present data indicated that the
experimentally-derived amplification rates are mainly of a
mass-flux nature
12
.
Experimental Approach
Test Conditions
3
The stagnation conditions for the present tests were a
temperature of 810° R and a pressure of 130 psia. The
measured freestream Mach number was 5.91. These
conditions yielded a freestream unit Reynolds number of
2.82x10
6
/ft.
Prior to the present tests, the characteristics of the
freestream flow were documented through a series of
pitot-probe and hot-wire measurements. The details of the
freestream measurements are presented in Refs. 8 & 12,
but a few salient features are presented here. The pitot-
probe surveys verified that the nozzle mean flow was
uniform within the flow test volume. Within this volume
the Mach number was 5.91 ± 0.08. Measurements in
cross-sectional planes indicated that the mean flow was
axisymmetric and varied by no more than 1% in the test
volume which encompassed the model. The RMS data
showed that the low-level disturbance field was
axisymmetric and was associated with sound-mode
generation of the nozzle wall turbulent boundary layer.
Measurements
The measurements were conducted in two stages. First,
model surface temperature and schlieren imaging
measurements were conducted, and then hot-wire
boundary layer surveys were conducted. The surface
temperature measurements were used to verify that the
model was in thermal equilibrium, and to estimate the
location of transition onset. The schlieren images were
used to verify the laminar-to-transitional state of the
boundary layer, and to identify the character of the
instability modes.
The hot wire boundary layer surveys were conducted at
17 streamwise stations, spaced 0.5" apart over the range
X=10.97" (R=1610) to X=18.97" (R=2120). At each
streamwise station, the wire was traversed perpendicular
to the cone axis of symmetry. The mean and rms
measurements were obtained at 13 points clustered near
the boundary layer edge. At each measurement point, 7
wire voltages were applied. The rms profiles were then
inspected to determine the maximum energy (rms)
location. Wave traces were subsequently measured at the
maximum energy locations using the largest, practicable
wire voltage. At this operating condition, the CVA is more
sensitive to changes in the mass flux as opposed to
changes in the total temperature
12
. In addition, the CVA
signal-to-noise ratio is improved.
Results
Surface Temperature Data
Fig. 2 presents the experimental and computational
surface temperatures. The temperatures are shown along
the left ordinate and the flared-cone surface coordinates
along the right ordinate. The experimental surface
temperature error is ±2°R (±0.0025 To
∞
). The
computational values
14
represent laminar adiabatic wall
temperatures. Over the range, R=690-1700, the flow is
laminar and the experimental data compare well with the
computational data. Further downstream, R=1800-2110,
there is sharp temperature rise region. This rise is
associated with transition since a transitional boundary
layer is heated and some of this heat is convected to the
model surface via turbulent-like vortices. An estimate of
transition onset was determined as the intersection point of
two straight lines passing through the laminar region and
sharp temperature rise region. Based on this criterion, the
transition onset is estimated to be in the range R=1960-
1990. The estimate compares well with linear stability
theory
9
which predicts an N-factor for the most unstable
frequency of about 8 at R=1975. Downstream of R=2110,
the temperature decreases due to the combined effect of a
relatively cold model base and, possibly, the flow field
tending to fully turbulent flow.
Schlieren Data
Schlieren data are presented in Fig. 3. These
measurements were conducted over the aft 3.5" of the
model which was positioned downstream of the nozzle
exit plane. A wavy structure can be identified near the
edge of the boundary layer. The wavelength of these
waves is measured to be approximately twice the
boundary layer thickness. These waves occur in wave
packets and are associated with second mode
disturbances
2,6
. The second mode disturbances are first
detected at about R=2025 according to a closer
examination of the video records used to construct Fig. 3.
This location is slightly downstream of transition onset as
estimated from the surface temperature measurements.
Boundary Layer Mean Data
The experimental and computational boundary layer
thickness distributions are presented in Fig. 4. Note that
the computational
14
boundary layer thickness distribution
was curve fit using a second order polynomial, and the
experimental error is ± 2% of the plotted values. Except
for a few locations over the range, R=1610-1915, the
experimental δ is slightly lower than the computational δ.
As discussed below this suggests a misalignment of the
model such that the boundary layer measurement ray is on
the windward side of the model. From R=1945 to R=2120,
the experimental δ becomes greater than the
computational δ, confirming the transitional nature of the
boundary layer over this region. For 1610 ≤ R ≤ 1915, the
close agreement between the laminar flow computational
predictions and the experimental data suggests that,
experimentally, the mean flow is laminar over this region
(i.e. no mean flow distortion). This laminar character is
seen more clearly with the aid of Figs. 5 and 6 which are
discussed next.
The experimental mean total temperature and mass flux
profiles are presented in Figs. 5 and 6 at 4 streamwise
locations. Also, laminar total temperature and mass flux
profiles, computed from the Navier-Stokes code of Ref.
14, are presented as the solid lines in Figs. 5-6. At
R=1785, the experimental and computational data
compare well; no effect of model misalignment is evident.
The good agreement with computational data at R=1785 is
typical of all total temperature data over the range, 1610 ≤
R ≤ 1915. This is consistent with the boundary layer
thickness, confirming the laminar flow region, R ≤ 1915.
At R=1945, the transitional nature of the boundary layer
becomes evident due to the slight total temperature
distortion from η=5.27 (0.706 δ) to η=6.62 (0.887 δ) and
mass flux distortion from η=5.61 (0.751 δ) to η=6.93
4
(0.928 δ). These distortions become more pronounced
further downstream. At R=2035, the total temperature is
distorted from about η=5.09 (0.734 δ) to η=6.39 (0.921 δ).
At the most downstream location, R=2120, the lower
portion of the boundary layer region is distorted from
η=4.49 (0.680 δ) to η=6.18 (0.935 δ), marking a "high
fluctuating disturbance" region. Overall, the mean flow is
distorted in the range, (0.71-0.93) δ, which is in the
vicinity of the critical layer.
The uncalibrated rms profiles are presented in Fig. 7.
The hashed region, from R=1610 to R=1750 represents the
measurement range over which the rms signal-to-noise
(S/N) ratio was ≤1. Thus, the data at these five streamwise
locations were not considered further in this study. Also
shown in Fig. 7 is the locus of the maximum disturbance
energy. The position of the maxima are at about 80 to 90%
of the boundary layer thickness which is in good
agreement with the eigenfunction maxima locations
predicted by stability theory. For the range, 1785 ≤ R ≤
1945, the S/N > 1, but no clear indication of rapid
maximum rms amplitude growth is evident. However, just
downstream, a rapid growth region occurs over the range,
1975 ≤ R ≤ 2120. The location of R=1975, is in good
agreement with the transition onset location estimated
from the surface temperature data.
The CVA calibration of Ref. 12 was used to convert the
data of Fig. 7 into rms mass flux and total temperatures.
These data are normalized by the mean mass flux and total
temperatures. For brevity, the normalized rms quantities
are termed rms fluctuations in the presentation of Figs. 8-
9. (Note that the instrumentation noise was not subtracted
for these data and thus only regions where the S/N > 1
represent the true fluctuation levels).
The mass flux and total temperature rms fluctuations
are presented as a function of R in Fig. 8. The rms
fluctuations are presented at the maximum energy
locations. At the most upstream location, R=1785, the
mass flux and total temperature rms fluctuations are 2.1%
and 0.5%, respectively. Over the region, 1785 < R ≤ 1945,
the mass flux and total temperature rms fluctuations
increase only slightly from their upstream values.
However, over this region, S/N≈1, and thus the actual
mass flux and total temperature rms fluctuations are lower
than the values shown. Further downstream, S/N >1, and
the fluctuation levels increase for both flow variables. At
the most downstream location, R=2120, the rms
fluctuations reach a maximum of 11% and 2.3% for the
mass flux and total temperature, respectively. Thus, from
R=1975 to R=2120, the mass flux rms fluctuation
increases by a factor of 5.2, comparable to the total
temperature increase of 4.6. However, considering each
location over the full R-range, the mass flux rms
fluctuation is a factor of 4-8.5 larger than the total
temperature rms fluctuation. Consequently, the second
mode disturbances are predominantly of a mass flux, or
acoustic, nature.
A typical set of boundary layer mass flux and total
temperature rms fluctuation profiles are presented in Fig. 9
at R=2035. This location is in the rapid disturbance
growth region. For both the mass flux and total
temperature, the rms fluctuation maxima occur at η=6.17,
or 0.889 δ. Thus, the rms maxima occur in the critical
layer region, (0.8-0.9) δ, as expected. In addition, the
calibrated maxima compare within 1% of the uncalibrated
maxima. Overall, for the entire streamwise range
surveyed, the uncalibrated maxima were within 0-3% of
the calibrated maxima.
Boundary Layer Fluctuation Data
The fluctuation spectra are presented in Figs. 10 and 11
at the maximum energy locations. Fig. 11 represents the
frontal view of the fluctuation spectra of Fig. 10. Prior to
discussing the specific instability waves of interest from
this data, the second mode fluctuations (f=210-290 kHz)
are first discussed.
As shown in Fig. 10, for R ≥ 1975, the second mode
amplitudes increase in the streamwise direction.
Furthermore, the frequency of the most amplified second
mode disturbances increases in the streamwise direction as
observed from Fig. 11. This observation verifies the
boundary layer tuning of the disturbances and also
confirms their second mode character
6
. Specifically, over
the range, 1975 ≤ R ≤ 2060, the frequency of the second
mode most amplified disturbances increases,
corresponding to the boundary layer thickness decrease
over this same range as observed in Fig. 4. Over the range,
2060 < R ≤ 2120 (last 3 streamwise locations), the second
mode most amplified disturbance frequency remains
constant at 254 kHz, suggesting a reduction in disturbance
growth rate over this range. This is consistent with the
“small” change in boundary layer thickness over this same
range as observed from Fig. 4. Overall, the boundary layer
tuning of the disturbances is consistent with the boundary
layer thickness data of Fig. 4.
Although the second mode disturbance growth rate
decreases over the last 3 streamwise locations, the
amplitudes grow to the last measurement station of
R=2120 as observed from Fig. 10. Since Kimmel
6,15
defines transition onset over a straight cone as the
streamwise location where the second mode amplitudes
reach a maximum before decaying, transition onset does
not occur in the present work. However, transition onset
estimated from the surface temperatures occurs in the
range R=1960-1990 as previously discussed. Thus, the
transition onset location as defined by Kimmel may not
work well for a flared-cone configuration due to the rapid
disturbance growth relative to the straight cone. (Kimmel's
definition seems more appropriate for defining the end of
transition as opposed to the beginning). In summary,
transition onset occurs in the vicinity of second mode
rapid disturbance growth, R=1975.
For 1945 ≤ R ≤ 2120 the most unstable frequencies in
terms of the maximum N-factor is in the frequency range,
245-255 kHz. Based on LST
10
, the most unstable
frequency range over the same R-range is 220-230 kHz.
Subsequent measurements presented in the next section
clearly indicate a most unstable frequency range of 218-
228 kHz, confirming that the frequency shift in the present
data is caused by model misalignment. This frequency
shift is attributed to a corresponding change in boundary
layer thickness.
5
The fluctuation spectra through the boundary layer at a
fixed streamwise location, R=2120, is presented in Fig.
12. (These data were measured using a different hot-wire
than all other data presented in this paper.) For R=2120,
the second mode frequency band, 210-290 kHz, is
constant throughout the boundary layer as expected. In
addition, the amplitude profile of the second mode band,
with respect to distance from the wall, is similar to the
stability theory eigenfunction profile. The second mode
amplitudes approach zero at both the wall and boundary
layer edge which is consistent with the prescribed
boundary conditions of a subsonic, second mode.
16
The amplification rates in 3 frequency bands of interest
are discussed in Figs. 13-22. The first frequency band of
interest is from 65 < f < 85 kHz. This frequency band is
associated with first mode disturbances but these
disturbance amplitudes are not clearly discernible in Fig.
10. The second band of interest, 110 < f < 130 kHz, is
associated with a sub-harmonic of the second mode. This
band is discernible in Fig. 10 over the last three R-
locations. As previously discussed, the last band of
interest, 210 < f < 290 kHz, is associated with the second
mode and is clearly discernible in Fig. 10. (Note that an
additional frequency band of interest, 0 < f < 80 kHz, is
not associated with Görtler vortices generated by the
curved flare since subsequent measurements over a
straight cone showed similar low-frequency growth. These
low-frequency disturbances may represent the footprint of
the freestream spectra. The growth of the low-frequencies
are consistent with the growth of the freestream
disturbances which would include the influence of the
Görtler mode on the nozzle wall.)
Figs. 13-19 present the spectra of the amplification
rates, fluctuation amplitudes, and instrumentation/CVA
noise amplitudes at 7 streamwise (R) locations on the
model flare. (Note that the right ordinate amplitude scale
varies from Figs. 13 to 19. In addition, the ratio of
fluctuation amplitudes to noise amplitudes in the spectra
are used to estimate the S/N ratio.) The amplification rate
data of Figs. 13-19 are discussed separately below in the
following order: first mode, second mode, second mode
sub-harmonic.
First Mode
In the frequency band, 65 kHz to 85 kHz,
the existence of the first mode is established from Figs.
13-19. The first mode remains unstable throughout the
entire streamwise range surveyed but the amplification
rate remains below 0.005. These observations compare
well with LST
10
. Since the oblique first mode disturbances
are most unstable in supersonic flows
16
, only a component
of the first mode waves are measured using the present
single hot-wire configuration. Thus, the 65-85 kHz band is
lower than the actual first mode frequency range, but the
degree of frequency shift cannot be determined from the
present data.
Second Mode
The frequency range, f=210 kHz to 290
kHz, is associated with the second mode. For R=1785, the
second mode is barely detectable near f=220 kHz. Further
downstream, at R=1850, the second mode becomes
unstable (-α
i
>0) in a small band around f=225 kHz. At
R=1945, the second mode amplification rates increase
substantially over a fairly large frequency band. The
amplification rates increase at the next location, R=1975,
but decrease montonically in the downstream direction for
R > 1975. However, the second mode remains unstable to
the last measurement location, R=2120. As seen clearly in
Figs. 15-18, the frequency band associated with the
second mode maximum amplification rate and maximum
amplitude do not coincide. This occurs since the unstable
second mode amplitudes shift steadily to higher
frequencies due to the overall thinning of the boundary
layer in the downstream direction (i.e. boundary layer
tuning). Thus, the maximum amplification rate is shifted
to the higher frequency side of the unstable second mode
frequency band. A similar observation was made by
Stetson
6
.
Sub-harmonic
The frequency band, 110-130 kHz,
associated with the second mode sub-harmonic was not
clearly identifiable from Fig. 12. However, the data of Fig.
12 were obtained using a hot-wire probe with a length-to-
diameter ratio that is approximately 25% lower than used
for the Fig. 10 data and thus the sub-harmonic may have
been attenuated. Note that unlike harmonics, which are
associated with nonlinearities
17
of the second mode, the
sub-harmonic is considered to be a separate mode of
oscillation similar to the secondary, sub-harmonic, helical
disturbances used as forcing frequencies in previous
PSE
18
and DNS
19
studies. For the range, R=1785-1945,
the S/N≈1 in the sub-harmonic frequency band. At
R=1945, the S/N is slightly greater than 1, and shows the
most upstream detection of the sub-harmonic. Slightly
downstream, R=1975, the sub-harmonic first becomes
unstable. In contrast, initial instability of the second mode
occurs at R=1850. Furthermore, R=1975 marks the
location of a rapid rise in amplification rates for the sub-
harmonic. In contrast, the second mode rapid rise in
amplification occurs at R=1945. The sub-harmonic
amplification rates increase for the next location, R=2005,
but decrease montonically for R > 2005. However, the
sub-harmonic remains unstable up to the last measurement
location, R=2120. Similar trends are observed for R ≥
1975 for the second mode. Thus, the overall downstream
character of the sub-harmonic is similar to the second
mode. However, the maximum amplification rate is
shifted downstream for the sub-harmonic relative to the
second mode.
The PSE study of Ref. 18 indicates that mild secondary
instability of the forced sub-harmonic helical mode occurs
at 10% mass flux fluctuations. From Fig. 8, the mass flux
is 10% or greater for the last 3 streamwise locations, R >
2060, corresponding to the region of the sub-harmonic
growth in Fig. 10. However, due to the obliquity of these
disturbances, only a component of the sub-harmonic is
measured in the present investigation. Thus, the possible
obliquity of the second mode disturbances in the present
experiment cannot be ascertained. As a result, previous
PSE
18
and DNS
20
studies, which use a helical pair of
second mode disturbances as forcing inputs may not be
suited for direct comparison. Rather, PSE or DNS studies
using a 2D second mode as forcing, or additional
experimental measurements, are needed for direct
comparisons.
