Upload
bhanu-prakash
View
75
Download
0
Embed Size (px)
Citation preview
RESEARCH ARTICLE
Stochastic estimation of flow near the trailing edgeof a NACA0012 airfoil
Ana Garcia-Sagrado • Tom Hynes
Received: 4 February 2010 / Revised: 21 December 2010 / Accepted: 10 March 2011
� Springer-Verlag 2011
Abstract A stochastic estimation technique has been
applied to simultaneously acquired data of velocity and
surface pressure as a tool to identify the sources of wall-
pressure fluctuations. The measurements have been done
on a NACA0012 airfoil at a Reynolds number of
Rec = 2 9 105, based on the chord of the airfoil, where a
separated laminar boundary layer was present. By per-
forming simultaneous measurements of the surface pres-
sure fluctuations and of the velocity field in the boundary
layer and wake of the airfoil, the wall-pressure sources near
the trailing edge (TE) have been studied. The mechanisms
and flow structures associated with the generation of the
surface pressure have been investigated. The ‘‘quasi-
instantaneous’’ velocity field resulting from the application
of the technique has led to a picture of the evolution in time
of the convecting surface pressure generating flow struc-
tures and revealed information about the sources of the
wall-pressure fluctuations, their nature and variability.
These sources are closely related to those of the radiated
noise from the TE of an airfoil and to the vibration issues
encountered in ship hulls for example. The NACA0012
airfoil had a 30 cm chord and aspect ratio of 1.
List of symbols
C Chord of the airfoil (m)
f Frequency (Hz)
h Trailing edge thickness (m)
kx Streamwise wavenumber (1/m)
p0, p0w Fluctuating surface pressure (Pa)
qL Linear wall-pressure fluctuating sources,
qL ¼ dudy
ov0
ox (1/s2)
qL/r Weighted linear wall-pressure fluctuating
sources, accounting for the distance to the
wall (1/m s2)
r Distance between the wall-pressure source and
the point of observation on the surface of the
airfoil (m)
S Saddle point
u0, v0 Streamwise and normal velocity fluctuations
(m/s)
u0s, v0s Stochastically estimated streamwise and normal
velocity fluctuations (m/s)
u0i,s Stochastically estimated velocity fluctuation
(m/s)
u0i Velocity fluctuation (m/s)
U1 Freestream velocity (m/s)
Ue Velocity at the edge of the boundary layer (m/s)
Uc Convection velocity (m/s)
urms Root mean square of velocity fluctuations (m/s)
V Vortex
x Streamwise distance from the airfoil leading
edge (m)
x0 Streamwise distance from the airfoil trailing
edge (m)
y Normal distance from the wall or from the
airfoil extended centre line (m)
z Lateral distance from the airfoil mid-span (m)
A. Garcia-Sagrado � T. Hynes
Whittle Laboratory, Department of Engineering,
University of Cambridge, 1 JJ Thomson Avenue,
Cambridge CD3 0DY, UK
Present Address:A. Garcia-Sagrado (&)
Applied Modelling and Computation Group,
Department of Earth Science and Engineering,
Royal School of Mines, Imperial College London,
Prince Consort Road, London SW7 2BP, UK
e-mail: [email protected]
123
Exp Fluids
DOI 10.1007/s00348-011-1071-9
Rec Reynolds based on the chord of the airfoil
d Boundary layer thickness (m)
d* Boundary layer displacement thickness (m)
s Time delay (s)
U Surface pressure power spectral density
(Pa2/Hz)
UpipjCross-spectra between surface pressure
fluctuations from microphones pi and pj
(Pa2/Hz)
Uup, Uvp Cross-spectra between streamwise and normal
components of velocity and surface pressure
[(m/s)Pa/Hz]
q Density (kg/m3)
LSE Linear stochastic estimation
MLSE Multi-point linear stochastic estimation
MSLSE Multi-point spectral linear stochastic estimation
MQSE Multi-point quadratic stochastic estimation
QSE Quadratic stochastic estimation
TE Trailing edge of the airfoil
1 Introduction
Understanding the relationship between the surface pres-
sure and velocity fields is essential to provide further
insight into the mechanisms responsible for flow-induced
noise and vibration and devise solutions to control the flow
field and minimise their undesirable effects.
Due to the impossibility to measure the pressure fluctu-
ations inside the flow because of the lack of non-intrusive
techniques, measurements of the pressure fluctuating field
have been confined to the wall. Wall-pressure measure-
ments have been done by researchers with the aim to
improve the understanding of the flow structures through
their wall-pressure manifestation (Blake 1970, 1975;
Brooks and Hodgson 1981; Daoud 2004; Hudy et al. 2003;
Gravante et al. 1998; Roger and Moreau 2004; Wark et al.
1998). Furthermore, wall-pressure measurements have been
performed in the literature (Brooks and Hodgson 1981;
Roger and Moreau 2004; Rozenberg et al. 2006, 2007; Yu
and Joshi 1979), in order to obtain surface pressure statistics
upstream of the trailing edge (TE) of an airfoil, which are
intimately related to the statistics of airfoil self-noise.
A few researchers have performed simultaneous mea-
surements of wall pressure and velocity such as Blake
(1975), Daoud (2004) and Goody (1999). Blake (1975)
reported simultaneous measurements of fluctuating surface
pressures on trailing edges of flat struts and of the fluctu-
ating velocities in the wakes that generated those pressures.
The aim was to investigate the influence of trailing edge
shapes on vortex induced vibrations of turbine blades, by
analysing the pressure field and the relationship of the
fluctuating surface pressures with the near wake region of
each trailing edge. Daoud (2004) investigated the pressure
and velocity fields downstream of a separating/reattaching
flow region on a splitter flat plate with fence. As Daoud
(2004) reported, separating/reattaching flows contain
highly energetic structures which generate large wall-
pressure fluctuations that are a direct representation of the
excitation forces produced by the turbulent flow on
the surface. If the excitation happens at the frequencies of
the surface’s resonant modes, considerable vibrations and
noise are generated. Daoud (2004) emphasised the impor-
tance of a better understanding of the wall pressure char-
acteristics to control such unwanted effects. Furthermore,
by measuring the velocity field at the same time, the tur-
bulent flow activity above the surface responsible for the
wall-pressure generation can be investigated. Goody
(1999) measured surface pressure and velocities over a
wing-body junction and a 6:1 prolate spheroid in order to
investigate three dimensional boundary layers. The corre-
lation between surface pressure and velocity was analysed.
However, measurements were only done with a single
pressure transducer instead of using multiple transducers
spatially separated and hence, the analysis of the pressure
field was limited.
In this paper, a stochastic estimation technique has been
applied to the simultaneous measurements of instantaneous
pressure and velocity near the TE of a NACA0012 airfoil,
downstream of a separated laminar boundary layer. From
the cross-correlations between the simultaneously measured
signals, regions of high correlation levels have been asso-
ciated with the location of the main pressure generating
flow structures (Garcia-Sagrado and Hynes 2011). How-
ever, the stochastic estimation technique has permitted a
deeper investigation into the flow structures associated with
the surface pressure generation. First used by Adrian (1977,
1979), this technique estimates the ‘‘pseudo-instantaneous’’
velocity field from its wall-pressure signature. Evolution in
time of the estimated velocity field provides a picture of the
convecting wall pressure generating flow structures and
allows information about the variability and nature of the
sources of the wall-pressure fluctuations.
2 Experimental setup
2.1 Airfoil model and experimental setup
Figure 1 depicts the experimental set-up where the airfoil
investigated has been placed at the exit (open jet configu-
ration) of an open-circuit blower type wind tunnel, sup-
ported by two perspex side walls.
The symmetric airfoil employed in the investigation was
a NACA0012 airfoil with a chord of 300 mm and an aspect
Exp Fluids
123
ratio of 1 that can be seen in Fig. 2. It consisted of three
parts of which the middle one, with a span of 200 mm, was
hollow. This part is also made of three different pieces
allowing the airfoil to be completely dismantled and pro-
viding access to the interior in order to place the instru-
mentation. The two lateral parts are solid and have a span
of 50 mm each.
Measurements of wall-pressure fluctuations are often
carried out in acoustically quiet wind tunnels or are treated
afterwards by applying noise cancellation techniques
(Agarwal and Simpson 1989; Helal et al. 1989; Naguib et al.
1996), in order to minimise the low-frequency contamina-
tion on the surface pressure signals by the facility back-
ground noise. In the present case, measurements were done
in the most quiet wind tunnel available with the internal
walls of the tunnel covered with a foam material (5 cm
thickness) that reduced the background noise of the facility
by up to 20 dB from 100 Hz onwards. This allowed the
measurement of the pressure signature of interest without
the need to apply a noise cancellation technique. Noise
cancellation techniques (Naguib et al. 1996) were applied to
the first set of data, and since it was found that there was no
further improvement by the application of the noise
cancellation methods, these have not been applied to the
data. Furthermore, a series of flat plate measurements with
embedded microphones were carried out to ensure that the
background noise of the wind tunnel with the foam material
was lower than the aerodynamic wall-pressure signature.
The exit of the tunnel has a rectangular cross section of
0.38 m 9 0.59 m (after the foam). The freestream turbu-
lence intensity of the wind tunnel is 0.4% allowing the
investigation of the flow around a NACA0012 in a smooth
non-turbulent inflow.
The results presented in this paper correspond to mea-
surements done at a Reynolds number of Rec = 2 9 105
based on the chord of the airfoil and a freestream velocity of
10 m/s. The TE thickness h was 1.6 mm and h/d* [ 0.3, d*,
being the boundary layer displacement thickness at the edge.
As indicated by Blake (1986), this corresponds to a blunt TE,
where vortex shedding is then normally observed from the
TE. Note that the NACA0012 airfoil was truncated near the
TE in order to have a thickness at the TE of 1.6 mm and
hence, a blunt TE. This resulted in a final chord C of 297 mm.
2.2 Model instrumentation and outline of surface
microphone array
The unsteady surface pressure measurements were per-
formed with microphones FG-3329-P07 from Knowles
Electronics that are 2.5 mm diameter omnidirectional
electret condenser microphones with a circular sensing area
of 0.79 mm.
These microphones have been embedded in the airfoil
under a pin hole of 0.4 mm diameter in order to minimise
attenuation effects at high frequencies due to the finite size
of the microphones. A microphone under a pin hole
arrangement is depicted in Fig. 3a. The geometrical
dimensions of the pin hole configuration (diameter, length
of pin hole) were selected such that the resonant frequency
associated with the arrangement (similar to a Helmholtz
resonator) was greater than 20 kHz which is outside the
range of interest. This was done by performing tests on a
flat plate with microphones under different pin hole geo-
metrical configurations. The dimensions of the microphone
and final pin hole configuration are indicated in Fig. 3c.
Fig. 1 Simplifying sketch of the exit of the wind tunnel and perspex
side walls holding the airfoil
Fig. 2 NACA0012 model used for the experiments: a NACA0012 assembled, b middle hollow part and lateral solid parts and c middle hollowpart made of three different pieces, one of them, an exchangeable TE
Exp Fluids
123
Additional microphones were located within the airfoil
in order to be able to measure the surface pressure fluctu-
ations very close to the TE. These microphones were
located inside the airfoil in a remote microphone arrange-
ment, since due to the reduced space close to the TE, they
could not be placed directly beneath the pin hole. They
were linked to the pin holes on the surface by plastic tubes
of 0.4 mm internal diameter that run along inner passages
and were continued ‘‘infinitely’’ to avoid reflections from
standing waves. The plastic tubes passed through the
interior of small boxes, each one containing a microphone
on one of their lateral sides. The boxes and microphones
were all inside the airfoil. The pressure fluctuations were
felt by the microphones through a small hole in the wall of
the plastic tube and in the wall of the box in contact with
the microphone.
A sketch illustrating this remote microphone arrange-
ment can be seen in Fig. 3b. With this method, it was
possible to measure the pressure fluctuations as close to the
edge as 2 mm (1 % of the chord), less than 1d*TE.
The surface microphone array is depicted in Fig. 4. The
positions of the microphone pin holes on the surface1 are
summarised in Table 1. Note that the black dots represent
the location of the pin holes but do not correspond to their
actual size (0.4 mm).
2.3 Measurement techniques
As mentioned earlier, the unsteady pressure measurements
were performed with FG-3329-P07 microphones from
Knowles Electronics. The sensitivity of the FG-3329-P07
microphone was provided by the manufacturer to be about
22.4 mV/Pa (45 Pa/V) in the flat region of the microphone
response (from 70 to 10,000 Hz approximately). From the
calibrations performed in the laboratory, the sensitivity of
the FG-3329-P07 microphones employed varied approxi-
mately between 20.2 and 23.5 mV/Pa in the flat region.
The microphones were powered by two units with 8
channels each containing the circuits needed for each
microphone (manufactured by the Electronics Develop-
ment Group at the Engineering Department of the Uni-
versity of Cambridge). The microphone signals were
amplified and low-pass filtered (at 30 kHz) before being
connected to the BNC-2090 connector panel. The sampling
frequency was 65,536 Hz, and a total of 223 = 8,388,608
samples were acquired.
A tube with a length of 110 mm was used for the cali-
bration of the microphones. The output from a white noise
Fig. 3 a Microphone under a
pin hole configuration, b remote
microphone arrangement and
c dimensions of microphone and
pin hole configuration
Fig. 4 Surface microphone array. The system of coordinates indi-
cated has its origin at the TE and at the airfoil mid-span
1 The position of the pin holes correspond to the microphone location
for those microphones in the pin hole configuration; microphones in
the remote microphone arrangement are placed further inside the
airfoil.
Exp Fluids
123
signal generator was connected to a loudspeaker that was
placed at one end of the tube. At the other end, in a cap
closing the tube, the FG-3329-P07 microphone and an
ENDEVCO 8507C-1 pressure transducer were placed
equidistant from the centre of the circular cap. The FG-
3329-P07 microphone was placed under a pin hole con-
figuration with the same dimensions as the one on the
airfoil. The ENDEVCO transducer with a diameter of
2.5 mm was placed flush mounted. The ENDEVCO known
calibration (slope of the linear Pa vs. V calibration) is
constant with frequency for the range of frequencies of
interest, i.e., up to 20 kHz. First of all, the performance of
the tube-calibrator was assessed by calibrating another
ENDEVCO pressure transducer whose calibration had
been previously obtained using a Druck DPI520 pressure
indicator as a pressure source. In this case, both END-
EVCO transducers were placed flush mounted on a cap at
the end of the tube.
Two tubes with two different diameters, 10 and
12.5 mm, were used for the calibration of the microphones.
According to acoustic theory (Fahy and Gardonio 2007),
plane wave propagation will occur in the ducts (tube) until
ka & 1.84, where a is the radius of the tube and k = x/c is
the wavenumber (c the speed of sound). Hence, the men-
tioned tubes provided a calibration of the microphones up
to f & 20 kHz when using the tube of 10 mm diameter and
f & 16 kHz with the tube of 12.5 mm diameter.
In order to calibrate the microphones connected to pin
holes very close to the TE, and hence, placed inside the
airfoil using the remote arrangement, the tube of 12.5 mm
diameter was employed. In this case, the calibration tube
was left open at the opposite end of the loudspeaker and it
was placed upside down covering the microphone to be
calibrated and an ENDEVCO flush mounted on a small flat
plate connected during the calibration to the TE of the
airfoil. This way, the microphones under the remote
arrangement could be calibrated in situ and their frequency
response (amplitude and phase versus frequency) could be
obtained. The attenuation and possible resonances induced
by the plastic tube used to connect the microphone with the
pin hole on the surface were accounted for by this in situ
calibration (calibration method described next). Examples
of the calibration of two of these remote microphones are
given in Fig. 5. Note the deviation of the frequency
response of the microphone under the remote arrangement
from the frequency response of the same microphone under
the pin hole configuration. The measured signal was first of
all converted into the frequency domain by means of cal-
culating its Fourier transform, resulting in an amplitude
(Volts) and a phase versus frequency. The microphone
frequency response obtained from the calibration was then
applied to the measured amplitude and phase for each
frequency. Then the inverse Fourier transform was per-
formed resulting in a signal with pressure units. Note that
the units of the microphone frequency response amplitude
are dB (20 log10 (A/Aref), where Aref is 1V/0.1Pa (1/0.1
V/Pa) and A is the amplitude of the calibration in V/Pa
(1/A Pa/V)).
The method employed in the calibration of the FG-3329-
P07 microphones is based on the calibration procedure
from Mish (2001). The calibration consists of taking two
different measurements (a and b) as it has been sketched in
Table 1 Positions of
microphones in the airfoil
model (chord C = 297 mm)
a Chordwise distance from the
LE to the microphone location
(to the pin hole)b Chordwise distance from the
TE to the microphone location
(to the pin hole)c Microphones placed using the
remote microphone
arrangement sketched in Fig. 3b
Microphone number xa/C Distance from
TEb, x0 (mm)
Distance from
mid-span, z (mm)
1c, 25c 0.99 2.0 0, 2
2c, 26c, 27c 0.98 5.0 0, 4, 10
3c 0.97 10.0 0
4, 15, 16, 17, 18, 19, 24 0.92 25.0 0, 5, 10, 15, 35, 65, -35
5 0.90 30.0 0
6 0.88 35.0 0
7 0.86 40.0 0
8 0.83 50.0 0
9 0.80 60.0 0
10 0.76 70.0 0
11 0.70 89.2 0
12 0.61 117.2 0
13 0.42 173.2 0
14 0.16 250.0 0
1psc 0.99 2.0 0
2psc 0.98 5.0 0
8ps 0.83 50.0 0
Exp Fluids
123
Fig. 6. In the first one, (a), the output signal from the white
noise source is measured at the same time as the output
signal from the ENDEVCO (with the BNC-2090 board).
Since the calibration of the ENDECVO is known, the
output signal from the speaker can be calculated. Once the
output from the speaker is known, since the input to the
speaker has also been measured at the same time, the
speaker response can be calculated.
In the second measurement, (b), the output signal from
the white noise source is measured at the same time as the
output from the FG-3329-P07 microphone. The frequency
response of the system formed by the speaker and micro-
phone can be calculated and since from measurement a),
the speaker response was obtained, the FG-3329-P07
microphone response can be calculated.
The above-described calibration method provided the
frequency response, i.e., amplitude and phase calibration of
each individual microphone used in the measurements of
the surface pressure fluctuations over the frequency range
of interest (20 Hz to 20 kHz). Note that these FG-3329-
P07 microphones clip if the signal is[30 Pa ([124 dB). It
was checked that the surface pressure fluctuations on the
airfoil under the flow conditions investigated were lower
than that limit.
A total of 223 = 8,388,608 samples were recorded in
order to characterise the wall-pressure field. The uncer-
tainty in the surface pressure spectra was mainly due to the
statistical convergence error, which is inversely propor-
tional to the number of records used. In order to reduce this
error, the spectra was calculated as the average of the
spectra of individual data records obtained from dividing
the pressure time series into a sequence of records. The
total number of records used was 1,024 resulting in an
uncertainty of about 3% (1=ffiffiffiffiffi
Nr
p;Nr being the number of
records).
The velocity measurements have been performed with
constant temperature hot-wire anemometry. The hot-wire
instrumentation included a fully integrated constant tem-
perature anemometer with built-in signal conditioning. The
AC and DC components were logged separately using
different pre-amplifier gains in order to ensure the adequate
resolution of the fluctuating AC component. This compo-
nent was obtained by band-pass filtering the signal between
1 Hz and 30 kHz. The hot-wire signal was then recon-
structed by adding the DC level to the zero-mean filtered
AC output and thereby enhancing the signal resolution.
The cross-wire probe that was used to simultaneously
measure the normal and streamwise components of
velocity was a subminiature boundary layer probe, espe-
cially designed by Dantec to given specifications. The
length of the wires was 0.7 mm with a diameter of 2.5 lm.
From specifications, the angle between each wire and the
horizontal axis is 45� and the angle between the wires is
90�. This subminiature cross-wire allowed measurements
as close to the surface of the airfoil as 0.3 mm, which is
approximately 3% of dTE or 15% of d�TE.
Boundary layer and turbulence intensity profiles from
the cross-wire probe were firstly compared with those
Fig. 5 Examples of the frequency response (amplitude and phase) from two of the microphones connected to pin holes very close to the TE and
thus located in the airfoil using the remote microphone configuration
Fig. 6 Method employed in the calibration of the FG-3329-P07
microphones (Following Mish 2001)
Exp Fluids
123
measured by a single hot-wire. This was a Dantec type P15
boundary layer probe with a tungsten element of 1.2 mm
length and 5 lm diameter set to an overheat ratio of 1.8.
The hot-wire probe was calibrated in the freestream
region of the test section by logging the voltage from the
probe together with the total and static pressures at
approximately the same position. These pressures and
hence the freestream velocity were measured with a pitot-
static probe that was placed at the same height as the hot-
wire in the test section and at a different spanwise location.
A best-fit calibration for King’s law (E2 - A = BUn) was
then found, where A, B and n are calibration constants and
A = Eo2 is the zero flow output voltage. The effect of air
temperature drift was taken into account with the correc-
tion described by Bearman (1971).
The surface of the airfoil was found by using a resis-
tance circuit that measured the electrical contact between
the probe and the surface. In order to take into account
the proximity of the surface and its effect on the cooling
of the wire, a traverse with no flow was performed to find
out the Cox (1957) correction to be applied to the mea-
sured data.
A yaw calibration was performed in a calibration tunnel
to determine the relationship between the effective cooling
velocity for each wire and the streamwise and cross-stream
velocity components u and v and to confirm the angle
between the two wires of the cross-wire configuration. The
calibration was carried out by changing the yaw angle of
the cross-wire probe over a certain range in a known
velocity flow (magnitude and direction) and recording the
output of the wires at every angle.
In a single hot-wire probe, the wire is perpendicular to
the flow direction experiencing the most cooling influence
and resulting in the maximum output voltage for a given
velocity. The angle between the velocity vector and the
normal to the wire in a plane that contains both, wire and
velocity, is zero. If there is an angle between the flow
direction and the normal to the wire, the wire mainly reg-
isters the cooling effect of the component normal to the wire
and very small amount of cooling results from the parallel
component. The velocity corresponding to the ‘‘net’’ cool-
ing influence is the effective cooling velocity, Ue.
The form of the yaw-response function used has been
that proposed by Champagne et al. (1967) and Champagne
and Sleicher (1967), that was previously introduced by
Hinze (1959). This function is expressed below as:
Ue ¼ UF2ð90� aÞ¼ U cos2ð90� aÞ þ k2 sin2ð90� aÞ
� �1=2 ð1Þ
where U is the flow velocity magnitude and a is the angle
between the velocity direction and the wire. Therefore,
90 - a is the angle between the velocity direction and the
normal to the wire, i.e., the yaw angle. The parameter k is a
constant that is determined from the yaw calibration, and
k2sin2(90 - a) is associated with the cooling influence of
the velocity component parallel to the wire.
A logging frequency of 65,536 Hz, equal to the sam-
pling frequency of the microphones, was used. For the
mean and rms profiles, a total of 219 = 524,288 samples
for the AC signal and of 217 = 131,072 samples for the DC
signal were acquired. When acquiring simultaneously
velocity and surface pressure data, 220 = 1,048,576 sam-
ples were acquired at each channel of the data acquisition
board. In order to calculate the cross-correlation between
velocity and pressure, a total of 256 records were used. The
velocity spectra were calculated by averaging 512 records,
resulting in an uncertainty due to statistical convergence
error of about 4%. For the cross-spectra between velocity
and surface pressure, a total of 2,048 records were
employed and the uncertainty is approximately 2%.
The velocity with the cross-wire probe was measured at
approximately 80 wall-normal (y) locations. These loca-
tions went from y = 0.3 mm to 40 mm with increments of
approximately 0.05 mm (0.007d) up to 0.1d, 0.1 mm
(0.014d) up to 0.2d, 0.5 mm (0.071d) up to d, 1 mm
(0.140d) up to 1.5d and 5 mm (0.710d) up to at least 2d.
For verification of the cross-wire calibration and mea-
surement procedure, the boundary layer and turbulence
intensity profiles at several positions were compared with
those measured with a single hot-wire. A good agreement
between the two measurements was observed with a
maximum error of 3% of U1 for the boundary layer profile
and of 1% of U1 for the turbulence intensity profile. The
plots have not been included for brevity.
To check for a possible interference caused by the
presence of the cross-wire probe on the wall-pressure
measurements, the microphones output (autospectra) was
compared with the cross-wire probe at different y locations
(not included for conciseness). The results with the cross-
wire located further away from the wall was considered as
the no interference case. The spectra corresponding to the
cross-wire position closer to the wall showed good agree-
ment with the other spectra, indicating that the microphone
measurements were not affected by the presence of the
probe in the flow.
The velocity measurements were taken in the mid-span
region of the airfoil after confirming that the same velocity
results were obtained at different spanwise locations within
that region.
3 Principle of stochastic estimation technique
In Garcia-Sagrado and Hynes (2011), a cross-correlation
analysis between the velocity in the wake and in the
boundary layer of the airfoil and the surface pressure
Exp Fluids
123
fluctuations in the region near the TE shed light on the
identification of flow structures associated with the surface
pressure fluctuations. However, that identification was
based on time-averaged statistical information which does
not disclose in detail the nature and variability of the
mechanisms of the wall-pressure generation. In order to
improve the understanding and provide additional infor-
mation about the relationship between the flow field and
the surface pressure, a stochastic estimation of the velocity
field associated with wall-pressure events was carried out.
With this technique, important information can be gathered
about the instantaneous dynamics and spatio-temporal
structure of the various events and their evolution (Gue-
zennec 1989).
The ‘‘quasi-instantaneous’’ velocity at different normal
positions over the surface of the airfoil, i.e., at the locations
of the cross-wire traverse, has been estimated from the
wall-pressure fluctuations as if the velocity at all the dif-
ferent positions over the wall had been measured simulta-
neously and at the same time as the pressure signals from
the microphones on the surface. Evolution in time of the
estimated velocity provides a picture of the quasi-instan-
taneous flow structures passing by the streamwise location
of the cross-wire and affords further information about the
wall-pressure generating mechanisms.
The turbulent velocity field, u0i;sðro þ Dr; t þ sÞ, has
been estimated from a known surface pressure signature or
event, p0wðro; tÞ, where ro ¼ ðxo; 0; zoÞ, with components xo,
0 and zo, in the streamwise, wall-normal and spanwise
direction, respectively, is the location of the pressure event,
subscript i refers to the velocity component and s denotes
stochastic estimation. The estimated mean-removed
velocity u0i;sðro þ Dr; t þ sÞ is obtained from a Taylor series
expansion in terms of the mean-removed surface pressure
event p0wðro; tÞ.u0i;sðro þ Dr; t þ sÞ ¼ Aiðro þ Dr; sÞp0wðro; tÞ þ Biðro
þ Dr; sÞp02w ðro; tÞ þ � � � ð2Þ
where Aiðro þ Dr; sÞ and Biðro þ Dr; sÞ are the estimation
coefficients for the linear and quadratic terms, respectively.
The objective of the stochastic estimation technique is
to obtain these coefficients Ai and Bi in order to be able to
calculate the velocity from Eq. 2. When only the linear
term in the equation is employed, the estimation is known
as Linear Stochastic Estimation (LSE), whereas when the
first two terms are included in the estimation, it is called
Quadratic Stochastic Estimation (QSE). Both can be
implemented using a single point of observation (data
from a single microphone or pressure transducer on the
surface of the airfoil) or multiple points simultaneously,
leading to what is called multi-point stochastic estimation,
linear (MLSE) or quadratic (MQSE). In the current
investigation, a multi-point estimation has been used
since, as proved by different authors such as Bonnet et al.
(1998) and Daoud (2004), it provides generally a more
accurate estimation of the dominant flow structures than
single-point estimations.
3.1 Single-point LSE
The single-point LSE provides a linear estimation of the
conditional velocity field from the surface pressure of a
single point of observation:
u0i;sðro þ Dr; t þ sÞ ¼ Ai;linðro þ Dr; sÞp0wðro; tÞ ð3Þ
In order to determine the linear estimation coefficients,
Ai,lin, the long-time mean squared error between the
measured velocity and its estimate, eiðro þ DrÞ, is
minimised, leading to:
Ai;lin ¼u0iðro þ Dr; t þ sÞp0wðro; tÞ
p20w ðro; tÞ
¼ u0iðro þ Dr; t þ sÞp0wðro; tÞp02w;rmsðro; tÞ
ð4Þ
Therefore, Ai,lin is equal to the ratio between the cross-
correlation of u0i and p0w, at the time delay s between the
estimate and the pressure event, and the square of the root-
mean-square of the surface pressure.
The equations to obtain the coefficients for the single-
point QSE can be found in Daoud (2004) or Naguib et al.
(2001)
3.2 Multi-point LSE (MLSE)
The multi-point LSE is based on a weighted linear
combination of surface pressure events at multiple
positions. The relationship between the estimated veloc-
ity and the surface pressure would be in this case (the
surface pressure p0w has been called p0 to simplify the
equations):
u0i;s¼Ai;1p01þAi;2p02þAi;3p03þAi;4p04þAi;5p05þAi;6p06þAi;7p07 ð5Þ
using seven observation points on the surface of the airfoil,
corresponding to the seven streamwise microphones closer
to the TE. In this case, Ai;1;Ai;2. . .Ai;7 are the unknown
coefficients, p01 is the microphone closer to the TE at 2 mm
upstream (x/C = 0.99) and p07 is the seventh microphone
upstream of the TE in the streamwise direction, at 40 mm
from the edge (x/C = 0.86). As in the single-point LSE, the
coefficients are determined by minimising the mean square
error between the estimate and the measured velocity
leading to:
Exp Fluids
123
Au;1
Au;2
Au;3
..
.
Au;7
2
6
6
6
6
6
4
3
7
7
7
7
7
5
¼
p01p01 p01p02 p01p03 . . . p01p07p02p01 p02p02 p02p03 . . . p02p07p03p01 p03p02 p03p03 . . . p03p07
..
. ... ..
. ... ..
.
p07p01 p07p02 p07p03 . . . p07p07
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
�1u0p01u0p02u0p03
..
.
u0p07
2
6
6
6
6
6
4
3
7
7
7
7
7
5
ð6Þ
Similar expression for the normal component of the
velocity v0.The inverse matrix on the right-hand side of Eq. 6 con-
tains data from the two-point correlation between the data
of the seven streamwise microphones closer to the TE. The
second matrix on the right-hand side represents the cross-
correlation between the velocity and the surface pressure
from all the microphones used in the MLSE. Note that the
cross-correlation values must not be normalised (i.e. they
must not be the cross-correlation coefficient values).
3.3 Multi-point spectral LSE (MSLSE)
Multi-point Spectral LSE (MSLSE), which is an extension
to the classical stochastic estimation technique described
above, was applied in this case (Tinney et al. 2006; Ewing
and Citriniti 1997).
The coefficients for the calculation of the stochastically
estimated mean-removed velocity are in this case a func-
tion of the frequency f and relate the Fourier transforms of
the estimated velocities with the Fourier transforms of the
surface pressure events:
u0fi;sðf Þ ¼ Afi;1ðf Þp0f 1ðf Þ þ Afi;2ðf Þp0f 2ðf Þ þ � � �þ Afi;7ðf Þp0f 7ðf Þ ð7Þ
Similarly as in the previous cases, the coefficients are
determined by minimising the mean squared error between
the estimate and the actual measured velocity, leading to a
similar expression to Eq. 6 to be solved for each frequency
but instead of with cross-correlations between the different
signals, with the cross-spectral density between signals
(Up0ip0j, Uu0p0 and Uv0p0). The solution from these equations
leads in this case, to an array of complex spectral
estimation coefficients Afi,j(f).
Afu;1
Afu;2
Afu;3
..
.
Afu;7
2
6
6
6
6
6
4
3
7
7
7
7
7
5
¼
Up01p0
1Up0
1p0
2Up0
1p0
3. . . Up0
1p0
7
Up02p0
1Up0
2p0
2Up0
2p0
3. . . Up0
2p0
7
Up03p0
1Up0
3p0
2Up0
3p0
3. . . Up0
3p0
7
..
. ... ..
. ... ..
.
Up07p0
1Up0
7p0
2Up0
7p0
3. . . Up0
7p0
7
2
6
6
6
6
6
4
3
7
7
7
7
7
5
�1 Uu0p01
Uu0p02
Uu0p03
..
.
Uu0p07
2
6
6
6
6
6
4
3
7
7
7
7
7
5
ð8Þ
Similar expression for the normal component of the
velocity v0.
As Tinney et al. (2006) summarised, the advantage of
the Multi-point Spectral Linear Stochastic Estimation
(MSLSE) is that for fields where the estimated condition is
separated in time (typically by a convection velocity:
s ¼ Dx=Uconv), the time lag s between the unconditional
source, p0j(t - s), and the conditional estimator, u0i(t), is
embedded in the spectral estimator.
It is important to remark that in the cases where the
unconditional terms are marginally correlated within
themselves, the quality of results expected from both the
MLSE and the MSLSE will be limited. This is particularly
the case in situations such as the present study where
surface pressure is used to estimate the velocity. There is a
high level of filtering that occurs between the pressure field
and the velocity field, that is, the pressure field is driven by
the most compact/coherent flow structures. Therefore, a
‘‘decent’’ correlation between velocity and pressure as well
as between the pressure field itself is necessary if good
results are to be obtained from stochastic estimation tech-
niques. The level of the correlation marks the success or
failure of the technique in identifying the dynamic flow
structure associated with the surface pressure generation.
4 Results
The laminar boundary layer over the surface of the airfoil
present at Rec = 2 9 105, separated at approximately
x/C = 0.65 (x & 193 mm) and reattached a short distance
upstream of the TE, at about x/C = 0.97 (x & 288 mm).
The results shown correspond to a cross-wire location at
5 mm upstream of the TE of the airfoil (x/C = 0.98, x =
292 mm), which is immediately downstream of the reat-
tachment point. The cross-wire was traversed normally to
the surface of the airfoil. The surface pressure signals used
for the multi-point stochastic estimation of the velocity
over the surface at that position correspond to the signals
from the seven streamwise microphones closer to the TE,
covering a region of 40 mm upstream of the TE, between
x/C = 0.86 to x/C = 0.99. The results from the multi-point
spectral LSE will be shown.
Figure 7 illustrates the MSLSE mean-removed velocity
vector field together with the pressure time series at
x/C = 0.97, from the microphone immediately upstream of
the location of the cross-wire at x/C = 0.98. The stochas-
tically estimated velocity vector has been plotted for two
consecutive time windows. The fluctuating surface pres-
sure has been normalised by p0rms. The horizontal axis
represents the normalised time. Figure 8 represents the
stochastically estimated velocity vector viewed in a frame
of reference moving with 0:6U1. For this, the mean
velocity has been added to the fluctuating components of
the velocity u0 and v0 and then, the convection velocity
Exp Fluids
123
0:6U1 has been subtracted from the horizontal velocity
component. The saddle points and vortices associated with
positive and negative pressure peaks, respectively, have
been indicated in the figure with letter S and letter V. They
are well identified when viewed in a frame of reference
moving with a similar velocity to that of the vortex cores
(Fiedler 1988; Vernet et al. 1999).
In turbulent boundary layer cases, the convection
velocity can be calculated from the wall-pressure wave-
number-frequency (kx-f) spectra that displays an inclined
ridge where most of the fluctuating energy is concentrated.
A line representing the peak locus of the spectrum ridge
passes through the origin and its slope (f/kx) indicates the
convection velocity of the dominant wall-pressure distur-
bances. In the current case, where the laminar boundary
layer separates, due to the dominant periodic disturbances,
no spectrum ridge could be observed in the kx-f spectra. On
the contrary, most of the energy was concentrated around
the main tonal peaks. No convection velocity could hence
be extracted in this way for this case.
To investigate the effect of the convection velocity, the
stochastically estimated velocity vectors were viewed in
different frames of reference, using different convection
velocities. It was observed how varying the velocity of the
frame of reference affected the location of the vortical
structures and saddle points. With lower constant convec-
tion velocity, the saddles and vortex cores appeared closer
to the wall whereas with higher values they moved further
away. However, as Adrian et al. (2000) observed, the fact
that the vortices are recognisable with the three convection
velocities indicates that it is not necessary to remove an
eddy’s exact translational velocity, sometimes, a window-
averaged streamwise velocity works well identifying a
majority of vortices in a given flow. In the current case,
0:6U1 seems to be a good averaged value. It will be seen
later, that with this convection velocity, when plotting the
0
1
2
3
4
5y/
δ*
0 10 20 30 40 50 60−5
0
5
τ Ue/δ*
p’/p
rms
(a)
0
1
2
3
4
5
y/δ*
60 70 80 90 100 110 120−5
0
5
τ Ue/δ*
p’/p
rms
(b)
Fig. 7 Surface pressure time series p0 at x/C = 0.97 and stochasti-
cally estimated mean-removed velocity vectors at x/C = 0.98 (5 mm
upstream of the TE) using MSLSE for two consecutive time windows
Fig. 8 Surface pressure time series p0 at x/C = 0.97 and stochasti-
cally estimated velocity vectors at x/C = 0.98 (5 mm upstream of the
TE) using MSLSE for two consecutive time windows, viewed in a
frame of reference moving with 0:6U1
Exp Fluids
123
velocity vectors superimposed onto the spanwise vorticity,
the centres of the vortices coincide with regions of high
vorticity.
Note that following Daoud (2004), the time s is defined
as s = -(t - t0) with t \ t0, t0 being an arbitrary time
offset. It is possible to display different time windows of
the data by selecting different values of t0. With this
operation, the normalised s values have been folded, that
is, they increase in the time-backward direction. Hence, by
reversing the time, the progress of information with
increasing time is from right to left. The motivation for this
is that it allows to view the vector field in a frame of ref-
erence where increasing s may be considered as increasing
downstream distance by using Taylor’s hypothesis of frozen
turbulence.
The stochastically estimated velocity field viewed in a
frame of reference moving with the velocity of the vortex
cores can be used to depict the spatial velocity field con-
sidering Taylor’s hypothesis of frozen turbulence (Taylor
1938). This hypothesis which in its more rigorous form is
only applicable to statistically stationary, homogeneous
flows, affords an approximation to obtain spatial informa-
tion from temporal data.
Following a similar analysis as Daoud (2004), it can
be argued that the applicability of the hypothesis holds
fairly well if as Taylor (1938) showed, urms/Uc � 1, Uc
being the convection velocity of the main flow struc-
tures. Furthermore, the hypothesis is appropriate over a
time window T that is much smaller than an eddy ‘‘turn-
over time’’, l/urms, where l is the characteristic length
scale of the dominant eddies. The latter means that there
is hardly any time for the eddy to evolve and change
state during T. Therefore, T should be much smaller than
l/urms. The dominant length scale l of the structures
associated with the wall-pressure generation was esti-
mated from the wall-pressure wavenumber-frequency (kx-
f) spectra, which has not been incorporated in the paper
for brevity. If kxpeak is the wavenumber associated with
the peak in the spectra, then l & 1/kxpeak. In this case,
kxpeak & 45 and l & 0.022 m. Examining the velocity
(turbulent intensity, urms=U1) in the region upstream of
the TE covered by the seven microphones used in the
estimation, a value of urms of about 0:1U1 is selected
(the turbulent intensity profiles were presented in Garcia-
Sagrado and Hynes (2011). Consequently, considering
that Ue � U1, the structural features can be considered
as spatial structures over a time window TUe/d* � (l/
0.1Ue)Ue/d*, hence, TUe/d
* � l/0.1d*. For an estimated
d* of 1.82 mm at this location (Garcia-Sagrado and
Hynes 2011), TUe/d* � 120. Thus, over a time window
TUe/d* & 12, the Taylor’s hypothesis can be considered
to hold reasonably well and the observed structures can
be regarded as spatial structures. This time window has
been indicated as a reference in the velocity vector plots
of Fig. 8.
In both Figs. 7 and 8, the periodic nature of the struc-
tures is clearly observed. The vortex cores seem to be at a
distance from the wall equal to y/d* & 2, and the saddles or
stagnation points result from the vortex interactions. As the
vortex cores convect past a point of observation on the
wall, they produce negative pressure and the saddle points
generate positive pressure values. In addition to the main
vortices and saddle points, there seems to be other small
vortices below the saddle points that are rotating in coun-
terclockwise direction, in contrast to the principal clock-
wise vortices.
The time between two vortices (or two saddle points) is
sUe/d* & 23, which is equivalent to a normalised fre-
quency fd*/Ue of 0.044. These observations are in agree-
ment with the results from the cross-correlations between
velocity and pressure presented in Garcia-Sagrado and
Hynes (2011) Furthermore, these results corroborate that in
this case, the surface pressure generation is dominated by
the passage of the periodic structures. The periodicity of
the structures can be further confirmed in Fig. 9, where the
surface pressure power spectral density referenced to
po = 20 lPa has been represented for streamwise micro-
phones covering a region between x/C = 0.61 and
x/C = 0.99. A main sharp peak can be seen at a frequency
close to 200 Hz at all microphones.2 This peak is related to
the periodic vortical structures originated in the separated
shear layer and is associated with the instabilities from the
laminar boundary layer, the so-called Tollmien–Schilich-
ting (T–S) instability waves that become amplified along
the separated shear layer. These disturbances interact with
the Kelvin–Helmholtz (K–H) instabilities from the sepa-
rated shear layer that could be related to the second peak
(&230 Hz). The peaks in the spectra seem to be part of a
complex mechanism where the different instabilities
interact. The peak close to 400 Hz corresponds to the first
harmonic of the fundamental peak.
In order to confirm that the convection velocity selected
to plot the velocity vectors in Fig. 8 is the velocity of the
vortex cores and that the circulatory patterns observed in
the vectors are indeed vortices, the vorticity which is a
frame of reference independent measure is plotted next.
Note that there can also be regions of high vorticity without
necessarily the existence of a vortex. Nonetheless, it is
generally expected that vortex cores will be associated with
regions of high vorticity.
Figure 10 shows the spanwise vorticity, xz;s ¼ ov0sox �
ou0soy ,
that has been calculated from the stochastically estimated
2 With the value of the displacement thickness d* at the location of
the cross-wire of approximately 2 mm and the velocity Ue of 10 m/s,
the normalised frequency fd*/Ue is 0.04.
Exp Fluids
123
velocity field by means of a central-finite-difference
scheme. Note that when calculating the termov0sox , the tem-
poral variation is converted into spatial variation by con-
sidering ov0
ox ¼ ov0
ototox and invoking Taylor’s hypothesis in
combination with the local mean velocity. The estimated
velocity vectors of Fig. 8 have been plotted superimposed
onto the contours of vorticity. The vortices have been rep-
resented by a circular arrow indicating the direction of the
rotation and the saddle points by a small circle. The vortex
cores identified in Fig. 8 coincide reasonably well with the
localised regions of high vorticity; hence, this confirms the
existence of vortices whose cores move with a velocity
equal or similar to 0:6U1, which was the velocity of the
frame of reference selected to ‘‘visualise’’ the vortices.
Poisson’s equation governs the pressure fluctuations for
incompressible turbulent flows:
1
qr2p ¼ � o2
oxioxjðUiUjÞ ð9Þ
where p is the pressure, q the fluid density and Ui is the
total velocity. By applying Reynolds decomposition and
for a two-dimensional flow such as the particular flow
under investigation, with only one important mean shear
component, that is homogeneous in the spanwise and
streamwise direction (the latter being an approximation for
boundary layer flows), the previous equation can be written
as:
1
qr2p0 ¼ �2
du
dy
ov0
ox� u0i;ju
0j;i ð10Þ
where u is the mean streamwise velocity, the prime denotes
the mean-removed or fluctuating quantities. The second
term on the right-hand side of the equation has been
expressed using tensor notation.
Equation 10 indicates that the surface pressure fluctua-
tions are a function of the ‘‘flow sources’’ in the turbulent
flow field. The first term on the right-hand side of the
Fig. 9 Surface pressure power spectral density referenced to
po = 20 lPa at different streamwise locations
Fig. 10 Spanwise vorticity and
associated stochastically
estimated velocity vectors using
MSLSE at x/C = 0.98 (5 mm
upstream of the TE) at a
particular time window, viewed
in a frame of reference moving
with 0:6U1
Exp Fluids
123
equation represents the turbulence–mean shear interaction.
It is linear with respect to the velocity fluctuations and is
sometimes called the rapid term because it responds faster
to changes in the mean flow. The second term on the right-
hand side represents the turbulence interaction with itself.
It is non-linear with respect to the velocity fluctuations and
is sometimes called the slow term because it responds to
changes in the mean flow only after the mean flow alters
the turbulence. Often, the two source terms are also
referred to as the mean-turbulent and the turbulent-turbu-
lent sources, respectively. These have been expressed
below as q(x,y,z,t) that represents the spatial distribution of
the strength of the flow pressure sources at time t.
qðx; y; z; tÞ ¼ �2du
dy
ov0
ox� u0i;ju
0j;i ð11Þ
Most investigations of surface pressure fluctuations in
wall-bounded flows consider only the linear term, and the
simplification is normally justified by the large value of
du/dy in comparison with the turbulent velocity gradients
of the non-linear term. An order of magnitude analysis
similar to the one done by Daoud (2004) can also be used
to confirm this.
Figure 11a illustrates the time evolution of the y distri-
bution of the linear pressure sources qL that have been
normalised by Ue2/d*2. It has been calculated from the
stochastically estimated or ‘‘quasi-instantaneous’’ velocity
field. It can be observed that negative pressure peaks
coincide with negative pressure sources and positive peaks
are associated with positive pressure sources. The sources
seem to be distributed across the boundary layer
(d & 3.8d*) and there seems to be good coincidence
between the negative pressure sources and the location of
the strong clockwise vortices observed in the flow structure
of Fig. 8. The positive pressure sources appear upstream
and downstream of these vortices, occupying the same
region of the boundary layer as the negative sources. This
scenario confirms that these dominant clockwise vortices
are responsible for the pressure generation in this particular
case. The term qv0/qx from qL is very strong in the region
near the cores of the main vortices and switches between
negative and positive values creating the negative and
positive pressure sources illustrated in Fig. 11a. The
strength of qv0/qx decreases away from the vortex cores,
however, near the wall, the values of the sources associated
with the wall-pressure generation are still very high. This is
due to the term du/dy from qL, i.e., the mean velocity
gradient, that increases substantially near the wall.
On account of the 1/r decay present in the integral
solution to the Poisson equation, the source term has been
weighted as such and the results are presented in Fig. 11b.
The results from Fig. 11b seem to indicate that from the
point of view of the pressure, the strongest sources are near
the wall, although since the pressure is the result of the
integration over the flow volume, weaker sources that
extend over larger volumes may be equally important as
stronger, more localised sources (Daoud 2004). Hence, it
can be concluded that in the present case, the wall-pressure
sources are distributed across a significant part of the
boundary layer.
5 Conclusions
The complex mechanisms present in the generation of
surface pressure fluctuations by a separated laminar
boundary layer convecting past an airfoil have been inves-
tigated by performing simultaneous measurements of the
surface pressure fluctuations and of the velocity field in the
boundary layer and wake of a NACA0012 airfoil, especially
in the region close to the trailing edge. These detailed
measurements have provided further understanding about
0 20 40 60 80 100 120-5
0
5
τ Ue/δ*
p’/p
rms
0
1
2
3
4
5y/
δ*
-0.2
0
0.2
qLδ*2/U
e2
(a)
-0.2
0
0.2
0 20 40 60 80 100 120
τ Ue/δ*
qL/r δ*3/U
e2
-5
0
5
p’/p
rms
0
1
2
3
4
5
y/δ*
(b)
Fig. 11 Surface pressure time series p0 at x/C = 0.97 and pseudo-
instantaneous linear pressure source, qL, at x/C = 0.98 (5 mm
upstream of the TE). a Contours of qL and b contours of qL weighted,
taking into account the distance to the point of observation of the
pressure
Exp Fluids
123
the flow structures responsible for the wall-pressure gen-
eration. These flow structures comprise the sources of the
wall-pressure field, which are closely related to the sources
of the so-called airfoil self-noise. Improving the under-
standing between the velocity field and the surface pressure
fluctuations is also very important for the flow-induced
vibration problems found in ship hulls or submarine peri-
scopes for example.
The stochastic estimation technique presented in this
paper provided a picture of the ‘‘quasi-instantaneous’’
(velocity vectors) flow structures convecting past the
location of a cross-wire and revealed information about the
variability and evolution of the sources of the wall-pressure
generation. From the stochastic estimation technique, due
to the dominance of the large-scale vortices from the sep-
arated shear layer, the pseudo-instantaneous linear pressure
sources (qL ¼ �2 dudy
ov0
ox) were found to be distributed across
the boundary layer.
Acknowledgments This work was funded by the UK Department of
Trade and Industry under the MSTTAR DARP programme. The
MSTTAR DARP was supported by Rolls-Royce plc and BAE Sys-
tems. The authors are grateful for fruitful discussions with Ahmed
Naguib and his students Mohamed Daoud and Laura Hudy from
Michigan State University and with Charles Tinney from University
of Poitiers (currently at the University of Texas at Austin) and
Lawrence Ukeiley from Florida State University. Special thanks for
their support to Phil Joseph, Neil Sandham, Tze Pei Chong and
Richard Sandberg from the University of Southampton and John
Coupland and Philip Woods from Rolls-Royce and BAE Systems,
respectively. The authors wish to thank Gavin Ross for his out-
standing technical help with the airfoil model instrumentation and the
experimental facility. Ana Garcia-Sagrado also gratefully acknowl-
edges the financial support from the Zonta International Amelia
Earhart Fellowship.
References
Adrian R (1977) On the role of conditional averages in turbulence
theory. In: Proceedings of the 4th biennal symposium on
turbulence in liquids. Princeton, NJ
Adrian R (1979) Conditional eddies in isotropic turbulence. Phys
Fluids 22:2065–2070
Adrian R, Meinhart C, Tomkins C (2000) Vortex organisation in the
outer region of the turbulent boundary layer. J Fluid Mech
422:1–54
Agarwal NK, Simpson RL (1989) A new technique for obtaining the
turbulent pressure spectrum from the surface pressure spectrum.
J Sound Vib 135((2):346–350
Bearman PW (1971) Correction for the effect of ambient temperature
drift on Hotwire measurements in incompressible flow. DISA Inf
11:25–30
Blake W (1970) Turbulent boundary-layer wall-pressure fluctuations
on smooth and rough walls. J Fluid Mech 44:637–660
Blake WK (1975) A statistical description of pressure and velocity
fields at the trailing edges of a flat structure. Technical report,
David W. Taylor Naval Ship R&D Center Report No. 4241
Blake W (1986) Mechanics of flow induced sound and vibration,
Volume I (General Concepts and Elementary Sources) and
Volume II (Complex Flow-Structure Interactions). Applied
Mathematics and Mechanics Volume 17-I and Volume 17-II
Bonnet J, Delville J, Glauser M, Antonia R, Bisset D, Cole D, Fiedler
H, Garem J, Hilberg D, Jeong J, Kevlahan N, Ukeiley L,
Vincendeau E (1998) Collaborative testing of eddy structure
identification methods in free turbulent shear flows. Exp Fluids
25:197
Brooks TF, Hodgson TH (1981) Trailing edge noise prediction from
measured surface pressures. J Sound Vib 78:69–117
Champagne FH, Sleicher CA (1967) Application of multi-point
correlation techniques to aerodynamic flows. Am Inst Aeronaut
Astronaut 28:177–182
Champagne FH, Sleicher CA, Wehrmann OH (1967) Turbulence
measurements with inclined hot-wires. Part. 1 Heat transfer
experiments with inclined hot-wire. J Fluid Mech 28:153–175
Cox RN (1957) Wall neighborhood measurements in turbulent
boundary layers using a hot-wire anemometer. A.R.C. Report
19191
Daoud MI (2004) Stochastic estimation of the flow structure
downstream of a separating/reattaching flow region using wall-
pressure array measurements, Ph.D. thesis, Michigan State
University
Ewing D, Citriniti J (1997) Examination of a LSE/POD complemen-
tary technique using single and multi-time information in the
axisymmetric shear layer. In: Sorensen, Hopfinger, Aubry (eds)
Proceedings of the IUTAM symposium on simulation and
identification of organised structures in flows. Kluwer, Lyngby,
pp 375–384
Fahy FJ, Gardonio P (2007) Sound and structural vibration: radiation,
transmission and response. Elsevier, Amsterdam
Fiedler HE (1988) Coherent structures in turbulent flows. Prog
Aerospace Sci 25:231–269
Garcia-Sagrado A, Hynes T (2011) Wall pressure sources near an
airfoil trailing edge under separated laminar boundary layers.
AIAA J (submitted)
Goody MC (1999) An experimental investigation of pressure
fluctuations in three-dimensional turbulent boundary layers,
Ph.D. thesis, Virginia Polytechnic Institute and State University,
Blacksburg, VA
Gravante SP, Naguib AM, Wark CE, Nagib HM (1998) Character-
isation of the pressure fluctuations under a fully developed
turbulent boundary layer. AIAA J 36(10)
Guezennec Y (1989) Stochastic estimation of coherent structures in
turbulent boundary layers. Phys Fluids A 1:1054–1060
Helal HM, Casarella MJ, Farabee T (1989) An application of noise
cancellation techniques to the measurement of wall pressure
fluctuations in a wind tunnel. Am Soc Mech Eng 22:14–22
Hinze JO (1959) Turbulence: an introduction to its mechanism and
theory. McGraw-Hill Series in Mechanical Engineering, New York
Hudy L, Naguib A, Humphreys W (2003) Wall-pressure-array
measurements beneath a separating/reattahing flow region. Phys
Fluids 15:706–717
Mish PF (2001) Mean loading and turbulence scale effects on the
surface pressure fluctuations occurring on a NACA0015 airfoil
immersed in grid generated turbulence, Master thesis, Virginia
Polytechnic Institute and State University, Blacksburg, VA
Naguib AM, Gravante SP, Wark C (1996) Extraction of turbulent
wall-pressure time-series using an optimal filtering scheme. Exp
Fluids 22:14–22
Naguib A, Wark C, Juckenhfel O (2001) Stochastic estimation and
flow sources associated with surface pressure events in a
turbulent boundary layer. Phys Fluids 13:2611–2626
Roger M, Moreau S (2004) Broadband self-noise from loaded fan
blades. AIAA J 42(3)
Rozenberg Y, Roger M, Moreau S (2006) Effect of blade design at
equal loading on broadband noise. In: Proceedings of the 12th
Exp Fluids
123
AIAA/CEAS aeroacoustics conference, Cambridge, Massachu-
setts. AIAA Paper No. 2006-2563
Rozenberg Y, Roger M, Guedel A, Moreau S (2007) Rotating blade
self noise: experimental validation of analytical models. In:
Proceedings of the 13th AIAA/CEAS aeroacoustics conference,
Rome. AIAA Paper No. 2007-3709
Taylor G (1938) The spectrum of turbulence. Proc R Soc Lond Ser A
164:476–490
Tinney C, Coiffet F, Delville J, Hall A, Jordan P, Glauser M (2006)
On spectral linear stochastic estimation. Exp Fluids 41:763–775
Vernet A, Kopp GA, Ferre JA, Giralt F (1999) Three-dimensional
structure and momentum transfer in a turbulent cylinder wake.
J Fluid Mech 394:303–337
Wark C, Naguib A, Ojeda W, Juckehoefel O (1998) On the
relationship between the wall pressure and velocity field in a
turbulent boundary layer. Am Inst Aeronaut Astronaut, AIAA
Paper No. 98-2642
Yu JC, Joshi MC (1979) On sound radiation from the trailing edge of
an isolated airfoil in a uniform flow. AIAA Paper 79-0603
Exp Fluids
123