Turbulence in the wake of a thin airfoil at low speeds

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Title:
Turbulence in the wake of a thin airfoil at low speeds
Series Title:
NACA TM
Physical Description:
38 p. : ill ; 27 cm.
Language:
English
Creator:
Campbell, George S
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

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Subjects / Keywords:
Downwash (Aerodynamics)   ( lcsh )
Airplanes -- Wings -- Testing   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Experiments have been made to determine the nature of turbulence in the wake of a two-dimensional airfoil at low speeds. The experiments were motivated by the need for data which can be used for analysis of the tail-buffeting problem in aircraft design. Turbulent intensity and power spectra of the velocity fluctuations were measured at a Reynolds number of 1.6 x 10⁵ for several angles of attack. Total-head measurements were also obtained in an attempt to relate steady and fluctuating wake properties. Mean-square downwash was found to have nearly the same dependence on vertical position in the wake as that shown by total-head loss. For this particular wing, turbulent intensity, integrated across the wake, increased roughly as the 3/2 power of the drag coefficient. Power-spectrum measurements indicated a decrease in frequency as wing angle of attack was increased. The average frequency in the wake was proportional to the ratio of mean wake velocity to wake width.
Bibliography:
Includes bibliographic references (p. 23-25).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by George S. Campbell.
General Note:
"Report date January 1957."

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Source Institution:
University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003875247
oclc - 156979694
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AA00009186:00001


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410 4









NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMO0RANDUM 1627


TURBULENCE IN THIE WAKE OF A THIN AIRFOIL AT LOW SPEEDS

By George S. Campbell

SUMMARY


Experiments have been made to determine the nature of turbulence
in the wake of a two-dimensional airfoil at low speeds. The experi-
ments were motivated by the need for data which can be used for analysis
of the tai-buffeting problem in aircraft design. Turbulent intensity
and power spectra of the velocity fluctuations were measured at a
Reynolds number of 1.6 x 10 for several angles of attack. Total-head
measurements were also obtained in an attempt to relate steady an
fluctuating wake properties.

Mean-square downwash was found to have nearly the sam dependence
on vertical position in the wake as that shown by total-head loss.
For this particular wing, turbulent intensity, integrated across the
wake, increased roughly as the 3/2 power of the drag coefficient.

Power-spectrun measurements indicated a decrease in frequency as
wing angle of attack was increased. The average frequency in the wak
was proportional to the ratio of mean wake velocity to wak width.


INTRODUCTION


The study of velocity fluctuations in the wake of an airfoil is
of interest in connection with the buffeting problem. In one forn of
buffeting, referred to as tail buffeting, the horizontal tail of the
aircraft is immersed in the wake of the wing for certain angles of
attack. If the wake contains sufficient turbulent energy, as it may
due to separated flow over the wing, the aircraft may experience un-
desirable load fluctuations. It might be mentioned that buffeting is
not always undesirable, as it is often useful as an advance warning
against dangerous stability deficiencies.

A recent book on aeroelasticity (ref. 1) has sum~marized the re-
sults of some investigations of tail buffeting. Fran this summary,
it appears that there has been considerable work done on buffeting








for hitgh angles of attack, but that there is little information avail-
abe on the possibility of buffeting at lower angles. The use of
increasingly thinner wing sections introduces the likelihood of leading-
edge sepaation at low angles of attack (ref. 2). Although this type of
separation usually disappears at transonic Mach numbers, it is generally
replaced by separation behind the main compression shock. In either
case, the occurrence of such separated flows at low angles of attack
would be expected to cause buffeting difficulties.

A useful approach for the study of most of the various types of
tail buffeting has been provided in reference 3. In this paper, buffet-
ing was considered to be the response of a linear system to a random
.forcing function. In order to predict the response of an elastic struc-
ture to turbulent velocity fluctuations, one must know the power spectrum
of the turbulence, the aerodynamic impedance of the lifting surface, and
the frequency response of the structure. Nonstationary wing theory may
be used to estimate the aerodynamic impedance functions of various lift-
ing surfaces. The determination of the frequency-response character-
istics of aircraft structures has received sufficient attention over a
period of years so that this portion of the buffeting problem may be
considered solved. The remaining quantity necessary for the analysis
of tail buffeting is the power spectrum of turbulence in the wake of the
forward lifting surface. Up to the present time, it has been necessary
to assume that the turbulence is isotropic or that it possesses a dis-
crete frequency corresponding to a Karman vortex street. However, it
is evident that measurements of the power spectra of turbulence behind
wings are necessary before the general statistical approach to buffeting
can be fully utilized.

The primary purpose of the present investigation has been the
measurement at low speeds of the power spectrum of turbulence behind a
thin, t~wo-dimensional airfoil. The tests were made by means of the
usual bot-wire techniques for measuring velocity fluctuations in turbu-
lent flow (ref. S). Angle of attack was varied from 0 to 22 degrees,
an the Reynolds number for most of the tests was about 1.6 x 10 It
might be expected that the resullts could be applied at higher Reynolds
numbers provided the wing exhibits the type of leading-edge separation
shown by- the airfoil in the present tests. Total-head surveys and
surface-tuft studies were made in order to determine the static character-
istics off the wing and its wake. Wherever possible, an attempt has been
made to relate the power spectrum of turbulence to the mean properties
of the flow.


A recent experimental investigation (ref. 4) confirms, in general,
th applicability of classical nonstationary wing; theory when the veloc-
ity fluctuations are of a random nature.


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This research was conducted at the California Institute of Tech-
nology under the guidance of H. W. Liepmann while the author was a
holder of a Howard Hughes Fbllowship.


APPARATUS AND TESTS


Wind Tunnel


The experiments of this investigation were made in the GALCIT:
20- by 20-inch low turbulence wind tunnel; a diagram of the unel, is
presented in reference 6. The turbulence level of the tunnel is about
0.03 percent.

Most of the present tests were conducted at velocities of about
50 feet per second. Free-stream velocity was, in general, measured
using a pitot-static tube and a micromanometer. At lower speeds,
however, it was found convenient to measure velocity by observing on.
an oscilloscope the frequency of vortices shed from a reference cyrlin-
der. A description of this procedure is given in reference 7.

The traverse mechanism for positioning hot-wire an total-head
probes is the same as the one described in reference 7.


Airfoil


A two-dimensional airfoil of 6-inch chord spanned the test section.
The airfoil section was a 1/8-inch thick flat plate with a blut trail-
ing edge and a rounded nose. The leading-edge radius was approximately
half the airfoil thickn~ess. The wing was made of steel and was tested
in a smooth condition.

All measurements were made behind the center section of th wing
span. Reynolds number for most of the tests was about 1.6 x 10 i.


Total-Head Surveys


A total-head survey of the wake was made one chord length behind
the wing trailing edge using a total-head tube of 0.108-inch outer
diameter. Profile-drag coefficient was obtained by integrating th
total-head profile


CD = dc (1)





MACA TM 18r27


This formula is expected to give quite satisfactory accuracy with the
survey plane one chord from the trailing edge (ref. 8).


Hot-Wire Apparatus


The usual methods of hot-wir anemometry (ref. 5) were used to
measure fluctuating velocities in the airfoil wake. Hot wires of 0.1
mil platinum or :platinum-rhodium! were used for all measurements. For
measurements of the longitudinal velocity fluctuation, u2, a 1/8-inch
long wire was soldered to the tips of the probe. Transverse velocity
fluctuations were obtained with a probe having two wires of 0.2 inch
length inclined at angles of +i 30 degrees from the stream direction.

A considerable saving in testing time was accomplished by aonitting
hot-wire calibrations. Reliable relative measurements of the turbulent
velocity fluctuations could be obtained in this manner provided the
hot-wie apparatus was linear. It is estimated on the basis of data
fran reference 9 that the downwrash probes of the present investigation
should be linear for fluctuations up to about + 25o. The overall
linearity of the hiot-wire apparatus was checked by observation of
oscilloscope traces of the hot-wire output. The presence of nonlin-
earities (such as those which could be introduced by excessively high
amplifier gains) generally resulted in flat tops to an otherwise random
oscillograph trace.

The frequency response of the amplifier used for the hot-wire
measurements was flat for frequencies of 1/2 to 25,000 cycles per
second. The post-anplifier was provided with re sis tance- capac itan ce
compensation which was adjusted byr the usual square-wave method. In
order to minimize noise a 10 kilocycle cut-off filter was used in the
present measurements.


Electronic Measurements


Intensity of th turbulent velocity fluctuations was determined
by means of an average-reading vacuum-tube voltmeter. Although the
voltmeter was calibrated to give RMS values for sinusoidal voltage
variations, the instrument did not provide true RMS intensities for
random inputs. On the basis of turbulence measurements with a statis-
-tical analyser (ref. 7), the difference between true and indicated
RM is expected to be less than 10 percent.

power-spectral density was measured with a Hewlett-Packard model
300A wave analyzer. A constant half-band width of 30 cycles per second,
based on an attenuation of Ir0 decibels, was used for all measurements.






NACA TM 16r27


The filtering characteristics of the wave analyzer are shown in figure
1 for th 30 cycle band width. The frequency range of the analyzer was
about 30 16,000 cycles per second. The output was read directly fran.
the instrument's voltmeter, and so the values obtained were subject to
the same errors for rando input as were the intensity measurements
describd in. th previous paragraph.


ACCURACY


Fan Frquency


Of the factors affecting the accuracy of measurements of power-
spectral density, the rotational speed of the wind-tunnel fan is one of
the most important. The motor and fan were quite well isolated from
the test section so that vibrations were not transmitted to the hot-
wire mount. HfoweJver, periodic pressure fluctuations were found to be
transmitted upstream, and so a test was made to determine the effect
of tunel fan rotation on powe~r-spectra measurements. Doawnwash veloc-
ities are of primary interest, and so downwash spectra are compared
in figure 2 for two fan speeds. The same free-stream velocity was
maintained by introducing a screen just ahead of the fan for one of the
tests. Without the screen, the peak in the downwash spectrum occur at
nearly the same frequency as the fan's second hanonic. However, the
downwash spectrum~ has practically the same shape when the fan and wig
peaks do not coincide, i.e., when the screen is added. Therefore, it
may be concluded that the fan is not the cause of the peak in the down-
wash spectrum, even though it does somewhat modify the spectran


Noise


The noise introduced by the measuring apparatus was generally
small and was subtracted from the total output. Near the center of the
wake, the ratio of signal to noise was greater than 100 to 1 for RMS
turbulent intensity. Measurements at 60 cycles per second were usually
high by about 3 percent due to pickup of line frequency. The greatest
uncertainty in the spectrum measurements occurred at the lowest fre-
quency, 30 cycles per second, due to the characteristics of the wave
analyzer.


Wall Interference


On the basis of an analysis by Glauert (ref. 10), the Strouhal
number of the vortex street obtained at an angle of attack of 21.7
degrees in the present investigation should be about 6 percent higher






NACA TM 16r27


than the value obtained in a stream of unlimited extent. This is also
the magniitude of the correction necessary to bring the present results
into agreement wit those of Fage and Johansen (ref. 11), obtained for
the sarme size model in a much larger tunnel. The corrections to
Strouhal number at lower angles of attack are proportionately smaller.
It appears that wall corrections for power spectra and turbulent in-
tensities are not presently available. In order to have consistency
beaten Strouhal numbers and power spectra, all data are therefore pre-
sented without applying an corrections rfor wind-tunn~el wall interference.


Hot-Wire LeRngth Corrections


The general effect of the finite length of a hot wire is to reduce
the voltage across th wire as a result of imperfect correlation of the
velocity fluctuations at different points of the wire. For isotropic
turbulence, estimates of the corrctionts to intensity may be quite
simply obtained (references 12 and 13, for example). In experimental
measurements of the spectrum of turbulence behind grids (ref. 6), cor-
rections to the power-spectral density were found negligible for reason-
able hot-wire lengths. Unfortunately, the estimation of length corree-
tions for the present investigation is greatly complicated by the use
of a two-wire probe in a field whih bears scant resemblance to isotropic
turbulence. In view of this uncertainty, length corrections have not
been applied to the data.


RES ULTS


Wake Energy D~istribution


Two measures of wake energy were obtained in the present investiga-
tion: total-head loss and th mean-square of the downwash fluctuations.
As total-head measurements are considerably simpler than hot-wire
stuies, it is desirable to determine whether properties of the turbu-
lent flow can be related to the simoler mean flow.

Presented in figures 3-7 are the wake profiles obtained from
total-head and hot-wire surveys at a distance of one chord length from
the wing trailing edge. The profiles have been normalized to unit
area; the position of the trailing edge relative to the wake center is
indicated on the curves. It is seen from these results that the shape
of the turbulent velocity profiles closely resembles that of the total-
head profiles. The main differences are the more gradual decay of the
turbulent velocities at the outer edges of the wake and the somewhat
sharper peaks for the total-head less.





NACA TM 16L27


Wake widths from figures 3-7 have been plotted as a function of
angle of attack in figure 8. In defining wake width, the edge of the
wake is arbitrarily taken as the point at which total-head loss or w
is one-tenth of the maximum value. It is seen from figure 8 that the
wake widths obtained frorm the total-head survey and the v measur-
ments are in close agreement for the angle-of-attack rage investigated.
It should be mentioned that wake widths based on profiles of the RMS

turbulent velocity, y would necessarily be larger thn the present
wake widths, which are based on the mean-square velocity.

A comparison of the total-head profiles for angles of attack from
0 to 14 degrees is presented in figure 9. A similar comparison for
the v measurements is given in figure 10. For both total head an
v the profile shapes exhibit approximate similarity for the angle-
of-attack range investigated.

Variation of the wing drag coefficient with angle of attack is
shown in figure 11. This result was obtained by integration of the
total-head profiles measured one chord length behind the wing trailing
edge. The abrupt rise in drag in the vicinity of 3 degrees angle of
attack apparently results frapn separation of the boundary layr at the
leading edge of the wing. It was observed during a limited tuft study
that flow was extremely irregular over the entire upper surface of the
wing at 6 degrees incidence. Flow over the wing was smooth at 2
degrees except for a small region of moderately irregular flow at the
leading edge.

Variation of the relative intensity of the downwash fluctuations
with angle of attack is presented in figure 12. These values were
obtained by integrating the w profiles measured one chord length
behind the wing trailing edge. The intensity of the downwash flue-
tuations increased sharply at the higher angles of attack, probably
due to the occurrence of leading-edge separation, as previously dis-
cussed.

Turbulent downwash intensity is shown as a function of drag
coefficient in figure 13. Integrated v intensity increased appraxi-
mately with the 3/2 power of the drag coefficient in these tests.
When the local intensity at a given point in the wake, such as the
wake center, is plotted against drag, the variation is nearly linear.
A large part of the increase in integrated intensity is due to the
broadening of the wake at the higher angles of attack.





NACA TM 1627


Power Spectra of the Downwash Fluctuations


Effect of angle of attack and wake position. The results of
measurements of the power spectra of the downwash fluctuations one chord
length behind the wing trailing edge are presented in figures 16 to 20
for angles of attack fran 0 to 16r degrees. In general, the power spectrum
was measured directly behind the wing trailing edge and also at the outer
edge of the wing wake at each angle of attack. The relative height of
the turbulent trace for the two positions in the wake may be seen frapn
the wake profile of RMS downwash at the top of each figure. The vertical
ties on the wake profiles show the position at which power spectra were
measured.

A constant half-band width of 30 cycles per second, based on L0
decibels attenuation, was used for all spectrum measurements. The filter-
ing characteristics of the wave analyzer are shown in figure 1.

From figures 16C to 20 it is seen that wing angle of attack has an
important effect on the power spectrum of the downwash velocity. The
spectra at zero incidence (fig. 16) is dominated by the discrete fre-
quency corresponding to a vortex street shed from the blunt trailing
edge of the wing. At 3-1/2 degrees angle of attack, the trailing edge
vortex street is still apparent, but the bulk of the turbulent energy is
spread over a broad band of frequencies.

With further increase in angle of attack, the downwash spectrum
ne~ar th wake center shows a progressive decrease in the predconinant fre-
quencies. The bulk of the turbulent energy is concentrated near the
wake center, and so a general conclusion frau this data is that wake fre-
quency decreases as wing incidence is increased.

By 16C degrees angle of attack (fig. 20), a vortex street is formed
behind the stalled wing, and the measured spectrum corresponds essen-
tially- to a delta function which is spread out by the finite band width
of the wave analyzer (ig. 1). At first thought, it seems unusual that
the frequency measured near the center of the vortex street is not double
the frequency at the wake edges because of the influence of both rows of
the vortex street. Therefore, a calculation of the velocities induced
by a vortex street was made using the equations of reference 16. Thel
results are presented in figure 21, along with the gecanetry of the as-
sumed vortex street. The upper graph in figure 21 gives the variation
of the longitudinal induced velocity, u, with distance along the vortex
street, and the lower graph presents the variation of induced downwash,
-w. The induced velocities were calculated along the center of the vor-
tax street (a = 0) and also along a line located above the vortex street
(at = b). A9s expected, the predominant frequency of the u velocity in
the center of the wake is double the frequency outside the vor-tex street.
However, the downwash velocity in $he wake center does not show fre-
quency doubling; this result is in agreement with the measurements of








the downwash power spectrum presented in figure 20.

Below th stalling incidence, the dona~sh spectru is much
broader near the center of the wake than at the outer edge of the
wake. Moreover, the predominant frequencies at the~ edge of the wake
ar practically indpendent of wing incide~nce for angles of attack
from 6 to 16 degrees. Because there is very little turbulent energy
in the outer edges of the wake, subsequent analysis of the data will
be, for the most part, based on the spectrum measurments directly
behid the wing trailing edge (2. = 0).

Relation between frequency and wake width. The data presented
in figures l to 20 show that the predominant frequencies in the wig
wake generally decrease as the wake width increases. In order to
illustrate this result more explicitly, it is convenint to define
an average frequency- for the turbulent power spectrum. The average
wake frequency, nay, is defined as the frequency which divides a
power spectra into two equl areas. In figure 22 the reciprocal of
the average wake frequency is ploted against an effective wake width,
defined as the wake width divided by average wake velocity. The ex-
periments show that th average wak frequency is inversely propor-
tional to the effective wiake width. In other words, a non-dimensional
frequency or Strouhal number based on wake width and wake velocity is
nearly constant. This result is in agreement with a recent correla-
tion based on free-streamline theory for a series of bluff bodies
(ref. 15).

A line labeled instability theory" in figure 22 was calculated
using the instability theory of wakes presented in reference 16. The
possibility of applying stability theory to a turbulent velocity
profile has never been fully discussed.2 Instability of laninar shear
flows generally implies the eventual formation of a turbulent flow.
When the flow is already turbulent, the meaning of an instability
calculation is far from clear. However, even in laminar flow only the
very beginning of instability can be handled by a linear theory; it
is not impossible that the application of similar considerations to
turbulent flowI could provide interesting results. For example, it is
known that Kgrm~n vortex streets and three-dimensional Taylor vortices
may exist even in turbulent flow. Therefore, it is interesting to
compute the frequencies predicted for neutral stability and to can-
pare these frequencies with the measured results, as in figure 22.
Instability theory predicts the same inverse relationship between
frequency and effective wake width shown by the experimental measure-
ments. Although the actual magnitude of the predicted frequency is


A few remarks concerning the existence of a large-scale structure
in turbulent shear flows have been made in reference 13.


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NACA TM 1427


in close agreement with th measured results, this is very likely coin-
cidental. First of all, the theory assumes neutrally stable disturb-
ances whereas amplified disturbanes would probably be more important
in any real flow. Also, different average frequencies could be ob-
taind fromn the measured spectra depending on the method chosen to
defin an average. Details of the stability calculation are presented
in Appendix B.

Relation between frequency and projected wing chord. It has been
observed in several investigations that the Strouhal number based on
the projected chord of a wing is nearly constant for angles of attack
above the stall (ref. 1). In order to determine whether such a constant
Strouhal numbr lis obtained for smaller angles of attack, the average
wake frequencies, nav, obtained from measured power spectra are shown
in figure 23 as a function of projected airfoil breadth. This projected
breadth is defined as the total airfoil projection, which includes the
finite thickness of the wing. Th line labeled "empirical" corresponds
to a constant projected Strouhal nube obtained by extrapolating the
.frequency measured for the stalled wing ( r = Lto)

The empirical extrapolation gives a reasonable prediction of the
average wake frequency except for an angle of attack of 3-1/2 degrees.
At this angle of attack the airfoil does not have the leading-edge
separation that occurred at higher angles, and so it is not surprising
that wake frequency is not related to projected chord. When frequency
is related to wake width instead of projected chord, the unseparated
flow condition does not present this difficulty. Therefore, it appears
that wake width is more suitable than projected chord for correlating
frequency measurements.


Power Spectra of the Longitudinal and Spanwise Velocities


Although downwash fluctuations are generally more effective as a
cause of buffeting loads, the power spectra of the longitudinal and
spanwise velocities are necessary for more complete understanding of
turbulent wakes. Longitudinal velocity spectra were measured for an
angle of attack of 6 degrees at several vertical positions in the wake,
and the results are shown in figures 21 and 25. Th corresponding
downwash spectra are presented in figures 26 and 27.
2 2
Looking act the data as a whole, the u and W' power spectra gen-
erally have pronounced maxima at values of no = 0.65 or 1.30. The
2 2 c
u and w spectra are similar to each other over the outer portion of
the wake, but the spectra are markedly different near the wake center.





MACA TM 1827


In order to determine wheher the win wae resembles a vorrtexc
street at this angle of attack, the velocites induced byr a v~ortex
street (fig. 21) were exressed in terms of their Fourier coefficients.
The resulting harmonic analysis of an idealized vortex street is
presented in figure 28 for several vertical locations above the center
of the vortex street. Th fist an second ~Fourier coefficients for
the a velocity are on the left of the~ figure, and th corresponding
downwash coefficients are on the right. Comparing th calculated
results with ~the measurements given in figues 2L to 27, it is seen
that the measured data do not show consistent agreement with the cal-
culations. Therefore, it seems likely that the struture of the wing
wake at low angles of attack is more complicated than that of a simple
vrore street.

n figure 29 the u2 spectra directly behind the wing trailing
edge is compared wit the spectrum of the spanrwise velocity component.
These spectra may be compared with the spectra for isotropic turbu-
lence. Tfhe longitudinal or a2 spectrum of isrotropic turbulence has
been obtained experimentally in the investigations of references 6
and 17. From such results th transverse spectra may be calculated
using the continuity relation for isotropic turbulence (ref. 18).
Measurements of the longitudinal spectrum may be approxiated by the
relation


LL 1
Fl(n) = (2)
oaD (1 +(2)


and the corresponding transverse spectrum is


2L (1 + 32
F2 (n) = p- (+2 2 23


The variable 5 is a non-dimensional frequency given by

oL



and L is the turbulent macroseale of the longitudinal velocity

Uo Fl(0)
L-- = a>





NACA TM 1427


The isotropic power spectra shown in figure 30 were calculated
for the same value of L as that measured in the wing wake, figure 29.
Comparison of figures 29 and 30 indicates that the u2 and v2 spectra
near the center of the wing wake are similar, respectively, to the
longitudinal and transverse spectra of isotropic turbulence. This
similarity was not shown at other vertical positions in the wake, al-
though part of the total turbulent energy may still be attributable to
an isotropic-like turbulence.


Effect of :Free-Stream Velocity


The results presented so far were obtained at a free-strean
velocity of about 50 feet per second. In order to determine whether
the results may be used for other free-stream velocities, representa-
tive tests were made at a lower velocity. These additional tests do
not provide significant information on the effect of Reynolds number
variation, but should facilitate interpretation of the results. For
example, it is generally assaued that aerodynamic frequencies are
proportional to velocity, and so this assumption should be checked.

Wake-energy distribution. The fluctuating-dow wash profile is
shown in figure 31 for velocities of 30 and IC9 feet per second. Although
the profile is somewhat steeper for the lower velocity, the wake width
is practically unaffected by the velocity change. The overall effect of
the velocity change on the profile is negligible.

Discrete frequencies. The effect of velocity on the periodic
wake behiind the blunt trailing edge of the flat-plate airfoil is shown
in figure 32 for zero angle of attack. At the higher velocities, fre-
nt
quency is proportional to velocity. The Strouhal number, p-, of the
Uo0
fluctuations, based on trailing-edge thickness, is 0.25, which is in
good agreement with the Strouhal number of 0.23 obtained in reference 19
.from schlieren observations of the vortex street behind a blunt trailing-
edge airfoil. For ve~lociti~es below about 20 feet per second (fig. 32),
the frequencies in the wing vake become nonlinear with velocity. This
corresponds to a decrease in Strouhal number for Reynolds numbers
below 10 with. Reynolds number based on trailing-edge thickness. The
Strouhal number of.a cylinder also shows a decrease for Reynolds num-
bers below 10 (See fig. 169, ref. 20.)

Velocity fluctuations in the wake were also periodic above the
stalling angle of the wing. The effect of free-stream velocity on wake
frequency is shown in figure 33 for two angles of attack. Over the






MACAI TM 1627


Reynolds number range of these tests, frequency is proportional to
velocity. The Strouhal number based on projected chord is 0.19 for
a; = 16.$o and 0.18 for a. = 21.7o. In the discussion of the accuracy
of measureents, it was noted that these frequencies are probably
slightly high du to wall interference. Nevertheless, th values are
well within the range of Strouhal numbers obtained in various inves-
tigations of vortex streets behind stalled plates and airfoils (ref. 1).

Power spectra. The power spectrum of downwash fluctuations
behind te wing at a moderate angle of attack is shown in figur 3L
for two vaue of the free-stream velocity. Roughly speaking, th
norma~lized power-spectral density is uniquely dependent on the usual
frequency parameter, n-, for the two velocities. Some of the dis-
Uoo
agreement between th two spectra very likely results fran. the replace-
ment of hot-wires during th intervtal between the two tests. In any
case, the overall agreement of the spectra appears quite satisfactory.

DISCUSSION


Prediction of RIIS Lift in Tail Buffeting


Formulas for the prediction of the RM lift fluctuations on a
tail surface located in a turbulent wake have been given in reference
21. However, in order to obtate explicit results, it was necessary
to assume tha the turbulence striking the tail surface was isotropic.
The effect of this assanpton ma be evaluated by using an actual
power spectrum measured in these tests to calculate RMS lift on a
hypothetical tail surface.

A convenient relation for calculating mean-squae lift was de-
rived in reference 21, and, except for a slight difference in notatian,
the relation is:



GL2 =P S ra (n) l(k) 2 dn (6)


The assumptions on which this equation is based are: (1) the tail
aerodynamics may be described by a two-dimensional admittance fune-
tion, /(k); (2) the scale of turbulence is large compared with the
tail span.

It is evident frau equation 6 that certain parameters, such as












































1 _CL
SPECTRUM --
2n


Wing 0.37
Isotropic 0.41


NACA TM 1627


velocity an king chlord, anat be assumed in order to calculate the
mean-square lift coefficient. Howevr, the minimum number of assump-
tions are mae if equation 6 is modified so as to incorporate a useful
similarity property of the power spectrum; namlyr, that


~= fn--
cc


as seen from figure 3t. Then equation 6 becomes


2 2


The object of the lift calculation may now be described more
explicitly. Using the same tail surface and aerodynamic admittance

function, CL2/ a 2 is to be calculated for a power spectrum measured
behind the wing (fig. 17, a = 0) and compared with a similar result
calculated for the transverse isotropic spectrum (fig. 30). The
aerodynamic admittance of a tail having ot/c = 0.2 was calculated using
equation 13 of reference 3, and the result is shown in figure 35. For
this tail surface, the comparison of RM lift for an actual wing down-
wash. spectrum with, the RN~-S lift for isotropic turbulence is as follows:


aOt no nc
' ~d -
oC U U


no >






NTACA TM 1t27


From this comparison, itt is evident that t~he exct shape of the
turbulent power spectrum has a relati~vely- smal effect on the mean
lift fluctuations experienced byr a tail surface. A simple closed-form

expression is available for calculating CL2/a 2 in isotropic tur-
bulence (equation 28 of reference 21), and the above comparison shows
that this equation should be quite useful, for predicting the RMIS lift
on a tail in the wak of a wing. However, in order to apply the
formula, it is necessary to estimate the macroscale of the turbulence
striking the tail surface. For the comparison just presented, the
turbulent scale was obtained from a measured u2 spectrum using equation
i;. Such results are not usually. available to designers, and so i~t is
necessary to have a simpler means of estimating the scale of turbulence
in a wing wake. A possible method for obtaining: such an estimate will
no be discussed.

The turbulent scale, L, may be used to define a characteristic
frequency by simple dimensional anakysis



2nL

It might be expected that this frequency would correspond to the pre-
dominant frequency in the turbulence, and this assumption ma be
checked using the data of the present tests. The characteristic fre-
quency calculated from the u" spectrum of figure 29 and equation 9
was found to be nobooa = 1.2k. This frequency corresponds almost
exactly to the predominant frequency in the v power spectrum, figure
17. Therefore, equation 9 provides~ a simple means of estimating tur-
bulent scale because wak frequencies can be obtained more easily
than a power spectrum. For examle, a tuft probe coud be placed in
the wing vake and th~e mean frequency coud then be obtained from
motion pictures of the tuft movement.


Buffeting Measurements with Detector Airfoi~ls


From the results of some earlier investigations of tail buf-
feting (ref. 22, 23), it was concluded that lift fluctuations may be
obtained well outside the limits of the total-head wake. This result
appears to be at variance with the present investigation and that of
reference 2L. Therefore, it seems worthwhile to discuss the reasons
for the different conclusions.

The fact that relatively strong lift fluctuations have been





NACA TM 1627


detected beyond the region of the total-head wake is not surprising
for several reasons. First, the intensity of the fluctuations observed
in references 22 an 23 was taken, roughly speaking, as the maximum
amplitude of the detector airfoils. Therefore, the data obtained in
this manner represent an upper limit of the lift fluctuations and do
not necessarily represent a statistical average of the fluctuations.
In the present investigation, it was generally observed that near the
edge of the total-head wake, occasional bursts of turbulence were found
to be notably larger than the general level of the turbulent velocity
fluctuations. Whn such bursts occur in the outer edges of the wake,
a wake width based on maximum values would be larger than the width
obtained from statistically averaged data.

Even in the absence of such bursts, the profiles in references
22 and 23 vould be broader than those of the present report as only
the latter are expessed in terms of mean-square fluctuations. The
profiles of references 22 and 23 correspond more nearly to RMS fluctua-
tion, and RMIS values must necessarily fall off more rapidly than mean
squares.

Finally i~t is noted that the predominant velocity fluctuations
near the center of the wake occur at higher frequencies than those
observed at the wak edges (figs. 26 andi 27). Because low frequency
ve~locity fluctuations are more effectively converted to lift fluctua-
tions (ref. 3), the lower frequencies foun in the outer portions of
a wake are less attenuated than the high frequency energy in the wake
center. This effect supplies another contributing cause for the breadth
of the fluctuating lift profiles presented in references 22 and 23.

In order to illustrate the effects just discussed, representative
data obtained in the present investigation have been shown in several
different forms in figures 36 to 38. Using the measured power spectra
for the downmwash fluctuations, figure 36, the frequency distribution
of RMS lift has been estimated for an airfoil located at various
heights in the wing vake. The simulated horizontal tail had a chord
equal to 20 percent of the wing chord. The form of the approximate
admittance function used to calculate lift spectra is shown in figure
35. On the basis of results from reference L, it is expected that
such an admittance function is representative of the admittance of
the sensitiv airfoils used by Ferri (ref. 23). The resulting fre-
quency distributon of RMS lift estimated in this manner is presented
in figure 38. A comparison of the relative intensities obtained from
the RMS lift spectra (fig. 38) an the conventional angularity spectra
(fig. 36) is as follows:













0 1.000 1 .000

.22 .305 .551

.68 .006 .051


NACA TM 1627


F'rom! this result, it is seen that tests using sensi-tive airfoils would
be expected to indicate greater wake widths than those obtained from
hot-wie surveys. Th occurence of bursts, as previously discussed,
could cause even greater differences between lift and angularity data.


CONCLUSIOI\S


1. The mean square of the downash fluctuations was found to
have approximately the same variation with vertical~position in the
wing wake as that shown by measurements of total-head loss.

2. The intensity of downrwash fluctuations increased rapidly when
the flow separated from the wing surface. For the present tests
integrated downrwash intensity increased with approximately the 3/
pover of wing drag coefficient.

3. Power spectra for the downwash velocity were generally broader
at the low angles of attack than at high angles. The average fre-
quency in the wae was proportional to the ratio of vake velocity to
wake ,ridth. Vake wridth appears to be a more suitable parameter for
correlating spectrum measurements than the projected airfoil chord.

L. The outer edges of the wing wadce contained relatively little
high-frequency energy, whereas the turulent energy near the wake
center was distributed over a fairly wide range of frequencies.

5. The vake structure at low angles of attack appears to be
much more complicated than for the stalled airfoil, for which a vor-
tex street forms behind the wing.

6. Power spectra measured at different velocities were found
to be similar when frequency and power spectral density were non-
dimensionalized by the ratio of airfoil chord to free-stream velocity,





NACA TM 1627


7. On the basis of calculated results, th exact shape of the
'turbulent power spectrum was found to hav a relatively small effect
on the RM lift of a tail surface located in the wake of the wing.
Formulas for predicting lift assuming isotropic turbulence should give
satisfactory results if the turbulent scale is correctly estimated.

APPENDIX A


Symbols

The following symbols are defined as they are used in the main
body of this paper. Certain letters havre been used with different
meaning within Appendix B in order to retain the notation of reference
16.

a distance between vortices in. the same row

b breadth of vortex street


CD drag coefficient, drab spa

OL lift cosefiient, liftspa

c airfoil chord

F(n) power spectral density of a random variable;


Fn) = R(c)eos 2nns &T

F1(n) power spectral density for longitudinal velocity in isotro-
pic turbulence

F2(n) power spectral density for transverse velocity in isotro-
pic turbulence

k reduced frequency, mer/2Ua

L turbulent macroscale, equation 5


frequency, cycles/second





NACA TN 1827


n the first harmonic frequency of a vortex street

n averae wake frequency*, defined as the frequency which
divides a power spectrum into two equal areas

1 2~m
q dynamic pressure, 2 o

R(z) time-correlation function of a random variable;


R(c) = u(t) u(t + z)/ u2

R(T)= co~n)cos 2nnz dn


TE trailing edge

t wing thickness

U0 free-stream velocity

n longitudinal perturbation velocity

v spanwise perturbation velocity

w vertical perturbation velocity

WT wake wridth

x distance along stream direction from wing trailing edge

z vertical distance above centerline of wing trailing edge

a angle of attack

/ vortex strength
nH difference between free-stream and local total head

height above wake center

5 non-dimensional frequency, "L/Uc


free stream density





NACA TM 16~27


.Z time

/(k) ratio of lift at frequency k to lift at zero frequency
ao circular frequency, 2nn.

APPENDIX B


Application of Stability Theory to the Prediction of
Vake Frequencies


In this appendix assumptions underlying the stability theory of
waks (ref. 16) are briefly outlined. Also certain additional formulas
are developed in order to facilitate comparison between theory and
experiment.
Hollingdale (ref. 16) considers a mean wake flow whose velocity
relative to the stream at infinity is




v (Bl)

With the assumption of small two-dimensional disturbances a linear,
homogeneous equaton for the disturbance velocities is obtained from
the incompressible Navier-Stokes equations. The stream function
may be assumed periodic, so that

(eiar(x-ct) (2


Th wave length of the disturbance is 2n/ar, and c is the wave velocity.
Thie basic equation. for determining the stability of small disturbances
in a vak or other region of shear flow is then


(U o)( .2 6) Un~ ( "v 2 a2B n + # ) (B3)

From the complete stability equation B3 Hollingdale has deter-
mined that the wave velocity c for an undamped disturbance is equal
to the value of U at the point of inflection of the velocity profile
U(s). This result is valid for-symmetrical or monotonic profiles in
which U approaches a constant for large s.






NACA TM 16L27


For sufficiently large Reynolds numbers the viscous terms of
equation B3 may be omritted:


(u c)( *" ak~g) und o


(sh~)


The velocity profile may be assumed to be of the form


U = A(1 + cos ks)


04 151 ME

Is I > n/k


(Bs;)


U = 0


where n/k is the wake semiwidth, subsequently denoted as W/2. The
constant A must equal the wave velocity in order that U = c at the
point of if~lection Un .I 0. For the particular form of profile
chosen the stream function must therefore satisfy the relations


*I + (k2 a2)8 = 0



Th bounda 7 conditions to be s
function in the region -},
large z. Soluions satisfying
equations B6 are


=i P cos 1 k2 a2

= Ge-aja


Is I > n/k


n/k


(B6)


satisfiedd by i are that it is an even

TZ and that it approaches sero for
these conditions and the differential


05Isl I Q

Isl> n/k


n/k


(B7)


These equations provide the characteristic equation for determrining
the wave-length parameter a if and gI are required to be continuous
at Isi = n/k:


2 2


(B8)





NACA TM 1k27


The parameter

a a v_,
k 2n/ar


is simply the ratio of the wake width to the wave length of the disturb-
ance. The imortant result from equation 88 is that this ratio is a
constant and, in fact, has the numerical value

a=0.926 (B10)


Disturbance waves of the type predicted by Hollingdale reach a
point which is fixed relative to the body producing the wake with the
frequency


n o: a (To A) (811)


where Too is the free-stream velocity and A is the wave velocity. In-
troducing Hollingdale's result for the proportionality between wave
length and wake width, equation B9, provides the expression for the
frequency of small periodic disturbances:



n = oo----- (812)


Although the vIalue of the wave velocity A may be determined as
the velocity at the point of inflection of the velocity profile, it is
difficult in practice to determine the inflection point of an experi-
mental velocity profile. Hence, it is desirable to have a more practi-
cal method of estimating wave velocity. The drag may be obtained by
integration of the total-head loss if the survey plane is a sufficient
distance from the wing trailing edge:



CD = ds ( 813)






NACA TK 1627


The relation between total-head loss and the velocity deficit U in the
wak may- be obtained from Bernoulli's equation by assuming the static
pressure in the wake is equal to the free-strean pressurre.


hR U U



Whn, equation. BS for the velocity variation, is substituted into
equations B13 and Bltk, a relation for the wave vrelocity in terms of
drag and wak width is obtained:



a2 B 1 1' -~ (s15)




REMECES


1. Fung, T. C.: An Introduction to the Theory of Ae~roelasticit~y.
John WJiley and Sons, Inc., 1955.

2. Lindsey, rW. F,~ Daley, B. M., and Humrphreys, M. D.: The Flow
and Force Characteristics of Supersonic Airfoil at High
Subsonic Speeds. NACA TN 1211, 1947.

3. Liepmann, H. W.: On the Application of Statistical Concepts
to the Buffeting Problem. Jour. Aero. Sci., vol. 19, no. 12,
pp. 793-801, Dec. 1952.

b.Lamson, P.: Measurements of Lift Fluctuations Due to Turbu-
lence. CIT, Ph.D. Thesis (to be published by the NACA), 1956.

5. Willis, J. B.: Review of Hot Wire Anemometry. Australian
A. C. A. 19, 1965.

6. Liepmann, H. W., Laufer, J., and Liepmann, K.: On the Spectrum
of Isotropic Turbulence. NACA TN 2k73, 1951.

7. Roshko, A.: On the Development of Turbulent Wakes tran Vorte
Streets. NACA TN 2913, 1953.





NACA TM 1627


8. Fage, AP., and Jones, L., J.: On the Drag of an Aerofoil for two-
D~imensionral Flow. Proc. Roy. Soc. of London, ser. A., vol. 111,
pp. 592-603, June 2, 1-926.

9. Zebb, Keir: Technique of Mleasuring Transverse Components of
Velocity Fluctuations in Tubulent Flow. CIT, A. E. Thesis, 1963.

10. Glauert, H.: The Characteristics of a Karman Vortex Street in a
Chnnl. of Finite Breadth. Proc. Roy. Soc. of London, ser. A,
vol. 120, pp. 3L-46, Aug. 1, 1928.

11. Fage, A., and Johansen, F. C.: On the Flow of Air Behind an Inclined
Flat Plate of Infinite Span. B~ritish ARC R and M 1106, 1927.

12. Dryden, H. L., Schubauer, G. B.,, Mock, W. C., Jr., and Skramstad,
H. KC.: Measurements of Intensity and Scale of Vind-Tunnel
Turbulence and Their Relation to the Critical Reynolds numberr of
Spheres. NACA Rep. 581, 1937.

13. Liepmann, H. W.: Aspects of the3 Tubulence Problem. Jour. Appl.
Math. Phy. (ZAMP), vol. 3, fasc. 5 and 6, pp. 321-362; 607-lr26,
1952.

14. Lad, Sir Horace: Hydodynamics. Dover Publications, p. 226, 1965.

15. Roshko, A.: On the Tiake and Drat of Bluff Bodies. Jour. Aero.
Sci., vol. 22, no. 2, pp. 124-132, Feb. 1955.

16. Hollingdale, S. H.: Stability and Configuration of the Wakes
Produced by Solid Bodies Moving through Fluids. Phil. Mag.,
7th series, vol. 29, pt. 1, pp. 209-257, Ma~r. 1980.

17. Dryden, H. L.: Turbulence Investigatioin at the National Bureau
of Standards. Proc. Fifth Int. Cong. Appl. Mech. (Sept. 1938,
Cambridge, Mass.), John Wiley and Sons, Inc., pp. 362-368, 1939.

18. Batchelor, G. K.: The Theory of Homogeneous Turbulence. Cambridge
Univ. Preas, pp. LS-L7, 1953.

19. Sumes, J. L., and Page, W. A.: Lift and Moment Characteristics
at Subsonic Mach H~umbers of Four 10-Percent-Thick Airfoil
Sections of Varying Trailing-Edge Thickness. NACA R~M A50JO9,
1950.

20. Goldstein, S., ed.: Modern Developments in Fluid dynamics.
Oxford Univ. Press, vol. 2, p. le19, 1938.






NACA TMjI 1827


21. Liepmann, H. W.: Extension of the Statistical Approach to Buf-
feting and Gust Response of Wings of Finite Span. Jour. Aero.
Sci., vol. 22, no. 3, pp.'197-200, j\arch 1955.

22. Duncan, W. J., Ellis, D). L., and Scruton, C.: Two Reports on
Tail Buffeting (1st Report). British ARC R and M 1657, 1932.

23. Ferri, Antonio: Preliminary Investigation of D~ownwash Fluctu-
ations of a High-Aspect-:Rtio Wing in the Langley l-]Foot
High-Speed Tunel [JACA RM L6H28b, 19L6.

2L. Sorenrson, R. M., TWyss, J. A, and Kyle, J. C.: Prel iminary
Investigation of the Pressure Fluctuations in the Wakes of
Two-Dimensional Fings at Low Angles of Attack. NACA RMJ
AS1G10, 1951.






NACA TM 1427


S0.8
O



0.7




O



0.5-




0.4




0.3














-1 0 -20 -1O O IO 20 30
CYCLES OFF RESONANCE, An, cps


Figure 1.- Filtering characteristics of Hewlett-Packard model 300A wave
analyzer for 30 ops half-band vidth.







NACA TM 1427










pt


P-9
LO

SII
Utl







co a
LL g II

O 90
aO co


B~s to it %
cc



eg 2 N
c c 5
** i
LD -L roa
oi~~ as




-0 u


8o o a
O o a o



*rl







28 NACA TM 11e27






d "d
Oh





o -8 8

a *Ir-1
o ad


O ,0




a a, i

o so




aa B g


a a






4 I
O Od
o






puo HVd








NACA TM 1627


di

'0







C
*r-4
c,




a,



10 (




ka
OO
he
Cin0
1 .s


O

2
(11


a


I IS
O
O o*


MlV


'U o m a


puo
m'







NACA TM 16 27


a
a. QI


r O



o
to c


co,


lo








oa


O
Y
O


St-

i-
I


a


puD
HV






NAC TM 1427 51











e hO


n-- a,



oo






,, >






OEE O


( J ( PHf a
puor
>at Y ,







32 NACA TM 1427




f@


a. g



08 B 8

Ut =
I






It- Mt e
a~G t o






10 *
dE
(n -

O







d --1F



4~~ ~ F 0

a~ HV







NACA TM 1427


II

*4








I>


O





to

O



Z






- 1




-0


t>


M























































III IIII1


NACA TM 1427i


O




o



O<






SI


a o.o oo
Oo no~--
S* O A & I


o B


04,
A n


oB

00












a
0
O do


o a
od

o 41

o, Q4


d
Qag
d0


d's
oe


O O
d <


a 0
d' 9
d

Ifli
*


t
d


aD re e a u
O dod(i~~


MV


row
d d


d o






NACA TM 142'{


ofo


One wao


ad



O


Od


0O


O o

r o


a 0
4 0




cos
84 4

0 Q
O~ d





O d



O a
B


d












00
1

Y


k
O
d
O
cn
k
rd
a
E
o
u
I
o



hi,
a


d a 6 P


EM


t rrc [
6 d 6


























































1


NACA TM 1427


Cl


b.
O







O


03'1#3131~3303 BtlL~a






NACA TM 11+27






co R

II


















o -

I (1
J 1>
ao

ao


-~ (1
L*H
o rlO

O~
b*H
OO &






~B NACA TM 1427






O



ro
~II



d II


d ar

o a




-z M



O ,--


0 m


d c~



-8~ o


0 a

oI M

4\ o

.oo
O .


Oc

O
4N




*i,-






















































0 1 L _
0 4 8 I 2 16 20
FREQUENCY, nc/U,



Figure 14.- RMS downwash intensity and v2 power spectrum measured
I chord length behind trailing edge; a, 0 ; U, = '52 fps.


N~ACA TM 1 27


t
i-
u,
r
LJ
c-
t:

ui
i
d


-02 -aI O 0.1
HEIGHT ABOVE TRAILING EDGE, zke


SO r-


Z/C
O


20E C


10






40 NACA TM 1427

















-0.2 -0.1 0 0.1 0.2
HEIGHT ABOVE TRAILING EDGE, z /c










-0.15













U 4 8 12 16 20
FREQUENCY, nc/U,


Figure 15.- RM downwash intensity an po Fwer spectrum measured
1 chord leng~th. behind trailing: edge; a, = 3030', Um = 52 fps.








NACA TM 1_427


r
c-
u,
z
w
c
z

ui
1E'
d


-0.2 O 0.2
HEIGHT ABOVE TRAILING EDGE, z /c


z Ic


n
0
re:
e
ri:

F4
t
u,
;r
8


a
o: 2
c
o
w
v,


2 4 6 8

FREQUENCY, nc/ U,


Figure 16. RMS~ downwash intensity and w2 power spectrum measured
1 chord length behind trailing edge; a, = 5015', 000 = 52j fps.





NACA TM 1427


F;
>-
m
w
z
ul;
Z"
Ir


-a4 O 0.4
HEIGHT ABOVE TRAILING EDGE, z/e


/ z/-0.


-O I2 5 4 5
FREQUENCY, nc/U,


Figure 1_7.- RMS downwash intensity and w2power spectrum measured
1_ chord length behind trailing edge; a, = 6020', U, = Sk fps.























-04 O 04
HEIGHT ABOVE TRAILING EDGE, z/c


NACA TM 142'{


[r;
1-

le
z
ui

d
O


16





12





-8





a4


0


-a4


3 4 5


FREQUENCY, ac/U,


v2 ]power spectrum measured
0, = Bo))', Umo = t$2 fps.


Figure 18.- RMS downwash intensity and
1 chord length behind trailing edge;






NACA TM 1427


r-
v,
z
w
r-
z

vj
1
d

O


-0.4 O 0.4
HEIGHT ABOVE TRAILING EDGE, z/c


16




no

X 12













a4







O,


z/c


- 0.4


FREQUENCY, nc/U,


Figure 19.- RI19 downwash :intensity and v2 power spectrum measured
1 chord length behind trailing edge; co = 10015', Umo = 5T fps for
a; = 0; a. = 10035', Umo = 48 fps For z/c = -0.4.


























' *


NA~CA TTM 1427


r
t:

w
c
z

vj

oi
O


- 0.4 O
HEIGHT ABOVE TRAILING


0.4
EDGE, z/c


04 0.8
FRE QUEN CY,


ne /U,


Figure 20O.- RMS downwash intensity and w2power spectrum measured
1 chord length behind trailing edge; a, = 140, Um = 48 fps for a = 0,
Um = 52 fps for z/c = -0.4.



















ra


NACA TM 1427


b= 0.281 a


z= b



z=0

- -----


2











O
-4


/b


O



o
o


TT
2nK x a


Figure 21.- The velocities .induced by a vortex street.





NACJA mTM 1427



14




1. 2














..J 04 -

O


0. -s


EXPERIMENT


INSTABILITY THEORY


/o'


0.2
EFFECTIVE


0.4
WAKE


0.6
WIDTH,


OS


Figure 22.- Dependence of average wake frequency on effective wae widthn.


















































2 O
+ cos


Figure 25.- Dependence of average wake frequency on projected airfoil
breadth.


1


N~ACA TMI 1427


1.4 r


1.2 C-


o







O


EMPIRICAL


I. OF


0.8 F


eEXPERIMENTAL


0.6 b


0.4 F


02! t


O'
O


0.5 1.0 1.5
PROJECTED AIRFOIL BREADTH, sin a























































OO


HIACA TM 14c27


ZERO FOR
2 /C =-0.4





0.3


16











0 1


O


FREQUENCY, nc/U,


Figure 24.- u2


power spectnrm; a = 6020',


U, = 54 fps, x = le.

























































I I2 3 4 5
FREQUENCY, nc/U,


NACA TM 1627


z/c

0.388


ZERO FOR

z/c = 0.388


0.22


o 12





c--
imj
z




a


0


0.12


Figure 25.- u2 power spectruza; 0, = 6020', Um = 54 fps, x = le.







NAC'A TM 1427


z/c





\- .4


ZERO FOR

z/C = -0.4


nl Ir
O







a


u4


-O. 2


III


I 2
FREQUENCY, nc/U,,


Figure 20j.- w2 power spectrum; a. = 020', Uoo = 54 fps, x = le.


























ZERO FOR

z/c = 048


NACA TM 1427


0.48


n 12




O

...

>-4


O


O 22


0.22


I 2 3 4
FREQUENCY, nc/U,


Um = Sk rps, x = le.


Figure 2'i.- w2


power spectrum a, = 6020',






NACA TM 1627


u VELOCITY


w VELOCITY


2 j

O

11

0
2







O O




2 -



OO 2
0


O2 -



I -

c O






3 -


2




OO
0


z=b


z=b


i 2


3b/2


3b/2









b/2








2_


1


b/2


3-

2-


I-

a
O


- ,
oL
0 I2


FREQUENCY RATIO, n/n,


Figure 28.- Calculated frequency distribution of energy for several ver-
tical positions in a vortex street.


1~2






NACA TM 1427



o






I a,





-0*

/a a
lld 2I- o
=1 a
1


a E II



zo a,

kO
c- 4I ~di
Z d O II

r ~do



rda

. 'o~


flI I 0 0

ata





N~ACA TM 1427 5







oO








go

II 1 4-
rr-





I Oa

to

I >- so a
I ; o oo X



9W /



O



I m






NACA TM 1427











II



cd


0.




O
1-- -


o





I
N





I


J


V)0 v


re -














































IO 20 30 40 50 61
FREE STREAM VELOCITY, U,, fps.


NVACA TMI 1427


1200 r


1000 1-


800 i-


600 C


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