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OCR for page 177
24th Symposium on Naval Hydrodynamics
FuLuoka, JAPAN, 8-13 July 2002
Hydrofoil Near-wake Structure and Dynamics at High
Reynolds Number
Dwayne A. Bourgoyne, Joshua M. Hamel, Carolyn Q. Judge,
Steve L. Ceccio, David R. Dowling
(Department of Mechanical Engineering, University of Michigan, USA)
J. Michael Cutbirth
(Naval Surface Warfare Center, Carderock Division, USA)
ABSTRACT
Methods to predict the hydrodynamic
performance of lifting surfaces at full-scale Reynolds
number (chord-based Rec ~108) have been limited by
the scarcity of controlled experimental data for
incompressible flow. This paper describes results
from the second and third phases of an experimental
effort to identify and measure the dominant flow
features near the trailing edge of a large hydrofoil at
Rec approaching 108. The experiments were
conducted at the US Navy's William B. Morgan
Large Cavitation Channel employing a two-
dimensional hydrofoil (2.1 m chord, 3.0 m span) at
flow speeds from 0.25 to 18.3 m/s, yielding Rec
ranging from 0.5 to 61 million. The hydrofoil section
is a modified NACA 16, designed to approximate the
cross section of a generic naval propeller blade, with
the capacity for testing different trailing shapes.
Bourgoyne et al. (2001A) reported observations of
the mean flow, turbulent flow statistics, and mean
surface pressure for the hydrofoil's baseline trailing
edge. Presented here are observations of the
hydrofoil's unsteady near-wake and the unsteady
surface pressures on the trailing edge for both a
baseline and a modified trailing edge configuration.
The results include PIV-acquired vector fields,
surface dynamic pressure spectra, and LDV-acquired
velocity spectra from the separating boundary layer
and near wake. Trailing edge geometry and
Reynolds number dependencies (Re-dependencies) of
the vortex shedding are demonstrated in
instantaneous vector fields and in velocity and
pressure spectra. Shedding intensity is found to
correlate with the mean suction side shear layer
velocity gradient and possibly with suction/pressure
side shear layer symmetry. Variation in the mean
suction side shear layer gradient is shown to be the
result of the Re-dependence of the laminar to
turbulent transition and growth of the suction side
boundary layer.
INTRODUCTION
Lifting surfaces are used both for propulsion
and control of ships and must meet performance
criteria such as lift, drag, vibration, and
hydroacoustic noise. Design tools suitable to predict
such criteria must handle complex flow phenomena
and manage the wide range of scales inherent in
marine applications. To date, the development of
such tools has been limited by the scarcity of
controlled experimental data for incompressible
flows with chord-based Reynolds numbers,
~ Rec _ Us ~ 108, C _ chord ~ exceeding several
million. A research effort was therefore initiated to
examine the flow over a large two-dimensional
hydrofoil (the HIFOIL project). A principal goal of
this effort is to provide a detailed set of
measurements that can be used to formulate and
validate numerical models for use at high Re.
The two-dimensional HIFOIL model has a
NACA-16 derived suction side surface and a beveled
trailing edge, features representative of a naval
propeller blade of moderate thickness and chamber.
This cross section is both application-relevant and
potentially rich in Re-dependent flow features. In
particular, this shape allows examination of wake-
vortex formation, or vortex shedding, and its Re-
dependence. Relatively small modifications to the
trailing edge geometry are known to lead through the
enhancement or suppression of wake vortices to
substantial changes in hydrofoil performance (Blake,
1986~. The application of a bevel (or knuckle) to the
trailing edge can modify the shedding of the wake
vortices. This, in turn, can substantially change the
magnitude and spectrum of the acoustic energy
1
OCR for page 178
generated by the wake. A high sensitivity of shedding
to trailing edge geometry suggests a similar
sensitivity to the properties of the trailing edge
boundary layers and initial separated shear layers.
Aspects of boundary layer growth, separation, and
shear layer development are potentially Re-
dependent. Thus, significant changes in Reynolds
number may lead to significant changes in the near
wake shedding.
Besides serving as a test case for numerical
models, trailing edge vortex shedding is a
phenomenon requiring greater fundamental
understanding. In addition to impacting lift and drag,
near wake shedding is one of the main hydroacoustic
noise sources from a fully submerged non-cavitating
lifting surface. Moreover, when there is feedback
between the vortex shedding drive (associated with
the Kelvin-Helmholtz instability) and a structural
vibrational mode, excessive noise and potentially
damaging vibration (singing) may occur. To date,
the details of this hydrodynamic forcing and
subsequent structural response are largely un-
documented at the Reynolds number of many marine
propulsion applications. Prior work on the shedding
problem has focused on non-lifting bodies fitted with
propulsor-like trailing edges and limited to Rec of
several million (Blake, 1986~. By incorporating a
realistic- suction side surface and hence a major
portion of the lift, the HIFOIL shape serves as the
next step toward a fully representative propulsor
model. It further extends the Rec of the available
data to near full-scale. Researchers have already
computed both the flow field and the noise generated
by minimally lifting surfaces (Wang et al. 1996,
Arabshahi et al. 1999) and plan to extend this work to
the HIFOIL shapes.
EXPERIMENTAL SET-UP
The hydrofoil was designed for use in the
U.S. Navy's William B. Morgan Large Cavitation
Channel (LCC) in Memphis, Tennessee. The largest
facility of its kind in the world, the LCC is a low
turbulence re-circulating water tunnel with a 3.05 m
x 3.05 m x 13 m test section capable of steady flows
from 0.25 to 18.3 m/s. The hydrofoil (Fig. la) has a
chord of 2.13 m, fully spans the test section, and
based on its max thickness (0.171 m) presents a flow
area blockage factor of 6%. Fig. la also depicts the
coordinate system used in presenting the data, with
the idealized sharp tip of the trailing edge defined as
the coordinate (x/C, y/C) = (1, 0~. Facility limits for
flow speed (18.3 m/s) and water temperature (104°
F), yield for the HIFOIL model a Rec as high as 61
million. A bolt-on trailing edge modifier was used to
vary bevel geometry and in turn vary the
characteristics of the vortex shedding produced. The
two bevel designs tested are shown in Fig.lb: the
more streamlined baseline trailing edge (bevel radius,
RB=76.2 mm and apex angle, p=44°), and a more
bluff modified trailing edge (RB=3 8.1 mm, ,B=56°~.
Both geometries generate suction side boundary layer
flow separation in the last 97% of chord. Both shapes
are known through Navy experience to generate
quasi-periodic vortex shedding without causing
excessive hydrofoil vibration.
uniform
inflow
along suction
x-axis I side
/ ~ = 0.171 m
_ / (0~561~)
— ~ ,, ~ ,-
~, //////////////
~ 1 ~ do. Y/C) = (1,0):
pressure |
x/C=0.28 flat bosom
C=2.1336m(7.00ft)
| baseline |
~ 1
0.03 -
0.02 -
0.01
o.oo -
+lc
(a)
| T edified
0.90 0.92 0.94 0.96
O
Figure 1. (a)Hydrofoil cross sectional geometry with
the coordinate system shown and the coordinate (x/C,
y/C) = (1,0) at the trailing edge; (b)Detail of the
trailing edge showing the original (baseline)
configuration and the modified configuration.
The hydrofoil was machined from a solid
Ni-Al Bronze casting and polished to a RMS surface
roughness of 0.25 ,um or less. Based on estimates
from a flat plate boundary layer flow, this represents
a k+ < 1 at the highest Reynolds number tested. Thus
the hydrofoil may be considered hydrodynamically
smooth (White, 1991). For the data presented here,
the hydrofoil boundary layer is not artificially
tripped, but was allowed to transition to turbulence
naturally. Due to the curved suction side surface, the
2
+x/C
OCR for page 179
hydrofoil generates considerable lift: approximately
588 kN at a flow speed of 18.3 m/s and 0° angle of
attack. Further detail on the experimental facility and
model is provided in previous work (Bourgoyne et
al., 2001A, 2001 B).
The test section flow velocity was set
through computer control of the rotational speed of
the LCC impeller and monitored with a stationary
single-component Laser-Doppler Velocimetry (LDV)
probe positioned more than two chord lengths
upstream of the hydrofoil's leading edge. This LDV
probe provided the upstream reference velocity, Uref,
used in the data reduction. To support Particle
Imaging Velocimetry (PIV), the channel's 5000 m3 of
water was seeded with approximately 20 kg of silver-
coated glass spheres of 16 Em mean diameter (Potters
Industries, SH400S33~. Due to test constraints, both
LDV and PIV were taken under identical seeding
conditions, though smaller seed is preferable for
LDV. Tunnel pressure was held constant and
sufficiently high to suppress cavitation. Tunnel
temperature was monitored, and water temperature
increased as much as 1.3°C/hr during tests at 18.3
m/s. The main impact of temperature variation was
to vary water viscosity and hence the Rec for a given
flow speed. Water temperatures for the data
presented here ranged from 27 to 33 °C, producing
the Rec ranges shown in Table. 1.
ref [m/s] 1 0 5 10 75 1 1-0 1 1.5 1 2.25 1 3.0 1 6.0 1 12.0 1 18.3
06 1 1 | 2 | 3 | 4 | 6 7 8 | 15 17730 34|46 52
Table 1. Rec vs. Uref over test temperature range of
27to33 °C.
Measurements of the flow on and near the
hydrofoil were made with an external two-component
LDV (main measuring plane at 1/4 span), static
pressure taps, dynamic pressure transducers, and a
two-component PIV system (measuring plane at 1/3
span). Accelerometers were mounted within the
hydrofoil to monitor vibration. In addition, a separate
investigating team measured the streamwise velocity
statistics of the suction side boundary layer at x/C =
0.43 using a miniaturized 1-component LDV probe
housed within the hydrofoil body (Fourguette, et al.,
2001~.
The external 2-component LDV system uses
Dantec FO probes with 111-mm beam spacing and
1600-mm (in air) focal length. This provides an in-
water probe volume of approximately 170 ,um in
diameter and 6 mm in length. Further details on the
external LDV, including estimation of uncertainty
and time-averaged statistics from velocity surveys,
may be found in prior work. (Bourgoyne, et al,.
2001A). The 2-component LDV was also used to
gather velocity spectra at selected locations in the
wake (Fig. 2, black square symbols). Dantec-
provided software (BSA Flow) was used to estimate
the power spectrum from the uncorrelated raw
velocity data-samples using sample/hold re-sampling
and Fast Fourier Transform (FFT) with Hanning
windowing. Effective low-pass filtering occurred as
a consequence of re-sampling data produced from
random particle arrival times. No high-pass filtering
was employed, but the maximum re-sampling
frequency was maintained at 2-3 times the mean data
rate as a compromise between frequency resolution
and high-frequency aliasing. Data acquisition and
processing parameters defining the uncertainty of
these results are presented in App. A.
co
~ CJ)
O O
11 11
.~1
Q ~
o o
. .
11 11
_~/C - +0.03
~ (DRAWN
,, TO SCALE)
Figure 2. Trailing edge measurement locations. The
region of PIV measurement (flow contours), the LDV
time-averaged measurement stations (vertical black
lines), and the LDV spectral measurement points
(black squares) are shown. Unsteady surface
pressures are measured at the model surface locations
labeled with letters A through H.
The static pressure measurements were
made using 30 measuring taps on the hydrofoil and a
Rosemont differential pressure transducer routed to
individual taps with a Scanivalve rotary sampling
valve. Dynamic surface pressure measurements were
made with an array of fifteen flush mounted pressure
transducers (PCB 138M101) located at the PIV
measuring plane and in the vicinity of the hydrofoil
trailing edge. These sensors were arrayed in an "L"
shape with lines of transducers set parallel (shown in
Fig. 2) and perpendicular to the flow direction.
Results presented in this paper are from the pressure
side transducer 'H-1', nearest the trailing edge. The
dynamic pressure signals were analogue band-pass
filtered from 2 Hz to 5 kHz and sampled at 10 kHz.
3
OCR for page 180
An array of eight accelerometers (Wilcoxon 754-1)
was mounted within the hydrofoil to monitor
vibration. Accelerometer frequency response was 2
Hz tol5 kHz at +/-3dB, were sampled at 10 kHz, and
were filtered by a signal conditioner (PR701A) with a
frequency response of 0.5 Hz to 1 kHz at -3dB.
Results presented in this paper are from
accelerometer 'A2' located with the dynamic
pressure sensors.
Instantaneous flow field measurements in
the vicinity of the hydrofoil trailing edge were made
using a LaVision Flowmaster-3S PIV system running
DaVis v5.4.4 software and utilizing two flash-lamp
pumped YAG lasers (800 mJ per pulse at 532 nary).
A laser sheet masked to approximately 3.2 mm
thickness was routed downward through the top of
the LCC test section to illuminate the suction-side
trailing edge and near wake. The pressure side of the
hydrofoil was not illuminated. Two 1280 by 1024
pixel cameras were operated in tandem to capture the
composite field of view depicted in Fig. 2. Images
were acquired at a rate of approximately 1 Hz. For
the vector fields presented in this paper, raw images
from a single camera were processed via cross-
correlation of 32 by 32 pixel interrogation areas with
50% overlap into two-dimensional velocity fields
with 1.7-mm grid spacing. Thus each vector is the
result of particle pair averaging over a cube of flow
measuring approximately 3.4 mm on a side. Further
information on the experimental instrumentation is
provided in prior work (Bourgoyne et al., 2001A,
200 1B).
RESULTS- PRESSURE COEFFICIENTS
Flow, pressure, and vibration measurements
were made at a variety of test conditions. Presented
here are results for test speeds from 0.5 to 18.3 m/s,
0° angle of attack, un-tripped boundary layers, and
both the baseline and modified trailing edge
geometries. In order to depict the dependence of the
vortex shedding on trailing edge geometry and Rec.
emphasis is given to the flow speeds of 0.5, 1.5, and
18.3m/s.
The pressure coefficient, Cp, on the surface
of the hydrofoil was acquired at speeds from 1.5 to
18.3 m/s for both the baseline and modified trailing
edges. Acquiring data at a lower speed was not
feasible with the pressure sensors used. A
representative Cp curve for the modified trailing edge
(data averaged over all the measured speeds) is
shown in Fig. 3. Also shown is the estimated location
of boundary layer laminar to turbulent transition for
each flow speed. For all speeds, transition was
computed from the mean Cp data of Fig. 3 using
Thwaites' method and the 1-step method of Wazzan,
et al. (White, 1991~. Application of the 1-step
Wazzan method is consistent with the low freestream
turbulence of the LCC and the smooth surface finish
of the hydrofoil. The transition results were validated
against measurements where data was available.
Specifically suction side boundary layer streamwise
velocities at x/C=0.43 were measured with the
onboard LDV for the baseline trailing edge at flow
speeds of 3.0, 6.0, 12.0, and 18. 3 m/s (Fourguette, et
al., 2001~. These boundary layer measurements
indicate a fully laminar condition at 3.0 m/s, a
transitional or turbulent condition at 6.0 m/s, and a
fully turbulent condition at the higher speeds.
0.6
-Cp
~ ~ 30 2.25 1.5~0.5
~''''''"'"''''''''',~.c ............................ :
A .~.' 1.0
~ 18.3
0.4 /
/., O
I
I ~
0.2 j ~
I ~ :
,
18.3
1
.
~L ...... .. _
n
-0.2
Spline of Cp
Transltlon
Cp d ata
............ Cp -- P f \
~ ~ 2 ef ~ ~
~ \.
~ ~~
~'
12 0~ 6 ° ~ 3.0 2;25 1.5 ~
0 0.2 0.4 0.6 0.8 1
x/
/C
Figure 3. Static pressures coefficients measured on
the modified hydrofoil averaged over flow speeds of
1.5 to 18.3 m/s. This data was splined and used to
compute the location of natural transition on the
hydrofoil.
A significant feature of the Cp curve is
immediately apparent: the location of suction side
transition at speeds of 1.5 m/s and below is roughly
fixed at the Cp "cliff" near x/C~0.75. This has
interesting implications concerning the Rec-
dependence of the suction side boundary layer
thickness near the trailing edge. Fig. 4a shows the
PIV-acquired mean boundary layer velocities at
x/C=0.94 for several flow speeds. The velocity, u, is
the magnitude parallel to the surface at the given x/C
and is normalized by utymax), where Ymax is the
measured point furthest from the hydrofoil surface
(and provides the best available estimate of the local
freestream). These profiles as well as those at other
flows speeds were used to calculate the normalized
suction side boundary layer momentum thickness,
0/C, for the modified trailing edge (Fig. 4b). These
values of 0/C indicate that the suction side boundary
4
OCR for page 181
layer thickness at x/C=0.94 is in some ranges
increasing with Rec and in other ranges decreasing
with Rec. This is due to the dependence of 8/C on
the combined effect of the laminar growth rate, the
turbulent growth rate, and the location of transition.
At speeds above 1.5 m/s, all of these variables are
Re-dependent. However, at 1.5 m/s and below, the
location of transition becomes approximately fixed
on the suction side near x/C ~ 0.75, and only the
growth rate variables remain functions of Re. Thus,
near 1.5 m/s, there is a trend reversal in the Rec-
dependence of the suction side boundary layer
thickness at the trailing edge.
0o35r
0.030
Y/
/C
n nn
`;~/ 0.0016
/C
0 OO14
n nn1'
0.025 _
0.5 m/s
1.0 mis
1.5 m/s
_ 18.3m/s
.
_~
- /
~ . /
/
. f
~ I
..................
0.0 0.2 0.4 0.6 0.8
U(y) I U(Ymax)
(a)
Rec x 1 o6 (30°C)
0 10 20 30
1.0 1.2
40 50
. _ ..... ..... .................. A
,
~ I ~—tl- U]dY
~ . ~
where U ~ U(Ymax )
............................... . . .
Theory
o Data
... ....... , ..... .... .. . . .
. . . . . . . . ..
0 001 0 -, 1 ! 1, 1, 1, 1, 1, 1 ~ 1 ! i ! 1, i, 1, 1, 1, 1 1 1, 1 1, 1,
0 2 4 6 ~ 10 12 14 16 18
Flow speed, Urd [mis]
O
Figure 4. The suction side boundary layer
near the trailing edge knuckle (x/C=0.94) with (a) the
PIV-acquired mean velocity component parallel to
the local surface for four speeds and (b) the measured
normalized boundary layer momentum thickness, 8/C
vs. flow speed, Uref, for the modified trailing edge
geometry. Also shown in (b) are the computed
normalized momentum thicknesses.
This proposed role of natural transition in
the Re-dependent behavior of 8/C at the trailing edge
is substantiated by theory. A prediction of 6/C at the
trailing edge, also shown in Fig. 4b, was made using
a turbulent boundary layer integral method. Taking
the 0/C at transition of Fig. 3a as an initial value, the
Karman Integral Relation was solved as described by
White (White, 1991), and employing White's relation
(eqn. 6-120) for the friction coefficient, Cf. Use of the
Thwaites result as an initial condition effectively
treats transition as a point, across which 0/C is
conserved but all other boundary layer quantities
change discontinuously. Despite this simplification,
the predicted trend of 0/C is within 10% of the
measured values at speeds below 3.0 m/s, and at least
follows the measured trend at the higher speeds. The
worsening agreement between measurement and
theory as the speed increases over 1.5 m/s is
attributed to the application of the less accurate
turbulent boundary layer relations over an increasing
fraction of the chord. Also, there is increasing error
in treating transition as a point; the point
approximation is nearer to the truth at the speeds
where transition occurs at the Cp cliff.
Thus we have established that the
momentum thickness of the suction side boundary
layer is strongly Rec-dependent and passes through a
minima near 1.5 m/s. The Rec-dependence of the
momentum thickness is explained by the combined
effect of the Rec-dependence of the location of
transition and of the laminar and turbulent boundary
layer growth rates. The Rec at which 8/C is minimal
is that at which natural transition is approaching the
cliff in the Cp curve at x/C~0.75. Following, data
will be presented that shows a relationship between
the strength of near wake vortex shedding and this
Rec-dependent boundary layer thickness at the
trailing edge.
RESULTS- DEPENDENCE OF SHEDDING ON
GEOMETRY
Figs. 5 and 6 show PIV-derived contours of
the mean streamwise velocity field, normalized by
the flow speed Urem1.5 m/s, for the baseline and
modified trailing edges. The mean is taken from 320
images. Flow is from left to right, and the trailing
edge is shown on the left side of the frame. Data at
the surface suffers from higher uncertainty due to
laser glare, so those values are blanked out (white
space). No data is available in the shadow below the
hydrofoil. Selected vector profiles, also of the
normalized streamwise mean, are included to show
the downstream evolution of the flow field. The
profile at x/C = 1.002 shows the initial shear layer
formed by the merging of the suction and pressure
s
OCR for page 182
0.03
0.02
o~ O
-0.01
-0.02
0.96 0.97 0.98 0.99 1
1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1
I ~ I I , , , 1 1 ~ ~ ~ ~ 1 1 1 1 1 ~ ,
= 1.5 m/s (Re ~ million)
~ . , ~
: ~
| VECTORS |
....................................................................
, , ,,,, I,, . ... - ·
0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1
X/C
Figure 5. Averaged contours and vector profiles of normalized streamwise velocity, U/Uref .
trailing edge at Uref = 1 .S m/s . (320 averages)
0.03
0.02
4.01
-0.02
Normalized
Mean
0.03 Streamwise
Velocity
0.02
0.01
O
-0.01
-0.02
1 11o
1 Loo
.9o
0.80
7o
1 0.60
5o
.4o
.3o
0.20
1 o1o
1 ooo
1 -0.10
or the baseline
0.96 0.97 0.98 0.99
...... ... ,0 ~ 1 .
I I 1 1 1 1
V ECTORS
1.02 1.03 1.04 1.05 1.06
0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1
X/C
Normalized
Mean
Streamwise
Velocity
1.10
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
4.10
Figure 6. Average contours and vector profiles of normalized streamwise velocity, u—/Uref, for the modified
trailing edge at Uref = l.Sm/s . (320 averages)
0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 Normalized
non . .,,,, .,,, ~ ., .,, .. ~ · ~ ~ ~ ~ ~ ~ ~~ 003 Vonticity
Fluctuation
F
am
ooze
0.01
o
-0.01
_ _
0.99 1 1.01 1.02 1.03 1.04
1.05 1.06 1.07 1.08 1.09 1.1
X/C
Figure 7. Instantaneous contour of normalized vorticity fluctuation, (a' - ~ )d /Uref, and vector field of
normalized instantaneous velocity fluctuations, [(u - u—), (v - v~l/Uref, for the baseline trailing edge at
Uref = 1.5 m/s .
0.02
0.01
O
.01
-0.02
-0.03
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
02
0.1
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
· -1 .0
6
OCR for page 183
0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 Nonnalized
,,, ~ , , 0.03 VoriticitY
Fluctuation
0.02
0.01
o
-0.01
-0.02
~.03
0 01
By, O
.03
0.99
1 n,
1.03
1.5 mis (Re ~ million) 0 l
VECTORS -
1.06
1.07 1.08
1 ~
-
-
1~ '-1
-
1.0
0.9
0.8
07
0.6
0.5
0.4
0.3
0.2
0.1
-0.1
-0.2
.3
~4
-0.5
-0.6
-0.7
-0.8
-0.9
-1 .0
Figure 8. Instantaneous contour of normalized vorticity fluctuation, (~-~d /Uref ~ and vector field of
normalized instantaneous velocity fluctuations, [(u - u ), (v - v)~/Uref, for the modified trailing edge at
Uref = 1 .5 m/s .
side boundary layers. The differences in this profile
between trailing edge geometries should be noted,
and will be central to the discussion of the shedding
behavior.
In Figs. 7 and 8, a representative
instantaneous velocity field is shown for the same
flow condition and geometries. Here, contours are the
difference of mean and instantaneous vorticity,
normalized using the reference velocity Uref and a
nominal pressure side boundary layer displacement
thickness, 3*/C= 0.002. (The same nominal value for
6*/C is used in all vorticity figures in this paper,
allowing direct comparison of vorticity magnitudes at
various flow conditions.) The vorticity is calculated
using Stoke's line integral, where vorticity at a given
grid-point is derived from its eight immediate
neighbors. Vectors show the difference of mean and
instantaneous velocity. Shedding structures are
clearly evident in Fig 8 for the modified trailing edge
at 1.5 m/s. This condition and geometry was the
strongest shedding condition tested as determined by
surface pressure spectra, and was of equivalent
strength to 1.0 m/s modified trailing edge condition
as determined by velocity spectra. By comparison the
vertical structures are largely absent for the baseline
trailing edge in Fig 7. In images where such
structures do appear, they tend toward higher x/C and
did not form the organized "vortex street" of Fig. 8.
The appearance at higher x/C of the first organized
vortices is consistent with an inverse relationship
between shedding strength and vortex formation
distance. While 1.5 m/s is the most illustrative case, a
7
-25
an
A
similar trend is observed at all test speeds: the
appearance of coherent shedding structures increases
in going from the baseline to the modified trailing
edge.
The vortex structures apparent in Figs. 7 and
8 should also be detectable in the spectral
characteristics of the near wake. Figs 9 and 10 show
LDV-acquired power spectra of the vertical
velocities, estimated from between four and six
individual Fast Fourier Transforms (FFT's).
10 - log~O(q)v)
~ s
-20
. ~ 0.5 m/s
0.75 m/s
1.0 m/s
1.sm/s
~ 3.0 m/s
: :::: 6.0 m/s
: :::; ~ 18.3 m/s
_ :: . 1:::: 1: ::
Baseline trailing edge
(xJC, y/C) = (1.07, 0)
~
f =flfs
OCR for page 184
ef [m/s] Vrrnsluref [%]
0.5 11 +/- 2
0.75 9 +/- 2
1.0 9 +/- 2
1.5 11 +/- 2
3.0 11 +r 2
6.0 10 +r 1
18.3 10 +r 1
Figure 9. Normalized power spectrum of vertical
velocity component, 4) v, at varying flow speed,
Uref, for baseline trailing edge.
~
10 log1O(~v)
-10
-15
on
-25
-30
a freestream velocity and a wake thickness yf, after
the fashion of Blake (1986) and defined below:
~ ~ Off
v~f)-W
(1)
~_ f
f—~
f (2)
s
f Uref
2;z · yf [Hz] (3)
0.5m/s
- 0.75 m/s
1.0 m/s
------ 1.5m/s
3.0 m/s
60m/s
0.5
Figure 10. Normalized power spectrum of vertical
velocity component, 0) v, at varying flow speed,
Uref, for modified trailing edge.
1 1.S
~
f =flfs
ef [m/s] Vrrnsluref
0.5 13 +/- 2
0.75 13 +/- 2
1.0 14 +/- 2
1.5 15 +r 2
3.0 12 +/- 1
6.0 11 +/- 2
18.3 11 +r 1
2 2.5 3
The data was taken for both trailing edge geometries
at x/C coordinates of 1.01 and 1.07, and at y/C
coordinates chosen by searching (within test time
constraints) for the location of maximum shedding at
that x/C. Data from (x/C, y/C) = (1.07, 0) is presented
here as a point of reasonably good comparison for all
speeds and both trailing edges. Both the power
density axis and the frequency axis are normalized by
Here, yf is taken to be a constant value of yf/C=O.O1.
This is approximately equal to the vertical distance
between the peaks in the Reynolds stresses measured
in the near wake, as described in Bourgoyne, et al.
(2001B). Under this normalization shedding appears
near a normalized frequency of unity. Further
183m/~ definition of the velocity power density, ~v, its
uncertainty, and the data acquisition parameters of
the individual FFT's are provided in App. A. The
relationship of the integral of the power spectrum to
the variance is given on the figure on which it is
plotted. The normalized v=,s values shown are
computed directly from this integral and compare
favorably with LDV time-averaged statistics taken at
the same location during earlier testing (Bourgoyne,
et al., 2001A). The uncertainties given with the
normalized v~',,s values provide an indication of the
uncertainty of the overall spectra. These values are
the RMS of the vrlI,s values computed from the
individual FFT's averaged to make the plotted
spectrum. Note the relative magnitudes of the 1.5
m/s shedding peaks for the baseline and modified
trailing edges. Also note that at all speeds, the
shedding is stronger on the modified trailing edge.
The shedding trends in the velocity spectra
are in agreement with an independent measurement
of the unsteady pressure on the trailing edge surface.
The power spectra shown in Figs. 11 and 12 were
derived from the flush mounted dynamic pressure
sensor H-1, located on the pressure side surface at
x/C = 0.99 (see Fig. 2~. These spectra are normalized
after Blake in a fashion similar to that used for the
velocity spectra:
`~ f ~ _ ~ p Is (4)
qref — 2 PUref [Pa] (5)
8
OCR for page 185
~
10 · loglo(~p)
-30 _
......
..
_
. , ~ ~ ............
-70
.......................................................
............... .... .. ..... .. .. . .
. ~ ~
'' P~''=l2''''''f4'
. q.ref
.
........................................................
Baseline Trailina Ed
l own decade filter applied
. ~
1 ou ~
Uref [m/s] Prrns/qinf [%]
1.0 2.5
1.5 1.0
3.0 0.6
6.0 0.5
12.0 0.4
18.3 0.4
1 .0 m/s |
1.5 mls
3.0 m/s
6.0 m/s
12.0 m/s
1 8.3 m/s
Figure 11. Normalized power spectra of surface
~
pressure, ~ p, at varying flow speed, Uref, for the
pressure side sensor H-1 at x/C=O.99 on the baseline
trailing edge.
~
10 · logo (gyp )
an
-50
~0
-70
~0
-90 -
ef [m/s] Prms/qinf [%]
1.0 2.4
1.5 1.1
2.25 0.6
3.0 0.5
6.0 0.4
12.0 0.4
18.3 0.4
. .. .. .. .
1.0 m/s
1.5 m/s
2.25 mls
- 3.0m/s
~ 6.0 m/s
- 12.0 m/s
18.3 m/s
10° 10'
~
f =flfs
Figure 12. Normalized power spectra of surface
pressure, 4> p, at varying flow speed, Uref, for the
pressure side sensor H-1 at x/C=O.99 on the modified
trailing edge.
Again, the relationship of the power spectrum to the
variance is given on the plot. These spectra are
estimated from approximately (3~106 data points,
partitioned with 50% overlap to provide 376 FFT's
each of 16,384 points, with an estimated spectral bin
uncertainty of approximately 12% (Vetterling, et. al.,
1992~. A linear least-squares fit has been subtracted
from each partition to remove any trend and zero the
mean. Neither acceleration contamination nor other
noise has been subtracted, and a 1/lOOth decade filter
has been applied to clarify the plots (with negligible
effect on peak magnitudes). The upward translation
of the curves as velocities fall below 3.0 m/s reflects
the increasing fraction of the pressure signal that
results from noise and therefore does not scale with
velocity.
The shedding peaks seen in the velocity
spectra on the modified trailing edge at speeds over
3.0 m/s are not distinctly visible in the pressure
spectra of Fig. 12. However, this may be attributed to
shedding of insufficient power to be detected above
the separated shear layer turbulence. As speed is
reduced to below 3.0 m/s, peaks rise above this
background level to reveal a maximum detected
shedding condition at 1.5 m/s. The curve at 1.0 m/s is
compromised by poor signal-to-noise including a
strong noise peak near a normalized frequency of 0.5,
and fails to confirm or discount the 1.0 m/s peak seen
in the velocity spectra. Data from speeds below 1.0
m/s is dominated by noise and is not presented. By
comparison, the pressure spectra for the baseline
trailing edge (Fig. 11) entirely lack peaks (except
those attributed to noise or vibration), indicating no
shedding detectable above the turbulence of the
separated shear layers. Further discussion of the
baseline trailing edge pressure spectra at speeds 3.0
m/s and above, including comparison with historical
data, is provided in (Bourgoyne, et. al., 2000B).
The correlation of shedding strength with
these trailing edge geometries is consistent with prior
9
OCR for page 186
work in the field. A comparison of shedding strength
from minimally-lifting bodies with various trailing
edge shapes (Table 11-2 in Blake, 1986) suggests that
symmetry between the upper and lower shear layers
plays an important role. This might be attributed to
the ability of the shear layers to cooperate and roll up
at a common frequency. Among the more symmetric
shapes, those expected to produce sharper mean shear
layer velocity gradients are also found to shed more
strongly. This trend makes sense from the standpoint
of shear layer stability. A correlation of mean shear
layer gradient to shedding strength is also found with
shedding behind circular cylinders. With this in mind,
Fig. 13 re-plots on common axes the normalized
mean streamwise velocity profiles at x/C=1.002 from
Figs. 5 and 6. (These profiles and those of Fig. 16
are normalized by the upstream flow speed, Uref, and
do not collapse to the same normalized velocity at the
edges of the boundary layers. This is attributed to the
hydrofoil lift and resulting potential flow effects in
the test section.) The profile for the modified trailing
edge (stronger shedding) has a slightly steeper
suction side gradient and has the greater symmetry
between suction and pressure side shear layers. This
coincidence, however, is not sufficient to
demonstrate causality, given that both the stronger
shear gradient and the increased shedding may be
separate effects of the trailing edge geometry.
However, data will be presented next which
strengthens the case for a mean shear layer gradient
and shedding correlation.
[iiillIlilllilllii:lll .!li.il:: illi,,l, il. {ill 'lilllilllli ,
it - - :~_ - ~~ - - -
0.02 - L
- :
-
0 01
nnn
-
.
.. ~ .
.
-0.01
~ . ~
.
. . ~ :
I Baseline TE
I Modified TE
2 ~
_ . , .... _,< .. . ..
- ~
If/ ~
y ~
.~ . .
_ . —_ =~
..
..
I ~ · I It ~ ' 'II ~ it ~ ' 'I !, ! 'I ~ tI ~ ~ ~ I!: .I I i ! |- I ~ ~ | `,
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
U/Uref
Figure 13. Comparison of normalized mean
streamwise velocity profiles, ~/Uref, for the
baseline and modified trailing edges at x/C=1.002
and Uref = l Sm/s
RESULTS- DEPENDENCE OF SHEDDING ON
REYNOLDS NUMBER
In addition to showing dependence of vortex
shedding on trailing edge geometry, Figs. 10 and 12
demonstrate a clear dependence in the velocity and
pressure spectra on flow speed. Consider the
modified trailing edge and its maximum shedding
condition of 1.5 m/s, shown in Figs. 6 and 8 as mean
and instantaneous flow fields. For comparison, Figs.
14 and 15 show the same plots for the lesser shedding
condition of 18.3 m/s on the modified edge. The
instantaneous field of 18.3 m/s clearly reflects the
lower level of coherent shedding that is seen at that
speed in both the velocity and surface pressure
spectra. A given instantaneous field for 0.5 m/s on
the modified edge is more difficult to visually
distinguish from that of 1.5 m/s. (No figure is
presented.) However, a visual survey of a large
sampling of images does suggest the lesser level of
shedding coherence seen in the spectra.
The mean velocity profiles at x/C=1.002 for
the speeds of 0.5, 1.5, and 18.3 m/s are plotted for
comparison as Fig 16. The relationship between
shedding strength and the mean suction side shear
layer gradient is evident. In order of increasing
shedding strength, the test conditions are 18.3, 0.5,
and 1.5 m/s. The same order of speeds applies to
increasing suction side shear layer gradient.
However, the pressure side shear layer gradient is
maximum for the 0.5 m/s case. This may suggest that
the pressure side shear layer is playing a lesser role
than the suction side in determining the shedding.
The pressure side shear layers are more similar and
for all test conditions are relatively steep compared to
the suction side profiles. Since the pressure side shear
layer is less stable, shedding may for these
geometries be governed by the stability of the suction
side layer. Alternately, the shedding strength may be
correlated not to the magnitude of the suction side
gradient, but rather to the symmetry between the
pressure and suction side shear layers. The 1.5 m/s
case is also the case of greatest symmetry. It is
difficult to visually judge whether 0.5 or 18.3 m/s
would take second place on grounds of symmetry.
For the data presented, the role of symmetry is
difficult to separate from the impact of the suction
side shear layer gradient.
Fig. 4a plots the suction side boundary layer
upstream of separation, and reveals the source of the
variation with Rec in the suction side portion of the
initial shear layer profile: the same variation is
apparent in the suction side boundary layers. Though
the data is not available, it is reasonable to expect that
the variation in the pressure side shear layers is
10
OCR for page 187
0.96 0.97 0.98 0.99 1
003
0.02
1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 Normalized
Mean
0.03 Streamwise
Velocity
1 1.10
1 loo
1 o9o
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
-0.10
-0.01 t:
nn'L . .. ... - ..
} , · I , . . . . . . . . , . . . . . . . . . , . . . . . . . . . . . , . , , . ,
0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1
xic
0.02
0.01
o
-0.01
-0.02
Figure 14. Average contours and vector profiles of normalized streamwise velocity, U/Uref ~ for the modified
trailing edge at Uref = 18.3 m/s . (320 Averages)
0.02
0.01
~, O
~ n1
-0.02
0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 Nonnalized
0~03 VonticitY
Fluctuation
0.02
0.01
O
-0.01
.02
-0.03
4.03 t1 ,, ' , 1 , 1 1,, 1 , I ~ 1 , 1 1, 1 ,,
0.99 1 1.01 1.02 1.03 1.04
18.3 m/s (Re =49 mlilion) o
1.06
1.07 1.08
1 O9
1 '
l
-
1
1 ~ :~:~
-
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-0.1
-0.2
~.3
-0.4
~.5
-0.6
-0.7
-0.8
-0.9
-1 .0
X/C
Figure 15. Instantaneous contour of normalized instantaneous vorticity fluctuation, (a~ - a~ )d /Uref, and vector
field of normalized instantaneous velocity fluctuations, [(u - u ), (v - v)l/Uref, for the modified trailing edge at
Uref = 1 8 .3 m/s .
i:, i I i I 1; 1, i i I ., I i | i I, i | l i i j i i: ~ | . ~ i . | 1 i I i | ~ | ~ ; | ~|'| ~ ~
· W /?
_ ~
0.02
- .
~1
-0.01
1 OSm/s ~
1.5 m/s \ :
—153nVs 1:
-0.2 0.0 0.2 0.4 0.6 0.8 1.0
U/Uref
Figure 16. Comparison of normalized mean streamwise velocity profiles, U/Uref ~ for varying flow speed, Uref ~ at
x/C=1.002 on the modified trailing edge .
11
OCR for page 188
likewise found in the pressure side boundary layers.
It has not been determined to what degree the
shedding behavior is influencing the upstream
boundary layers. However, the mean profile
variation observed is consistent with the Re-
dependent location of transition and boundary layer
growth rates discussed earlier. Hence the Re-
dependence of the shedding may be linked through
the mean suction side boundary layer profile to the
Re-dependence of transition and growth rate of the
suction side boundary layer.
SUMMARY AND CONCLUSIONS
In summary, selected results have been
presented from the second and third phases of the
High Reynolds Number Hydrofoil Project, for which
flow features of two trailing edge geometries were
measured at Rec from 0.25 to 61 million.
Dependence of trailing edge vortex shedding on both
trailing edge geometry and Rec is demonstrated in the
PIV vector fields, the LDV-acquired wake velocity
spectra, and in the dynamic surface pressure spectra.
For both geometry and Rec variation, shedding
strength was correlated with the predominant slope of
the initial mean suction side shear layer, and possibly
with the symmetry between the initial suction and
pressure side shear layers. Beyond the potential
importance of suction/pressure side symmetry, the
pressure side shear layer did not appear to play as
significant a role as the suction side in governing
shedding behavior. Re-dependence of the shear layer
gradients is a result of the Re-dependence of the
naturally transitioning and developing boundary
layers. 1.5 m/s emerged as both a maximum shedding
case and the speed below which suction side
transition becomes fixed at the adverse pressure
gradient cliff in the Cp curve. Similar behavior would
be expected of actual naval propulsors, though
surface roughness would push the fixed-transition
condition to lower speeds. These findings invite
further study on the relationship of the mean shear
layer profiles to shedding behavior behind turbulent
hydrofoils. The influence of the location of transition
on the shedding strength raises practical questions
concerning tripping practices in model testing and the
appropriate application of those results to the full
scale.
ACKNOWLEDGEMENTS
The authors of this paper wish to the
acknowledge the contributions of Shiyao Bian and
Kent Pruss of the University of Michigan; William
Blake, Ken Edens, Bob Etter, Ted Farabee, Jon
Gershfeld, Joe Gorski, Tom Mathews, David
Schwartzenberg, Jim Valentine, Phil Yarnall, Joel
Park, and the LCC technical staff from the Naval
Surface Warfare Center - Carderock Division; and
Ki-han Kim, Pat Purtell and Candace Wark from the
Office of Naval Research In addition, the authors
wish to thank the Office of Naval Research for
supporting this research effort under contract nos.
N00014-99-1-0341, and N00014-99-1-0856.
REFERENCES
Arabshahi, A., Beddhu, M., Briley, W., Chen, J.,
Gaither, A., Janus, J., Jaing, M., Marcum, D.,
McGinley, J., Pankajakshan, R., Remotigue,
M., Sheng, C., Sreenivas, K., Taylor, L., and
Whitf~eld, D., "A perspective on naval
hydrodynamic flow simulation," 22nd
Symposium on Naval HydrodYnamics ,National
Academy Press, Washington, DC, 1999, pp.
920-934.
Blake, W.K. Mechanics of Flow Induced Sound and
Vibration~ Vol. 2~ Academic Press, Orlando,
1986.
Bourgoyne, D. A., Ceccio, S. L., Dowling, D. R.,
Jessup, S., Park, J., Brewer, W., and
Pankajakshan, R., "Hydrofoil Turbulent
Boundary Layer Spearation at High Reynolds
Numbers," 23n~ Symposium on Naval
HydrodYnamics Val de Reuil France National
, , ,
Research Council, 2001A.
Bourgoyne, D. A., Judge, C. Q., Hamel, J. M.,
Ceccio, S. L., and Dowling, D. R., :Lifting
Surface Flow, Pressure, and Vibration at High
Reynolds Number," Proceedings of the 2001
International Mechanical Encineerinc
Conference and Exposition, N.Y., NY, Amer.
Soc. of Mech. Eng., 2001B.
Fourguette, D., Modarass, D., Taugwalder, l.,
Wilson, D., Koochesfahani, M., and Gharib,
M., "Miniature and MOEMS Flow Sensors,"
AIAA paper no. 2001 -2982, 2001.
Vetterling, W. T., Teukolsky, S. A., Press, W. H., and
Flannery B. P. Numerical Recipes in C
, , ,
Second Edition, Cambridge University Press,
Cambridge, U.K., 1992.
Wang, M., Lele, S.K., and Moin, P., "Computation
of Quadrapole Noise Using Acoustic Analogy,"
AIAA Journal, Vol. 34., 1996, pp. 2247-2254.
White F.M. Viscous Fluid Flow 2nd Ed. McGraw
, . . .
Hill, Inc., New York, 1991, pp. 433-435.
12
OCR for page 189
APPENDIX A: ERROR ANALYSIS OF THE
VELOCITY SPECTRA MEASUREMENTS
The spectral analysis of the velocity
fluctuations in the hydrofoil wake was performed
using data provided by the LDV and data reduction
capabilities of the BSA Flow software provided by
Dantec. The spectrum tool within the BSA software
estimates the power spectral density from the raw
data-samples using sample/hold re-sampling and
FFT-techniques.
The BSA Flow software allows for three
user inputs for the power spectra: spectral samples,
maximum frequency, and filter settings. The
spectral-samples input determines the number of
discrete frequencies at which the power spectral
density is estimated. As the spectral analysis is FFT-
based, the software limits this user defined setting to
an integer power of 2. The maximum frequency
determines the highest frequency at which the power
spectral density is estimated, and it was maintained at
approximately 2 ~ 3 times the mean data rate for the
data reported in this study to eliminate aliasing.
Although not a user defined variable, the mean data
rate acts as a third variable necessary to define the
spectrum because the combination of the random
seeding particle arrivals for the LDV and the sample-
and-hold re-sampling technique acts as a first order
low-pass filter. The cut-off frequency for this
pseudo-filter occurs at n/2~, where n is the mean
data rate. Lastly, the data were filtered with a
Hanning window with a filter width of W = 0.2. A
summary of the spectral samples, maximum
frequencies, and data rates for the data provided in
Figures 9 and 10 is provided in Table 2.
Velocity
Uref
(m/s)
.
0.5
0.75
.0
1.5
3.0
6.0
18.3
Spectral
Samples
N
8192
8192
1 6384
16384
32768
32768
65536
The power spectral density, PSD, is defined by
Equation (6) where ~v(6 is the PSD, T is the
maximum lag time, V is the Fourier transform of the
velocity time series, u(t), as defined by Equation (7),
and V* is the complex conjugate of V.
As (f ) = T V (f, T)V(f, T) (6)
T
V(f, T) = |(v(t) - v )e-i27'f'dt (7)
o
The physical relationship between the
variance, shown in terms of the RMS value (within
the flow and the power spectrum) is given by
Equation 8.
,
vials= 2 J{~v~f)4f (8)
o
The definition of the spectra above is based
on the assumed knowledge of the true continuous
signal vets. However, in the actual LDV-based
experiment, the time history records are not
continuous. For an LDV-based experiment, a
velocity sample occurs whenever a seeding particle
passes through the measuring volume. Because of
the randomness of this occurrence, the time history of
the flow is provided by a discrete representation with
sequential arrival times of varying incremental time
steps. Without the knowledge of a continuous time
history, only an estimate of the spectra is possible.
The accuracy of the spectra decreases significantly
above the mean data rate. Because the LDV
measurements are non-continuous, The Dantec BSA
Flow software employs the use of the sample/hold
method as described by Equation (9). The sampled
Baseline Trailing Edge
Maximum
Frequency
(s-l)
500
700
1000
1500
3000
l 7000
~ ioooo
Data
Rate
n
( -1y
251
342
388
593
1121
3138
6366
Cut-Off
Frequency
n/2n
( -1'
40
54
62
94
178
499
1013
Spectral
Samples
N
8192
8192
16384
16384
32768
32768 .
65536
Modified Trailing Edge
Maximum ~ ~
Frequency Data Rate
n
-1' ~ (s l)
300 156
500 ~ 213
800 ~343
1500 531
3000 - ~ 869
7000 ~ 2591
10000 ~ ~ 5190
Cut-Off
l Frequency
l n/2
l (s-l)
25
34
l
l 55
85
138
412
826 _
Table 2. A summary of the spectral samples, maximum frequencies, and data rates for the LDV-acquired
velocity spectra of Figs. 9 and 10.
13
OCR for page 190
and held signal is re-sampled at regular intervals to
provide a series of evenly distributed velocity
samples.
{vreSamptt) = V(ti)|(ti ' t ~ ti+l)) (9)
Inaccuracies due to this simplistic approach occur
when two neighboring true samples are further apart
than the time between re-samples. For the spectra
presented, the re-sampling rate, defined as twice the
maximum frequency, is maintained at approximately
3 to 6 times the data rate. Assuming a Poisson
distribution for the particle arrival within a
homogeneous, random distribution of seeding
particles in the fluid, the loss of information due to
re-sampling is 1 ~ 4%. The loss information during
the hold periods acts like a first-order low pass filter
attenuating the spectrum at frequencies above f=
n/2~. For the data provided in Figures 9 and 10,
the attenuation of the signal above the cut-off
frequency should result in a slightly lower calculated
RMS level as compared to the actual RMS. The re-
sampled data results in a 0.6% reduction, for 1.0 m/s
with the modified trailing edge, in the calculated vrT,~s
using the spectrum as compared to the actual vr~,,s
calculated using the raw time series data.
With the re-sampled data, the integral in the
Fourier transform expressed in Equation (7) can be
replaced with the summation given in Equation (10)
where the term T/N = At is the re-sampling interval,
N is the number of spectral samples, and each value
of k represents a different frequency: fk = k/T, k = 0,
1,2, ...,N/2.
Ok N ~(vn v)exp(—it—) (10)
This value is calculated using FFT-algorithms, and as
before, the power spectrum estimate, shown in
Equation (11), is derived by the multiplication of the
Vk and its complex conjugate.
5~)k(fk) T Vk Vk (11)
APPENDIX B.: THE RIGID HYDROFOIL
APPROXIMATION
It is important to establish that the spectral
peaks attributed in this paper to rigid hydrofoil fluid
dynamics are not in fact due to resonances of a non-
rigid structure. Such vibration would generate
pressure spectral peaks (1) through inertial effects
within the pressure sensor (independent of fluid
pressures), (2) by generating surface pressure as the
flow is forced by the surface motion, and (3) by more
complex fluid-structure interactions. Figs. 17 and 18
compare the spectra of the pressure side pressure
transducer 'H-1' at 99% of chord with that of a
nearby accelerometer 'A2', at flow speeds of 18.3
and 1.5 m/s, respectively.
~
10 · log10 (~p )
10 loglo((~)a) f[Hz]
-20
-30
-40
-50
-60
-70
-80
-90
X3 ~
I. . , —.
= 2 1,a( f 3
. ~ . .
~ . Q ...
.
Pressure Sensor H-1 . .
Accelerometer A-2
.. . .
. . . . . . .
~
f =flfs
Figure 17. Comparison of normalized power spectra
~
of surface pressure, 1) p, and normalized power
spectra of foil acceleration, ~ a ~ for the pressure side
sensor H- 1 at x/C=0.99 and nearby accelerometer A2,
at flow speed Uref = 18.3 m/s on the modified
trailing edge. At this flow speed fs = 137 Hz.
14
OCR for page 191
~
10 · log10 (hap )
~
JO log10 ((Da)
-20
-30
-40
-50
-60
-70
-80
-90
f[HZ]
1o1 1o2
, ., . 1' , ~ l .' "1
.
. . on . . - .
, aims, 1 2 ~ ma ~ f )
. ..g..., , ~ 'O' ...~..
. ~ ~ - ......................... ` ~
. . . . . ~
. ~
. . . . . . . .
. ~ I\~ . ~ ~ .
- ~ Pressure Sensor H-1 ~ ~ l
Accelerometer A-2 Al
_ ~ 1 . ,: ., . ~ . . . ..
-100 I iiti,il_1 i ' ' i '1'0O 11 31
~
f =flfs
Figure 18. Comparison of normalized power spectra
~
of surface pressure, ~ p, and normalized power
spectra of foil acceleration, ~ a ~ for the pressure side
sensor H-1 at x/C=O.99 and nearby accelerometer A2,
at flow speed Uref = 1.5 m/s on the modified trailing
edge. At this flow speed fs = 11 Hz.
Normalization of the accelerometer spectra follows a
form similar to that used with the velocity and
pressure spectra:
~ ~ ~-f
(f ) = a s (12)
g
The relationship between the acceleration spectrum
and the variance is given on the plot. These power
spectra were produced in the same way as the
pressure spectra presented earlier, except in this case
no filtering was done. For convenience, the x-axis is
given in both Hz and normalized frequency. The
accelerometer selected is both the one nearest to the
sensor and the one showing the highest accelerations.
Both Figs. 17 and 18 confirm that the acceleration
contamination effects are not present at significant
levels in the frequency ranges of interest (near a
normalized frequency of unity). Peaks in the
accelerometer spectra at 18.3 m/s (Fig. 17) are
reflected in the pressure spectra only at 7, 10, and 18
Hz (marked with asterisks). 18 Hz corresponds to the
lowest natural frequency of the hydrofoil (beam
bending between spanwise supports), and 7 Hz is
both near both the channel impeller blade pass
frequency and the channel flow-circuit lowest-mode
acoustical frequency. Higher frequency peaks in the
accelerometer spectra are not reflected in the pressure
spectra, including the second natural frequency of the
hydrofoil near 30 Hz. (The 30 Hz mode shape is a
pitching motion about the spanwise mounts and
would be particularly interactive with the vortex
shedding.) Fig. 18 shows that at the maximum
shedding case of 1.5 m/s, there is no correlation
between the accelerometer and the pressure sensor,
and there are no acceleration peaks near the shedding
frequency. Maximum RMS acceleration levels at the
four test speeds are provided in prior work
(Bourgoyne, et al., 2001B) and the highest reported
level is roughly O.lg
A lack of fluid structure interaction is
further supported by order-of-magnitude arguments
comparing hydrofoil motion to the measured velocity
fluctuations near the hydrofoil. Cyclic vertical
hydrofoil motion or pitching about the spanwise
supports would potentially generate an oscillation in
the vertical flow velocities in the immediate vicinity
of the leading and trailing edges. Conservatively
assuming that the induced flows near the hydrofoil
are of the same velocity as the hydrofoil surface, and
assuming that the entire 0.1 g of acceleration is
confined to the peak in Fig. 17 at 30 Hz, the trailing
edge tip flow oscillations would be on the order of
0.1% of flow speed (Ure=18.3 m/s). Vertical velocity
fluctuations measured with LDV at the baseline
trailing edge tip (Bourgoyne, et al., 2001A) are
approximately 6% of the flow speed, so that the
estimated flow generated by hydrofoil vibration
would be roughly 1% of the actual measured vertical
flow velocity fluctuation. These levels rapidly
decrease with decreasing flow speed.
15
Representative terms from entire chapter:
shear layer
