initial circulation and peak vorticity behavior of vortices shed
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NASA / CR--2001-211144
Initial Circulation and Peak Vorticity Behaviorof Vortices Shed From Airfoil Vortex Generators
Bruce J. Wendt
Modem Technologies Corporation, Middleburg Heights, Ohio
August 2001
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NASA / C Rw2001-211144
Initial Circulation and Peak Vorticity Behaviorof Vortices Shed From Airfoil Vortex Generators
Bruce J. Wendt
Modem Technologies Corporation, Middleburg Heights, Ohio
Prepared under Contract NAS3-99138
National Aeronautics and
Space Administration
Glenn Research Center
August 2001
Acknowledgments
The author would like to acknowledge the mechanical and operational support provided by Winston Johnson,
Sam Dutt, and Bruce Wright (NASA Glenn), and Mar), Gibson (QSS Group, Inc.). The experiments were long anddifficult and would not have been possible, but for this dedicated team. The author appreciated the midstudy
support provided by engineers at Lockheed Martin in Fort Worth, Texas, and the funding provided byDavid Sagerser (NASA Glenn's Special Projects Branch) and the Office of the Chief Scientist at
NASA Glenn, Dr. Marvin Goldstein.
NASA Center for Aerospace Information7121 Standard Drive
Hanover, ME) 21076
Available from
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22100
Available electronically at http://gltrs.grc.nasa.gov/GLTRS
INITIAL CIRCULATION AND PEAK VORTICITY BEHAVIOR OF VORTICES SHED
FROM AIRFOIL VORTEX GENERATORS
B.J. Wendt*
Modem Technologies Corporation
Middleburg Heights, Ohio 44130
Abstract
An extensive parametric study of vortices shed from
airfoil vortex generators has been conducted to determine
the dependence of initial vortex circulation and peak
vorticity on elements of the airfoil geometry and
impinging flow conditions. These elements include the
airfoil angle-of-attack, chord-length, span, aspect ratio,
local boundary layer thickness, and free-stream Mach
number. In addition, the influence of airfoil-to-airfoil
spacing on the circulation and peak vorticity has been
examined for pairs of co-rotating and counter-rotating
vortices.
The vortex generators were symmetric airfoils
having a NACA-0012 cross-sectional profile. These
airfoils were mounted either in isolation, or in pairs, on
the surface of a straight pipe. The turbulent boundary
layer thickness to pipe radius ratio was about 17 percent.
The circulation and peak vorticity data were derived from
cross-plane velocity measurements acquired with a seven-
hole probe at one chord-length downstream of the airfoil
trailing edge location.
The circulation is observed to be proportional to the
free-stream Mach number, the angle-of-attack, and the
span-to-boundaD" layer thickness ratio. With these
parameters held constant, the circulation is observed to
fall off in monotonic fashion with increasing airfoil aspect
ratio. The peak vorticity is also observed to be
proportional to the free-stream Mach number, the airfoil
angie-of-attack, and the span-to-boundary layer thickness
ratio. Unlike circulation, however, the peak vorticity is
observed to increase with increasing aspect ratio, reaching
a peak value at an aspect ratio of about 2.0 before falling
off again at higher values of aspect ratio. Co-rotating
vortices shed from closely-spaced pairs of airfoils have
values of circulation and peak vorticity under those values
found for vortices shed from isolated airfoils of the same
geometry. Conversely, counter-rotating vortices show
enhanced values of circulation and peak vorticity when
compared to values obtained in isolation.
The circulation may be accurately modeled with an
expression based on Prandtl's relationship between finite
airfoil circulation and airfoil geometry. A correlation for
the peak vorticity has been derived from a conservation
relationship equating the moment at the airfoil tip to the
rate of angular momentum production of the shed vortex,
modeled as a Lamb (ideal viscous) vortex. This technique
provides excellent qualitative agreement to the observed
behavior of peak vorticity for low aspect ratio airfoils
typically used as vortex generators.
Introduction
Modern design characteristics of aircraft engine inlets,
diffusers, and associated ducting include large amounts of
streamwise curvature coupled with rapid axial variations in
cross-sectional area. The flow performance of these
components is degraded by the development of strong
secondary flows and boundary layer separations. Surface
mounted vortex generators are an effective means of
alleviating these problems. Recent experimental work at the
NASA Glenn Research Center in Cleveland, Ohio has
explored the strategy of using vortex generator induced flows
to counter deleterious secondary flow development and
boundary layer separation inside diffusing S-ducts of various
geometries. _-5 Dramatic improvements in exit plane total
pressure recovery and flow distortion have been
demonstrated. Figures 1 and 2 provide an example.
The duct in Figure 1 was a highly offset S-duct having
a "biconvex" aperture for an inflow (or throat) cross-section.
This biconvex shape transitioned to a circular engine face of
diameter D over an axial distance L of about 2.5D. The area
ratio (throat to engine face cross-sectional area) was about
1.2. This duct geometry represented a subsonic inlet
proposed for use in a compact unmanned combat aircraft
where stealth considerations required large amounts of
streamwise curvature to screen the engine face from all lines
of sight through the inlet cowl-plane aperture. Figure 2
illustrates the experimentally-determined profiles of total
pressure at the engine face for both the baseline duct and the
duct with two arrays of surface-mounted vane vortex
generators. The throat Mach number was 0.65 and the
upstream reference total pressure was atmospheric. Figure 1
indicates the approximate axial location where the vortex
generators were applied. 4 The numbers given in Figure 2
indicate a 5% improvement in total pressure recovery and a
56% drop in circumferential distortion intensity. That such
dramatic improvements can be obtained in this highly
*NASA Resident Research Associate at Glenn Research Center.
NASAJCR--2001-211144 1
contouredandthree-dimensionaldiffuserisstrongtestamentto thepotentialflowcontrolavailablein simplevortexgeneratorvaneswhen used fidlowing the strategy of
seconda O, flow management. Formal elements of this
strategy have recently been developed computationally by
Anderson and Gibb w8 using the RNS3D code, a p,'u'abolized
Navier-Stokes solver. The experimental results given in
Figure 2 were initially designed computationally using the
Falcon code, a Reynolds-averaged Navier-Stokes solver 9 in a
procedure detailed in Reference 4. Conmaon to both the
RNS3D and Falcon flow solvers is the manner in which the
vortex generator flow control is modeled. The vortex
generators ,are not modeled as part of the flow surface grid,
rather they m-e represented by modeling the convective
influence of the vortex shed from each vortex generator at an
"initial" location in the flow field grid downstream of where
the airfoils would be located in a corresponding experimental
arrangement. The basis of this convective influence is found
in the helical pattern of swirling flow; either the secondary
velocity or streamwise vorticity can be used.
The advantage to modeling the secondal); velocity or
streamwise vorticity distribution induced by the vortex
generator is that a simple exponential expression accurately
captures the secondary velocity or streamwise vorticity
distribution of the shed vortex. To demonstrate this, consider
Figure 3. Figure 3 illustrates the secondary velocity field of
an embedded vortex (data on the left, model on the right)
shed from an airfoil vortex generator. This vortex generator
was one of 12 symmetrically placed vortex generators
spanning the inside circumference of a straight pipe. The data
was acquired in the cross-plane one chord-length downstream
of the vortex generators. The model was constructed from a
summation of 24 terms (12 embedded vortices plus 12 image
vortices), each term being proportional to the expression:_°" z5
(1)
where r i is the measured initial circulation of the ith vortexi
or image, COma_ is the measured initial peak vorticity, and ri
is the distance from the cross-plane grid point to the center of
vortex i. The parameters F i i, COma_ , and the Center_location
of the vortex are known as "vortex descriptors" following the
work of Russell Westphal) _ Any such model requires a
description of the vortex strength (here Fi) andi
concentration (here COmax , but viscous core radius R i is
another option). The model above is based on the classical
"ideal viscous" or Lamb vortex. The axial location where the
model is placed (the initial location mentioned earlier) is the
position at which the formative or "roll-up" processes of the
vortex are complete. Circulation and peak vorticity have
reached maximum values here and further downstream
development leads to decay in both quantities. Previous
studies 1°'_2 have indicated the initial location occurs about
one chord-length downstream of the airfoil trailing edge.
From this point on, the terms F and O_,n_._will represent these
initial values of the vortex descriptors.
Many studies have demonstrated the remarkable
similarities between the Lamb-based model and the cross-
plane structure of embedded vortices shed from various types
of vortex generators. Pauley and Eaton 13 demonstrated this in
a study of delta wing vortex generators, Wendt et al._4 in a
stud)' of blade-type vortex generators, Wendt and Hingst t5 for
low-profile wishbone vortex generators, and more recently,
Wendt eta/. 1°'12']6 on _,mmetric airfoil vortex generators.
To complete the modeling of a vortex generator we
must show how the descriptors of the model vortex depend
on the geometry of the vortex generator, the influence of
neighboring vortex generators, and the impinging flow
conditions. For airfoil vortex generators we might express
this dependence as:
V= V (c.h,a, profile, s,3, M, Re, ...)
co_,, = co (c,h,a, profile, s,3, M,Re,...) (2)
where "profile" refers to the shape of the airfoil cross-section,
c is the chord-length, h is the span (the airfoil dimension
normal to the duct flow surface), (z the angle-of-attack, s a
spacing variable representative of the distance between
vortex generators in an array formation, _ the boundary layer
thickness, M the free-stream Mach number, and Re represents
the Reynolds number. The mathematical descriptions of
vortex generators in use at the present time only crudely
approximate the functional dependence indicated in
Equation (12). As a means of improving current vortex
generator models this study was undertaken to examine the
effect of vortex generator geometry, spacing, and impinging
flow conditions for one type of syrnmetric airfoil vortex
generator (NACA-0012). This paper reports the results of a
series of experimental tests covering a range of vortex
generator aspect ratio (- h/c), h/_), _ and M. Spacing effects
were examined by considering co-rotating and counter-
rotating pairs of vortex generators. The vortex generators
were mounted either in isolation, in pairs, or (in one instance)
in a symmetric counter-rotating array spanning the full
circumference of the pipe. Duct or pipe flow conditions were
subSOnic v¢ith a nominal boundary layer thickness to pipe
diameter ratio, 8_G 0.09. Three dimensional mean flow
velocity data were acquired in a cross-plane grid always
located one chord-length downstream of the vortex generator
trailing edge(s). This was done using a rake of seven-hole
probes or a single seven-hole probe tip. This work represents
an extension of the work conducted in Reference 12. In
Reference 12 a total of 14 test cases were considered. An
additional 88 test cases are considered (and fully illustrated)
in this report. Comparisons of the experimental results with
expressions derived from inviscid airfoil theory and a
conservation relationship are conducted to establish the
NASA/CR--2001-211144 2
F andto,,,a_dependenceonairfoilgeometryandimpinging
flow conditions. Closed-form functional relationships are
obtained for this limited set of parameters and are described
here.
Facilities and Procedures
Test Facility
This study was conducted in the Internal Fluid
Mechanics Facility (IFMF) of NASA Glenn. The IFMF is a
subsonic facility designed to investigate a variety of duct
flow phenomena. The facility, as it is configured for this test,
is illustrated in Figure 4. Air is supplied from the surrounding
test cell to a large settling chamber containing a honeycomb
flow straightener and screens. At the downstream end of the
settling chamber the airstream is accelerated through a
contraction section (having a cross-sectional area reduction
of 59 to 1) to the test section. The test section duct consists of
a straight circular pipe of inside diameter D = 20.4 cms. After
exiting the test section duct, the airstream enters a short
conical diffuser and is routed to a discharge plenum, which is
continuously evacuated by the laboratory's central exhauster
facilities. The Mach number range in the testsection duct is
between 0.20 and 0.80 with corresponding Reynolds
numbers (based on pipe diameter) between 1.0 and
4.0 million. Mass flows are between 3 and 7 kgs/sec. More
information on the desigu and operation of the IFMF may be
found in the report of Porro et al. r
Research Instrumentation and Test Parameters
Figure 5 is a detailed illustration of the various test
section components.
A short section of straight pipe (labeled "inlet pipe" in
Figure 5) connects the exit of the facility contraction to the
duct segment containing the vortex generator airfoil(s). Static
pressure taps located on the surface of the inlet pipe allowed
the nominal core Mach number in the test section to be set
and monitored.
The duct portion containing the vortex generator
airfoil(s) was referred to as the "vortex generator duct.'" The
inside surface of the vortex generator duct (and hence the
attached vortex generator) could be rotated about an axis
coinciding with the test section centerline. Full (360 °)
rotation was possible. The rotation was driven by a motor and
gear located in an airtight box above the vortex generator
duct.
Single-Airfoil Vortex Generator
A typical single-airfoil vortex generator is illustrated in
Figure 6. These vortex generators consisted of an airfoil-
shaped blade (with a NACA-0012 profile) mounted
perpendicular to the surface of a base plug. The surface of the
base plug was contoured to the inside radius of the vortex
generator duct. Both blade and plug were machined from an
aluminum alloy using a wire-cutting (electric discharge
machining) process. Table 1 provides the detailed geometry
of all single-airfoil vortex generators used in this stud)'.
Airfoil Array Configurations and the Dottble-Ai_fi_il Vortex
Generator
A single circumferential row of vortex generator
mounting locations was contained in the vortex generator
duct as illustrated in Figure 5. Figure 7 illustrates a cross-
section of the rotating flow surface (inner cylinder) of the
vortex generator duct in the vicinity of the vortex generator
mounting locations. Twelve mounting locations were spaced
30 ° apart and thus covered the entire inside circumference of
the pipe. The spacing between vortex generators in array
configurations is given in nondimensional form, s/c, where s
is the arc-length (measured along the flow surface) between
vortex generators at the mid-chord location, from profile
centerpoint to profile centerpoint. For test conditions
considering the variation in pipe core Mach number, vortex
generators were tested in a full counter-rotating array
(12 airfoils spaced 30 ° apart at the mid-chord position on the
airfoil). Table 1 provides the geometry of the single-airfoil
vortex generators used in this array. As indicated in Figure 7,
an additional 4 mounting locations created a 120 ° sector with
9 mounting locations 15 ° apart. This provided for two
spacing conditions (15 and 30 ° ) for vortex generators tested
in pairs. Table l provides the geometry of the single-airfoil
vortex generators used in this manner.
Airfoils with a chord-length of 1.355 cms were
constructed in the two-to-a-plug configurations illustrated in
Figure 8. Figure 8 diagrams the three different possibilities:
co-rotating, counter-rotating up, and counter-rotating down
pairs. The "up" and "down" designations refer to the
direction of secondary flow occurring between the vortex pair
shed from the vortex generator (e.g., a pair of vortices
produced by a counter-rotating up vortex generator has
secondary flow, between cores, directed away from the flow
surface). Table 2 provides the geometry of all double-airfoils
tested. Note that the description of vortex generator pairings
appropriate for double-airfoils (i.e., co-rotating, counter-
rotating up, etc.) apply to pairs created with single airfoils as
well (Table I).
brstrumentation
The coordinate system used in this study originated in
the vortex generator duct. z = 0 coincided with the trailing
edge of the vortex generator or vortex generator array. All
surveys conducted in this stud), were carried out in the
r-0 cross-plane at an axial location z = 1 chord-length.
The duct segment downstream of the vortex generator
duct is stationary (non-rotating). This test section segment is
referred to as the "instrumentation duct" in Figure 5. The
flow field measurements are acquired in this duct (at the
12 o'clock position) through the use of either a radially-
actuated rake probe (with 4 seven-hole probe tips, as in
Figure 5) or a single radially-actuated seven-hole probe tip.
NASA/CR--2001-211144 3
Theseprobetipswerecalibratedin accordancewiththeprocedureoutlinedbyZilliacJsTheflow,anglerangecoveredincalibrationwas_+40°inbothpitchandyaw.Uncertaintyinflowanglemeasurementwas_+0.7° ineitherpitchoryaw,forflowanglemag-nitudebelow35° (pitchandyawflowanglemagnituderarelyexceeded35° in this study).Thecorrespondinguncertaintyin velocitymagnitudewasapproximately+ 1% of the core velocity, vzc.
Figure 9 is a detailed sketch of the single seven-hole
probe tip. The tip could slide in the z coordinate, to be pinned
at any one of seven different locations (for the seven different
airfoil chord-lengths examined in this study). The probe tip
was always located 1 chord-length downstream of the vortex
generator, or vortex generator array, trailing edge. The
axially-fixed rake probe Was used only if the vortex
generators had a chord-length of 4.064 cms.
To acquire data in the r-0 cross-plane the rake probe, or
single probe, was actuated over a certain radial segment
,along the vertical line extending from duct wall to pipe
center. The vortex generator duct flow surface and vortex
generator(s) were then rotated an increment in
circumferential position, A0, and the radial survey repeated.
In this manner the rake probe would acquire pie-shaped
pieces of the flow field, and the single probe would acquire
ring-segment-shaped pieces. The circumferential and radial
extent of the survey grid was determined by the requirement
that the viscous core of the vortex be fully captured. This
allowed the calculation of F and ¢Om_,as explained below. In
most cases, the circumferential extent of the survey grid was
about 15 °, and the radial extent wa_ about 0.700 cms. Grid
resolution was A0 = 1° and Ar = 1.3 nml.
Experimental Results
Tables 3-6 and Figures I 1-22 summarize and illustrate
the results of this study. Tables 3-6 list the test conditions for
ever), flow field result illustrated in Figures 11-,._, and
tabulate the vortex descriptors determined for each captured
vortex. The pertinent airfoil geometry, airfoil array
configuration, and vortex descriptors are also included on
Figures 11-22. The figures illustrate a total of 90 surveyed
flow fields. Each illustrated flow field consists of a cross-
plane vector plot of secondary velocity (on the left hand
side), a contour plot of primary velocity ratio (in the middle),
and a contour plot of streamwise vorticity (on the right hand
side). The cross-plane Survey sector is generally of Sufficient
size to capture the core of the resultant tip vortex. The radial
axis of each plot represents distance from the wall, R-r,
where R = 10.21 cms was the pipe radius. The
circumferential axis represents angular position in de_ees. In
the interest of saving space in the figures, only the plots of
secondary velocity have the survey limits in (r, 0) marked.
The vortex descriptors originate from the secondary
velocity data acquired in the cross-plane. The secondary
velocity data is first converted to streamwise vorticity data
following the relation:
_ Ov_ v o 1 8v(.o. - -- 4 (3)
8r r r 80
where (v,., v 0 ) are the secondary components of velocity in
the radial and circumferential coordinates, respectively.
Finite difference formulas are used to represent the spatial
derivatives in Equation (3). In the plots of streamwise
vorticity, contour lines surrounding unshaded regions
represent negative values of vorticity, lines surrounding
shaded regions represent positive values of vorticity. The
contour increment for lines not labeled is + 3000 sec -_.
(.q,,_, is located at some grid point having coordinates (r,, 0,).
The vortex circulation F was calculated by first isolating the
region of core vorticity in the data field. This was done by
plotting the "bounding value" contour of core vorticity c%, =
0.0lOOmed, as is done in each plot of streamwise vorticity
given in Figures 11-22. The circulation was then calculated
according to the relation:
core areainMdc
bo_mdingcoatour
_ d(area) (4)
where o_ is the streamwise vorticity at each elemental cross-
plane area within the co,b contour. In all cases considered
here, the core has a well-defined circular shape, and so F is
easily determined. Uncertainty estimates for all listed
descriptors are given in Tables 3-6. These are derived by
combining the uncertainties in measured velocities and probe
placement in accordance with the procedure outlined by
Moffat. _9
Isolated Vortices
Figures 11 through 19 illustrate the flow fields in the
vicinity of a tip vortex shed from a single airfoil mounted in
isolation. In all cases the vortex shed is a clockwise vortex.
Figure 11 illustrates the effect of increasing the airfoil angle-
of-attack, Figure 12 illustrates the effect of increasing the
pipe core Mach number, and Figures 13 through 19 illustrate
the effect of increasing the airfoil span (or alternatively, the
airfoil aspect ratio 8h/rtc) for each of six different airfoil
chord-lengths.
Of particular interest are the contour plots of primary
velocity ratio VJVzc. These plots illustrate the vortex-airfoil
wake-boundary layer interaction as well as some unexpected
features of the tip vortex core. For the smaller chord-len_hs
of 0.85, 1.36, and 2.03 cms, the core appears as a small
circular wake region with no other discernible structure. This
holds true over the entire range of aspect ratio examined,
0.6 _< AR < 6.0. For chord-lengths at or greater than
2.54 cms, the core shows additional structural features.
NASA/CR--2001-211144 4
Threedistincttypesof corestructurearenotedandoccuraccordingtoairfoilaspectratio:
1) ForARintherange0.3< AR < 1.2 (refer to Figures
16a-c, 17a-c, 18a-d, and 19a-c) the core region
appears as a small circular wake buried within the
boundary layer. On the left side of the core, the
boundary layer is thinned in the downwash of the
vortex, on the right side it is thickened by the upwash.
From the left side, a tongue of high-momentum fluid
,xa'aps under the core region and extends up the right
side, creating a steep gradient in the axial velocity
field in the upwash of the vortex.
2) For AR in the range 1.2 < AR _<2.5 (refer to Figures
16d-e, 17d-f, 18e-f, and 19d-h) the vortex interaction
with the boundary layer is considerably less.
A portion of the upper boundary layer fluid has been
convected up along the right hand side of the vortex.
Structure in the core is apparent (refer to Figure 19g
and 19h in particular). Within the circular confines of
the core boundary, a crescent-shaped wake region
surrounds a smart circular region of jetting flow.
3) For AR greater than about 2.5 (refer to Figures 16f-i,
17g, and 18g) the core region is a strong circular
wake with no other apparent structure. Extending
below the core is the remainder of the airfoil wake.
Co-Rotating Pairs
Table 4 and Figures 20a-j present the results for
co-rotating vortices shed from a pair of airfoil vortex
generators of identical geometry. The spacing between
airfoils (the distance s measured between the centerlines of
the airfoil profiles) is nondimensionalized with the airfoil
chord-length c (see Table 4). Since a co-rotating pair of
vortices represents an asymmetric secondary flow pattern,
both vortices were captured in the experiment. For the
smaller spacing ratio of s/c = 0.67, the left hand and right
hand vortices were captured together or plotted together
(Figures 20a-e). For s/c = 1.34, the left hand and right hand
vortices are plotted separately (Figures 20g-j). When
compared to vortices shed from airfoils mounted in isolation,
co-rotating vortices produced from pairs of airfoils of
identical geometry are seen to have smaller values of
circulation and peak vorticity. The measured discrepancy is
between about l0 and 20%.
Counter-Rotating Up and Down Pairs
Table 5, Table 6, Figures 21a-i, and Figures 22a-i
present the results for the two types of counter-rotating
vortices shed from a pair of airfoils differing only in the sign
(plus or minus) of cx. Counter-rotating vortices represent a
symmetric secondary flow pattern and so (generally
speaking) only one vortex of the pair was captured in the
experiment. The exceptions are at the smaller value of s/c,
where some or all of the neighboring vortex was captured.
When compared to vortices shed from airfoils mounted in
isolation, counter-rotating up and down vortices are seen to
have enhanced values of circulation and peak vorticity (with
discrepancies in the range of about I0 to 25%).
Analysis and Modeling
A portion of the results listed in Table 3 are plotted in
Figures 23 to 31. The dependence of F and COma,on (x
(keeping M, h/8, and ,MR constant) is given in Figure 23.
Both F (left hand plot) and o_,,_ (right hand plot) increase in
approximate linear fashion with increasing c_. The fall-off in
o3,m_ at large _x is likely due to a flow separation on the airfoil
for angles _eater than about 16 degrees. The dependence of
F and O_ma_on M (keeping c_, h/_5, and AR constant) is given
in Figure 25. Again, the overall character of this dependence
is a linear increase in the descriptors with pipe core velocity.
The dependence of F and o_...... on AR (keeping c_, M, and h/8
constant) is given in Figures 25, 27, 29, and 31. Here we see
F fall off in monotonic fashion with increasing aspect ratio.
The behavior for mma_ is distinctly different, however. The
peak vorticity first climbs to a peak and then falls off with
increasing aspect ratio. The dependence of F and o_n_, on h/8
(keeping c_, M, and AR constant) is illustrated in Figures 26,
28, and 30. As expected, both F and oJ,,m_ increase with
increasing h/6. Peak vorticity, however, tails off at higher h/8
(Figure 30) while F apparently does not. Also plotted on each
of Figures 23-31 are the results of analytical modeling
equations developed briefly below.
Circulation
The dependence of vortex circulation on vortex
generator geometry and impinging flow conditions can be
approximated with an expression developed by Prandtl. :° For
a finite span wing of elliptical planform and symmetric cross-
section, the circulation developed about the wing in the plane
containing the wing cross-sectional profile is:
l- = T( l'zc O_ Co
1+ -AR
(5)
where AR = 8h/(m:). It should be noted here that, although
this definition of AR applies to wings of elliptical planform,
it is also used in this study to represent the AR of the
rectangular planform airfoil. If we now assume that the
circulation developed about the wing cross-section is turned
into the stream, Equation (5) forms the basis for an
expression approximating the circulation of the shed tip
vortex:
klo_v_ c f
1+ k2 I_AR
(6)
NASA/CR--2001-211144 5
wherekl andk2 replacetheconstants";_"and"2" inEquation(5),andthehyperbolictangentfunctionisusedtorepresenttheretardinginfluenceoftheboundarylayer.TheconstantskI throughk4aredeterminedusingaleast-squaresregressionwiththedataplottedinFigures23-31.Thevaluesaredeterminedtobe:
kl = 1.61,k2=0.48,k3= 1.41,k4=1.00.
Asthefiguresindicate,Equation(6)providesanexcellentcorrelationfor theinitialcirculationbehaviorof thetipvortices.
Peak Vorticity
In remarkably, accurate fashion, Prandtl's inviscid
analysis provides us with an estimate of shed vortex
circulation. 1" can be viewed as the integation or summation
of the streamwise vorticity field. Peak vorticity, on the other
hand, is an element of the streamwise vorticity field's
distribution or "structure". Nothing of this structure can be
discerned from the inviscid analysis. In fact. it may be noted
that the inviscid model of a vortex has an infinite (and
therefore unphysical) value of peak vorticity at the core
central location (and a zero value everywhere else). Thus, to
begin our analysis for co,,_,, we discard the inviscid model
and consider, instead, a viscous vortex. Perhaps the simplest
model of a viscous vortex is the Lamb vortex. The Lanab
vortex is obtained through classical solution techniques of
the Navier-Stokes equations simplified to represent two-
dimensional axisymmetric flow. 2_ Simply stated; a Lamb
vortex is a time dependent axisymmetric viscous flow that
starts as a potential vortex. The velocity at the origin (r = 0,
center of the vortex) is forced to zero at time t = 0. The
resulting solution describes the subsequent decay in terms of
angular velocity' v0. In radial coordinates this is:
/ {6})F 1.0 - expv° = 2/'/""'7
(7)
where F here is the limiting value of vortex circulation
(as r --) _,), v is the coefficient characterizing the viscous
diffusion of the flow, and r is the radial coordinate, measured
from the center of the vortex (crossplane origin). As before,
let z represent the streamwise or axial coordinate. Replace t
with the strearnwise convective displacement t = z/vzc (with
the origin of the streamwise coordinate coinciding with t = 0)
and differentiate v0(r,z) to obtain the streamwise vorticity
field:
, _ 2
= v= F expl-V:c r _.co: 4rcvz [ 4v z J
(8)
The peak vorticity is thus: ¢.o,,=, = v_cF/(4rtvz). Substitute for
v and t in Equation (7) to find:
v 0=- l.O-exp -rrco._ r-"2nr F
(9)
or, in other words, the Lanth vortex in terms of circulation
and peak vorticity. _4 This form (see also Equation 1) aids the
modeling process, since (as we have seen from the previous
section) it requires information (or data) about the vortex in
the vicinity of the core only,, where the circulation and peak
vorticity are entirely contained. If, for example, we chose to
write Equation (9) in terms of the conserved properties of
vortex crossplane angular momentum or kinetic energy, a
researcher would be required to obtain information over the
entire flow field.
To proceed with the analysis for COmax, we make the
following two assumptions:
1) At the "initial" streamwise location (in this study we
assume the z = i chord-length position) vortex
structure can be accurately represented by the Lamb
vortex model. As indicated earlier in the
"Introduction and Back_ound'" section, many
previous studies exist to support this assumption.
2) The conserved property of vortex angular momentum
can be related to a pressure force moment formulated
near the airfoil tip.
By integrating a suitable version of the Lamb vortex to obtain
an expression for angular momentum, and through the use of
Prandtl's equation for F (Equation 6) we will isolate O)max in
ternas of vortex generator geometry and impinging flow
conditions. Lets start first by developing the conservation
relationship.
Estimating Tip Vortex Angular Momentum Production
The rate of vortex angular momentum production is
equated to a moment summation taken at the airfoil tip.
Reference 22 is a very good experimental description of the
vortex roll-up phenomena in the vicinity of the airfoil tip. In
Reference 22, the airfoil is a large scale half-wing model
(similar to the models tested in this report) with a
NACA-0012 sectional profile. Let H be the angular
momentum of the shed vortex, CF a characteristic force
tending to rotate the fluid at the wing tip, and CMA a
characteristic moment arm through which the fluid is rotated.
Then:
_,(rnoments at the tip) = CF x CMA =
OH H(10)
CF is related to the lift force on the airfoil and is
estimated by integrating the 2D inviscid pressure. , . ._
dxstnbutlon: -3
NASA/CR--2001-211144 6
ACp=4.0(sina)_-I ill)
overthesurfaceoftheairfoil:
, -y,,_
(12)
where y, represents the lower limit on the span component of
integration, and 9 represents the fluid density. By setting
y, = 0 we confine our analysis to "small" aspect ratios. This
follows because, as the aspect ratio of the airfoil increases
and the span grows out of proportion to the chord, the lift
forces generated near the root (or wall location) have less of
an influence at the tip. Thus, at larger aspect ratios, Equation
(9) with Yn = 0 tends to overestimate CF.
CMA is taken to be the radius of the shed vortex. We
approximate this by an expression for rm,_, the radial location
at which vo (from Equation 9) is maximum:
FCMA = r._ = "_.. 7E (Drm x
(13)
Equation (13) for CMA describes what is also knovm as the
"Rankine" core radius of a viscous vortex.
Integrate Equation (12), substitute this expression with
Equation (13) into Equation (10) to obtain:
X(,.o.,<,,,, a, ,/p)_ V . (14)pv:< pdz : _2zrOam,_
We nov,' turn our attention to integrating the Lamb vortex
model for angular momentum H.
Angular Momentum of the Constrained Lamb Vortex
The angular momentum in a region of the polar
crossplane bounded by i"1:0 < r/_< r and 0:0 < 0 < 2z is
evaluated with the expression:
2x r
tt = I l (YX_') 2,'rrl drl dO (15)pd: o o
where i; represents the secondary or crossplane velocity
vector. Applying Equation (15) to Equation (9) provides the
angular momentum profile of the Lamb vortex:
I-"--rH -F . l-exp . (16)p dz 2 z (o_ F
It can be seen from this expression that the angular
momentum (per unit axial dimension) is unbounded in r, or
in other words, H(p dz)---)o_ as r ---)oo Thus, before we
can use the Lamb vortex in this analysis, a meaningful way to
constrain it (in r) must be found. One approach :a is to define
a model discontinuous in v0. Figure 32 diagrams this
construction. Figure 32 is a plot of vo versus r as defined by
Equation (9) for indicated values of circulation and peak
vorticity. Define a velocity ratio 4:
I' 0g =-- (17)
U0 trax
where v0,,m is the maximum angular velocity of the Lamb
vortex over the range r: 0_< r < oo. As Figure 32 indicates,
this velocity ratio is realized at two radial locations on the
Lamb vortex profile, rl and r2. Choose r2 to be the outer
boundary of this discontinuous vortex model, referred to here
as the "constrained Lamb vortex". Note, from Figure 32, that
by choosing r2 and _ << 1.0, all of the viscous core and a
good portion of the inviscid outer region of the Lamb vortex
are retained. Now r2 can be approximated:
r2= 1 ) ]'[_/2 F(1 - e -I' : /t"09,,_,
I (18)
Apply Equation (15) within this boundary to find:
H F-"- [n-l+<-"] (19)
pdz 2zm..,
where:
1
= 2¢: (1- e-"-" )2" (20)
Equation (19) is simplified to:
H =r -_(fl-1) (21)
p dz 2 zr co,=_
Note that the angular momentum of the constrained Lamb
vortex is proportional to F2/(.0rna,. Other vortex models have
been examined (both continuous and discontinuous in Vo) to
determine the angular momentum relationships and these are
also found to be proportional to F2/(0m_x.24
Returning now to Equation (14):
H =_ro:chv_.2 F
NASA/CR--2001-211144 7
wereplacethelefthandsidewithEquation(21)andsolvefor(,Oma x to find:
r _(/3-i):
2zr O_-c-h-vE_(22)
The constant _ from Equation (17) was deternfined by a
least squ,'u:es recession from the plotted peak vorticity data
on Figures 23 through 31. The ,,','flue was determined to he
= 0.29. The peak vorticity plots (right hand side of Figures
23 through 31) illustrate the correlation of Equation (22) with
the given value of 4. Overall, a very good qu,-ditative
a_eement with the data is obtained.
Summa_'
An extensive paranaetric stud), of vortices shed from
,airfoil vortex generators has been conducted to determine the
dependence of initial vortex circulation F and peak vorticity
co,,,_, on airfoil angle-of-attack cz, chordlength c, span h,
impinging external Mach number M, and boundary layer
thickness 8. In addition, the influence of airfoil-to-airfoil
spacing on F and eo,,_, has been examined for pairs of airfoils
producing co-rotating and counter-rotating vortices.
The vortex generators were symmetric airfoils having
NACA-0012 cross-sections. These airfoils were mounted
either in isolation or in pairs on the surface of a straight pipe.
The turbulent boundary layer thickness to pipe radius ratio
was 8/R _ O. 17. The circulation and peak vorticity data were
derived from cross-plane velocity measurements acquired
with a seven-hole probe at one chord length downstream of
the airfoil trailing edge location.
F is observed to be proportional to M, ct, and h/6. With
these parameters held constant, F is observed to fall off in
monotonic fashion with increasing airfoil aspect ratio
AR = 8h/_c. O)ma,_ is also observed to be proportional to M,
oq and h/8. Unlike F, however, e0m_ is observed to increase
with increasing aspect ratio, reaching a peak value at
AR -- 2.0 before falling off and rising again at higher values
of AR. Co-rotating vortices shed from closely-spaced pairs of
airfoils have values of F and e0m_, under those values found
for vortices shed from isolated airfoils of the same geometry.
Conversely, counter-rotating vortices show enhanced values
of F and tOma_when compared to values obtained in isolation.
F may be accurately modeled with an expression based
on Prandtl's relationship between finite airfoil circulation and
airfoil geometry. A correlation for (Omax has been derived
from a conservation relationship equating the moment at the
airfoil tip to the rate of angular momentum production of the
shed vortex, modeled as a Lamb (viscous) vortex. This
technique provides excellent qualitative a_eement to the
observed behavior of O)m_.x for low aspect ratio airfoils
typically used as vortex generators.
References
_Reichert, B.A. and Wendt, B.J., "Improving Diffusing
S-Duct Performance by Secondary Flow Control," AIAA
Paper 94-0365, Jan. 1994.
2Foster, J., Okiishi, T.H., Wendt, B.J., and Reichert, B.A.,
"Study of Compressible Flow Through a Rectangular-to-
Semiannular Transition Duct," NASA CR-4660, Apr. 1995.
3Wendt, B.J. and Dudek, J.C., "Development of Vortex
Generator Use for a Transitioning High Speed Inlet," Journal
of Aircraft, Vol. 35, No. 4, August 1908, pp. 536-543.
4Hamstra, J.W., Miller, D.N., Truax, P.P., Anderson, B.A.,
and Wendt, B.J., "Active Inlet Flow Control Technology
Demonstration," 22nd International Council of Aeronautical
Sciences, tCAS Paper 6112, August 2000.
5Wendt, B.J., "Suppression of Cavity-Driven Flow
Separation in a Simulated Mixed Compression Inlet,"
NASA/CR--2000-210460, September 2000.
6Anderson, B.H. and Gibb, J., "Application of Computational
Fluid Dynamics to the study of Vortex Flow Control for the
Management of Inlet Distortion," AIAA Paper 92-3177,
July 1992.
7Anderson, B.H. and Gibb, J., "Vortex Generator Installation
Studies on Steady State and Dynamic Distortion," AIAA
Paper 96-3279, July 1996.
SAnderson, B.H. and Farok_hi, S., "A Study of Three
Dimensional Turbulent Boundary Layer Separation and
Vortex Flow Control Using the Reduced Navier-Stokes
Equations," Turbulent Shear Flow Symposium Technical
Report, 1991.
9Bender, E.E., Anderson, B.H., and Yagle, P.J., "Vortex
Generator Modeling for Navier-Stokes Codes," Symposiumon Flow Control of Wall Bounded and Free Shear Flow,
1996 Joint ASMEIJSME Fluids Engineering Conference, San
Francisco, CA, July 18-22, 1999.
l°Wendt, B.J., Reichert, B.A., and Foster, J.D., "The Decay
of Longitudinal Vortices Shed from Airfoil Vortex
Generators," AIAA Paper 95-1797, June 1995.
NWestphal, RV., Pauley, W.R. and Eaton, J.K., "Interaction
Between a Vortex and a Turbulent Boundary Layer-Part 1:
Mean Flow Evolution and Turbulence Properties," NASA
TM-88361, January 1987.
l'-Wendt, B.J. and Reichert, B.A., "The Modelling of
Symmetric Airfoil Vortex Generators," AIAA Paper 96-
0807, June 1996.
t_Pauley, W.R. and Eaton, J.K., "The Fluid Dynamics and
Heat Transfer Effects of Streamwise Vortices Embedded in a
Turbulent Boundary Layer," Stanford University Technical
Report MD-51, Stanford, CA, August 1988.
NASA/CR--2001-211 144 8
14Wendt,B.J.,Greber,I.,,andHingst,W.R.,"TheStructureandDevelopmentofStreamwiseVortexArraysEmbeddedina TurbulentBoundaryLayer,"AIAAPaper92-0551,JanuarS 1992.
lSWendt, B.J. and Hingst, W.R., "Flow Structure in the Wake
of a Wishbone Vortex Generator," AIAA JottlvTal, Vol. 32,
November 1994, pp. 2234-2240.
_6Wendt, B.J. and Reichert, B.A., "Spanwise Spacing Effects
on the Initial Structure and Decay of Axial Vortices," NASA
CR-198544, November 1996.
WPorro, A.R., Hingst, W.R., Wasserbauer, C.A., and
Andrews, T.B., "The NASA Lewis Research Center Internal
Fluid Mechanics Facility," NASA TM-105187, September1991.
_sZilliac, G.G., "Modelling, Calibration, and Error Analysis
of Seven-Hole Probes," Eaperiments in Fhdds, Vot. 14,
1993, pp. 104-120.
_QMoffat, R.J., "Contributions to the Theory of Single-
Sample Uncertainty Analysis," Transactions of the ASME,
Vol. 104, June 1982, pp. 250-258.
2°Prandtl, L., "Applications of Modem Hydrodynamics to
Aeronautics," NACA Report 116, 1921.
21Panton, R.L., hwompressihh, Flow, 1st Edition, John Wiley
and Sons, New York, 1984. See Chapter 11.
2-'Chow, J.S., Zilliac, G.G., and Bradshaw, P.,
"Measurements in the Near-Field of a Turbulent Wing-Tip
Vortex," AIAA Paper 93-0551, January 1993.
-'_Karamcheti, K., "Principles of Ideal-Fhdd Aerodynamics,"
1st Edition, John Wiley and Sons, New York, 1966. See
Chapter 17.
24Wendt, B.J., "Initial Peak Vorticity Behavior for Vortices
Shed from Airfoil Vortex Generators," AIAA Paper 98-
0693, January 1998.
25Bray, T.P., "A Parametric Study of Vane and Air-jet Vortex
Generators," Cranfield University Engineering Doctorate,
BCC/DERA, October 1998.
NASA/CR--2001-21 1144 9
Flow
vane vortexgenerators
throat cross-section
Figure 1 - The Ultra-Compact Two-TurnS-Diffuser
Ptotalcontours of
reference Ptotal
exit plane or engine facecontour increment = 0.01 cross-section
Baseline (no VGs) Vane Span = 2 mmRecovery = 0.910 Recovery = 0.957DC60 Distortion = 0.390 DC60 Distortion = 0.170
Figure 2 - Vane vortex generators dramatically change total pressurecontours at the engine face for the two-turn S-diffuser.
NASA/CR--2001-211144 10
degrees degrees
100 90 80 100 90 8011o _ I -- / 11o t
, \ illx" • ", .... _,,_::::_ ..... ; ,_ , / ....... "::'-:_'"_"HH 't
(d/2-r) cms ._ 30 m/sec-_ (dl2-r) cms ._ 30 mlsec-_>
data model
Figure 3 - A comparison of the secondary velocity field of an embedded vortex;data on the left, and a model constructed from the superposition of Lamb vortices
on the right.
Main Plenum _i T- eest Secti°n Duc_'_Exhaust Regi°n "_1
Fiberglass Bellmouth Contraction I _ /] / _ vacuum exhaust line I
ambient air ,,- r I \ [ _ / \/
I_in [ .... T ............ -'_':_,7"" _1 _ _ / ELi Xll*'illl- _ i coarsemesh r -J i i II III I_1 _ / I I I
I_d_l II _ I / ,--_ ,, " III ,_ IL[I,, _ I _ i II I \_1 II I _ honeycomb_ ,, "'- -111I._-41r-'q_ll----_11- ;._--'_--'I:_
,_JebO<_;tl II I _ , --_ I _l i i -- i --ixl-- - _ - ,-I-Ii- --i r _ HIP I,!1,-,- - --I1 "t',=',,s'_r " "_, /_._cZh_>_l II I lip" , _ , , .-II_I llii.---#l lUlHii-.r._lll-----..4.1.1. _..'-....S:l
_IJL i ,o,,, : fine mesh_ :: ," IlL _ _ I - II I piog_aiveI
..................................
Circular Aperture Flange
Figure 4 - The Internal Fluid Mechanics Facility of NASA Glenn
InletPipe
Vortex Generator Duct ' "t_ Instrumentation Duct
Motor and Gearbox | Radial Actuator
__i I _i_ Movable Blocks for Axial Positioning
/
Ip. / o'Rad'a'AOuatorP=h/
ti//_ _/ _LL///[/////////////I
Vortex Generators and Plugs "'"
\
_/l/llllillllillllllllllllll_
Flow Static aps _ _ _ i Rake
Access Hatches
Figure 5 - A cut-away sketch of the test section duct.
NASA/CR--200I-211144 l I
° [ LIh's°an'II ;I
Front View! |
Side View
-c, chord
_/
Top View Single Airfoil VG
Figure 6 - The geometry of thesingle-airfoil vortex generator.
Table 1 - Summary of single-airfoil geometry, covering all models tested.
chord c span h angle-of-attackOL
(cms) (cms) (degs)
0.85 0.21 160.85 0.51 160.85 1.02 160.85 1.52 160.85 2.03 16
1.36 0.28 161.36 0.51 161.36 0.81 161.36 1.02 161.36 1.52 161.36 1.63 161.36 2.03 161.36 2.54 161.36 3.05 161.36 3.56 16
2.03 0.51 162.03 1.02 162.03 1.22 162.03 1.52 162.03 2.03 162.03 2.44 16
2.54 0.51 162.54 0.64 162.54 1.02 162.54 1.52 162.54 2.03 162.54 2.54 162.54 3,05 162.54 3.56 162.54 4,57 16
notes chord c span h angle-of-attackOC
(cms) (cms) (degs)
3.05 0.51 163.05 0.76 163.05 1.02 163.05 1.52 163.05 1.83 163.05 2.03 163.05 3.66 16
3.56 0.51 163.56 0.89 163.56 1.02 163.56 1.52 163.56 2.03 163.56 2.13 163.56 4.27 16
4.06 0.51 +/- 164.06 1.02 84.06 1.02 124.06 1.02 +/- 164.06 1.02 204.06 1.52 +/- 164.06 2.03 164.06 2.44 164.06 2.54 +/- 164.06 3.05 164.06 3.56 16
Notes:
a) used in co-rotating pairsb) used in counter-rotating up pairsc) used in counter-rotating down pairsd) used in full counter-rotating arrays.
notes
a. b, c
d
a, b, c
a. b, c
NAS A/CR--2001-211144 12
90 degI
j"
Blank Plugs 0 deg
Figure 7 - The r-theta crossplane looking upstream into the vg duct.
Co-rotating double.
Counter-rotating down double. -- t i_
Counter-rotating up double. _S-li _Top Vie
Figure 8 - The geometry of the double-airfoil vortex generators (c = 1.36 cms).
Table 2 - Summary of double-airfoil geometry, covering all models tested
chord c span h 0_ double-airfoil type(cms) (cms) (deg)
1.36 1.02 161.36 2.03 16
1.36 0.51 +/- 161.36 1.02 +/- 161.36 2.03 +1- 16
1.36 0.51 +/- 161.36 1.02 +/- 161.36 2.03 +/- 16
co-rotating double airfoilco-rotating double airfoil
counter-rotating down double airfoilcounter-rotating down double airfoilcounter-rotating down double airfoil
counter-rotating up double airfoilcounter-rotating up double airfoilcounter-rotating up double airfoil
NASA/CR--2001-211144 13
Computer-driven' A actuator provides
radial traverse of
_ ! ! J !
Instrumentation duct i '1 ',
vooox0ooo_,or_o..,_ ,_ ._.
-'/l' I '_ ,, Ii
Fto_ _ _._ _ Radially-actuated probe stem
,Probe tip can translate axially as indicated II Flex-hose _ .........
by arrows connections
Sliderwith pin-holestoflx _ _,, - =_'_'9 !
probe tip at correct axiallocation
Figure 9 - The single seven-hole probe shown in a cut-away view of the testsection. The tip could be adjusted in the axial component to acquire data one
chord-length downstream of the vortex generator airfoil.
Rake-probe
duct circumferential
translation
duct circumferential
translation
• ___
4'4, '1' '1' _' 'lk
÷ * ÷ * ÷ i _ each radial line has
", Single tip 1 between 10 and 20 points
The line at 90 degrees was the "active" survey line. The probe tip(s) acquiredata on this line.
_-_"_ each radial line has 80
points
To center of duct
Figure 10 - The survey grids used by the seven-hole probe rake and single tip.
NASA/CR--2001-211144 14
Table 3
Results for Isolated Airfoils, and Airfoils Tested in Full Counter-Rotating Arrays.
O_ c h AR hh5 v c Mach
deg cms cms m/sec
8 4.06 1.02 0.64 0.57 85 0.2512 4.06 1.02 0.64 0.57 85 0.2516 4.06 1.02 0.64 0.57 85 0.2520 4.06 1.02 0.64 0.57 85 0.25
F GOma x
m 2/sec 1/sec
See
Fig.
-0.291 -18193 11a-0.471 -24346 11b-0.643 -33189 11c-0.940 -34055 11d
16" 4.06 1.02 0.64 0.57 85 0.25 -0,722 -35714 12a16" 4.06 1.02 0.64 0.62 129 0.40 -1.I49 -56831 12b16" 4.06 1.02 064 0.65 187 0.60 -1.724 -84000 12c
16 0.85 0.21 0.64 0.12 85 0.25 -0.156 -20297 13a16 0.85 0.51 1.53 0.29 85 0.25 -0.240 -22340 13b16 0.85 1.02 3.06 0.57 85 0.25 -0.253 -30881 13c16 0,85 1.52 4.58 0.86 85 0.25 -0,297 -33368 13d16 0.85 2.03 6.11 1.14 85 0.25 -0.307 -38301 13e
16 1.36 0.28 0.64 0.19 85 0.25 -0.223 -29194 14a16 1.36 0.51 0.96 0.29 85 0.25 -0.342 -23734 14b16 1.36 0.81 1,53 0.46 85 0,25 -0,315 -42509 14c16 1.36 1,02 1.91 0.57 85 0.25 -0.315 -35415 14d16 1.36 1.52 2.86 0.86 85 0.25 -0.429 -37397 14e16 1.36 1.63 3.06 0.91 85 0.25 -0.419 -52295 14f16 1.36 2.03 3.82 1.14 85 0.25 -0.485 -50676 14g16 1,36 2.54 4.78 1.43 85 0.25 -0,428 -44093 14h16 1.36 3.05 5.73 1.71 85 0,25 -0.476 -60375 14i16 1.36 3.56 6.68 2.00 85 0.25 -0.446 -63832 14j
16 2.03 0.51 0.64 0.29 85 0.25 -0.322 -27772 15a16 2.03 1.02 1.27 0.57 85 0.25 -0.435 -43956 15b16 2.03 1.22 1.53 0.69 85 0.25 -0.493 -52181 15c16 2.03 1.52 1,91 0.86 85 0,25 -0,552 -57278 15d16 2.03 2.03 2.55 1.14 85 0,25 -0.596 -56235 15e16 2.03 2.44 3.06 1.37 85 0.25 -0,667 -44619 15f
16 2,54 0.51 0.51 0.29 85 0.25 -0.450 -22914 16a16 2.54 0.64 0.64 0.36 85 0.25 -0.421 -31387 16b16 2,54 1.02 1,02 0.57 85 0.25 -0,525 -43913 16c16 2.54 1.52 1.53 0.86 85 0.25 -0,626 -57104 16d16 2.54 2.03 2.04 1.14 85 0.25 -0.719 -73333 16e16 2.54 2.54 2.55 1.43 85 0.25 -0.781 -58750 16f16 2.54 3.05 3.06 1.71 85 0.25 -0.825 -42748 16g16 2.54 3.56 3.56 2.00 85 0.25 -0.862 -46882 16h16 2.54 4.57 4,58 2.57 85 0.25 -0,866 -70289 16i
16 3.05 0,51 0.42 0.29 85 0.25 -0.399 -23345 17a16 3.05 0.76 0.64 0.43 85 0.25 -0.530 -30924 17b16 3.05 1.02 0,85 0.57 85 0.25 -0.591 -40099 17c16 3.05 1.52 1.27 0.86 85 0.25 -0.704 -53251 17d16 3.05 1.83 1.53 1.03 85 0,25 -0.776 -59852 17e16 3.05 2.03 1.70 1.I4 85 0.25 -0.817 -58695 17f16 3.05 3.66 3.06 2.06 85 0.25 -0.985 -35505 17g
* Data from full counter-rotating airfoil arrays. +k 5% +# 7%
NASA/CR--2001-211144 15
Table 3 - Continued
Results for Isolated Airfoils, and Airfoils Tested in Full Counter-Rotating Arrays.
0,. c h AR hi5 v c Mach
deg cms cms m/sec m2/sec
See
Fig.
16 3.56 0.51 0.36 0.29 85 0.25 -0.451 -21114 18a16 3.56 0.89 0.64 0.50 85 0.25 -0.662 -36897 18b16 3,56 1,02 0.73 0.57 85 0.25 -0.648 -40085 18c16 3.56 1.52 1.09 0.86 85 0.25 -0.780 -54079 18d16 3.56 2.03 1.46 1.14 85 0.25 -0.944 -59439 18e16 3.56 2.13 1.53 1.20 85 0.25 -0.941 -67331 18f16 3.56 4,27 3.06 2.40 85 0.25 -1.170 -45710 18g
16 4.06 0.51 0.32 0.29 85 0.25 -0.429 -19956 19a16 4.06 1.02 0,64 0.57 85 0,25 -0.713 -39161 19b16 4.06 1.52 0.96 0.86 85 0.25 -0.824 -50476 19c16 4.06 2.03 1.27 1.14 85 0,25 -0.961 -60834 19d16 4.06 2,44 1.53 1.37 85 0.25 -1.044 -63493 19e16 4.06 2.54 1,59 1.43 85 0.25 -t.045 -66113 19f16 4.06 3,05 1.91 1.71 85 0.25 -1.210 -66124 19g16 4,06 3.56 2.23 2.00 85 0,25 -1.243 -69638 19h
+/- 5%
Table 4
Results for Co-Rotating Pairs
Left Hand S,/C V Machc h AR h/_ or zcdeg cms cms Right Hand m/see
16 1.36 1.02 1.91 0.57 LH16 1.36 1.02 1.91 0.57 RH
16 1.36 2.03 3.82 1.14 LH16 1.36 2.03 3.82 1.14 RH
16 4.06 0151 0.32 0.29 LH16 4.06 0.51 0.32 0.29 RH
16 4.06 1.52 0.95 0.86 LH16 4.06 1.52 0.95 0,86 RH
16 4.06 2.54 1.59 1,43 LH16 4.06 2.54 1.59 1.43 RH
16 4.06 0.51 0.32 0.29 LH16 4.06 0.51 0.32 0.29 RH
16 4.06 1.52 0.95 0.86 LH16 4.06 1.52 0.95 0.86 RH
0.67 85 0.250.67 85 0.25
0.67 85 0.250.67 85 0.25
0.67 85 0.250.67 85 0.25
0.67 85 0.250.67 85 0.25
0.67 85 0.250.67 85 0.25
1.34 85 0.251.34 85 0.25
1.34 85 0.251.34 85 0.25
1.34 85 0.251.34 85 0.25
16 4.06 2.54 1.59 1.43 LH16 4.06 2.54 1.59 1.43 RH
+/- 7%
£ O_ma x See
m2/sec 1/sec Fig.
-0.284 -35786 20a-0.305 -32902 20a
-0.398 -35994 20b-0.437 -55409 20b
-0.397 -17349 20c-0.462 -18203 20c
-0.667 -40248 20d-0.742 -42782 20d
-0.879 -48733 20e-0.954 -55171 20e
-0.480 -19411 20g-0.482 -19853 20f
-0.794 -49131 20i-0,794 -43568 20h
-1.021 -62701 20k
-1.012 -56314 2_
+/- 5% +/- 7%
NASAJCR--2001-211144 16
Table 5
Results for Counter-Rotating Up Pairs
O. c h AR
deg cms cms
16 1.36 0.51 0.9516 1.36 0.51 0.95
16 1.36 1.02 1.9116 1.36 1.02 1.91
16 1.36 2,03 3.8216 1,36 2.03 3.82
16 4.06 0.51 0.3216 4.06 0.51 0,32
16 4.06 1.52 0,95
16 4.06 2,54 1.59
16 4.06 0.51 0,32
16 4.06 1.52 0.95
16 4.06 2.54 1,59
LeftHand Seehi8 or Sic v c Mach r C0max Fig.
Right Hand m/sec m2/sec 1/sec
0.29 LH 0.67 85 0.25 -0.269 -28385 21a0,29 RH 0.67 85 0.25 +0.275 +28276 21a
0.57 LH 0.67 85 0.25 -0.402 -36723 21b0.57 RH 0.67 85 0.25 +0.403 +40026 21b
1.14 LH 0.67 85 0.25 -0.567 -39639 21c1.14 RH 0.67 85 0.25 +0.545 +45039 21c
0.29 LH 0.67 85 0.25 -0.580 -21634 21d0.29 RH 0.67 85 0.25 >+0,422 +19828 21d
0.86 LH 0.67 85 0.25 -1.120 -56758 21e
1,43 LH 0.67 85 0.25 -1.340 -66147 21f
0.29 LH 1.34 85 0.25 -0.499 -18690 21g
0.86 LH 1.34 85 0.25 -0.916 -49315 21h
1.43 LH 1.34 85 0.25 -1.201 -69724 21i
Table 6
Results for Counter-Rotating Down Pairs
Left Hand S/COt c h AR hi8 ordeg cms crns Right Hand
16 1.36 0.51 0.95 0.29 LH 0.6716 1.36 0.51 0.95 0.29 RH 0.67
16 1.36 1.02 1.91 0.57 LH 0.6716 1.36 1.02 1.91 0.57 RH 0.67
16 1.36 2.03 3.82 1.14 RH 0.67
16 4.06 0.51 0.32 0.29 RH 0.67
16 4.06 1.52 0.95 0.86 RH 0.67
16 4.06 2.54 1.59 1.43 RH 0.67
16 4.06 0.51 0.32 0.29 RH 1.34
16 4.06 1.52 0.95 0.86 RH 1.34
16 4.06 2.54 1.59 1.43 RH 1.34
+/- 5% +/- 7%
v c Mach -_ COmax Seem/sec m2/sec 1/sec Fig.
85 0.25 +0,280 +29840 22a85 0.25 -0.290 -31244 22a
85 0.25 +0.401 +30432 22b85 0.25 -0.371 -45129 22b
85 0.25 -0.494 -29692 22c
85 0.25 -0.516 -21336 22d
85 0.25 -1.064 -56439 22e
85 0.25 -1.326 -68872 22f
85 0.25 -0.465 -19842 22g
85 0.25 -0.917 -53157 22h
85 0.25 -1.215 -67317 22i
+/- 5% +/- 7%
NASAJCR--2001-211144 17
102 o 81 °
_":cmsX" "i'!!!_" !!'i"i;!'/':/' __
R-r ( \..: -" _.:-;: :" ';-'::: i : : \ normalized axial veloc'ty, _1 streamwlse vorticlty
2.92 _ / /
a) alpha = 8 degrees, circulation = -0.291 m2/sec, peak verticity = -18193 sec-1
104 °
0.00"_
R-r (cms)
,, ,,, "._i_--_'R ....... ,
,.![ /j,.,:.
'-i-:iii!ii)iiii!;;!//:. .-:::;::::::::::;:-::
3.18t;.-; ..................: . _
b) alpha = 12 degrees, circulation = -0.471 m21sec, peak vorticity = -24346 sec-1
o 81 °
• , ,,, ,: ,' ,\:, ,,, Ik_5_'.2_f/,,, ,v.. , ,,:;,_--_---{-_¢,,
(cms 0.91"o.94/
i.:-'1112::::::'. " - .- : .... :::-:--_--:.-.- . .
3.05 _- .I-
c) alpha = 16 degrees, circulation = -0.643 m2/sec, peak vorticity = -33189 sec-1
79° a
\ .. \
" -3250
c-1)
33
!02°--- _79° .4 4
' _ C_x. -350 (see-l)
350 "-3350
R-r (cms)
: ..... :::::;:;::...:: :
d) alpha = 20 degrees, circulation = -0.940 m2/sec, peak vorticity = -34055 sec-1
Figure 11 - Velocity and streamwise vorticity results for variation in airfoil
angle-of-attack.
NASA/CR--2001-211144 18
81° 4.. [.
\ o:_ _ \ ,oo -'--" \ .s,oo
TM --_,_, normalized axial velocit7 _ streamwise vortiii_ (sec'l)
vz / Vzc/ /
a) Mach = 0.250, circulation = -0.722 m2/sec, peak vorticity = -35714 sec-1
103 ° -._82° .t z
",t ttttt ,,'/R-r(cms) ' ' " ............ _'
:,:. ::1111---i!7_7!i : :3.05 _ _
b) Mach = 0.400, circulation = -1.149 m2/sec, peak vorticity = .56831 sec-1
103 °
0.00_
tttttt R-r '',, ' ttt:::::;''"
' ,, ',', ......... ;;,'$
, ', ',,:,!'"i--i!;7;7, ; ,"3.o5:J,_ ..............
82 °1
-800 (sec-l)
f f
c) Mach = 0.600, circulation = -1.724 m2/sec, peak vorticity = -84000 sec-1
Figure 12 - Velocity and streamwise vorticity results for variation in pipecore Mach number.
NASA/CR--2001-211144 19
°137
ms)l'_ , ., / t Z'"V' "_. ", _, ,
R-r{c , , , _ %-eL"%_ _ ' ,
z • t1 54
normalized axial velocity,
90 ° 30 m/see _ I vz / vzc I streamwtse vorticity
I.,, t N ....
: :2 - .... " " 200 "_'7-X"-32ooR-r (cms} _ -200 (sec-1)
t o.91_ t1.1o.... . • _
a) span = 0.21 cms, circulation = -0.156 m2/sec, peak vorticity = -20297 sec-1
95° 83°_. _ • , .j0.00-_1_" _
I ....... _.-,. 7 1 _ \_ i J -.i. \-_
/ , , , _ _-- ) ] , ' ' I / t._\\G / NI _V
i ......... : : ,' ' iI0.9__ i -3200' v\2001.I0"--_ ........... , "-'---
b) span = 0.51 cms, circulation = -0.240 m2/sec, peak vorticity = -22340 sec-1
95 ° 83 °._L__,.-L .-L._
c) span = 1.02 cms, circulation = -0.253 m2/sec, peak vorticity = -30881 sec-1
0.64"
R-r (cms
95 °
; : g 2,2__; ; .....
,_.tJ_4--_;ll _',. , _____ _ , , , ,
t t )_ \',_._--'=" )•,,.. ¢ i ' '
2.16""--'_
L
J__
v
:] :....
_1&..
<-3300
-300 (sec-1)
d) span = 1.52 cms, circulation = -0.297 m2/sec, peak vorticity = -33368 sec-1
1.14- -]- .L
..i--_'.--"7_
R-r(cm._" _ / _ /.'9_'__ _ , ,_ ztT_\ I .I.J. _ _ , ,#
2.67-
e) span = 2.03 cms, circulation = -0.307 m2/sec; peak vorticity = -38301 sec-1
.97 _ -400 (sec-1)
" _ __
Figure 13 - Velocity and streamwise vorticity results for variation in airfoilspan, with chord-length = 0.85 cms.
NASAJCR--2001-211144 20
90 ° 30 m/s_
0.00 I I.... _ 78 _
R-r(cms)[ i ! i i[/'_" '..,,.,,,/,..
normallzed axial vetoc?_y,
Vz / _c
r 0 91 t
. 0.94 --.....
streamwtse vo_Icity
_'_C_ _%300 {sec-1)
_ -33003OO
a) span = 0.34 cms, circulation = -0.223 m2/sec, peak vorticity = -29194 sec-193 ° ._
0oo# --, _ _ _24_u __.___ . J_ _.
1.10 "- - . , , ,
b) span = 0.51 cms, circulation = -0.342 m2/sec, peak vorticity = -23734 sec-1So
/H000-
: ....
, , , , t_2,A,/_ ' ,
_.4_.__-_-_... : :: , , ' ,
c) span = 0.81 cms, circulation = -0.315 m21sec, peak vorticity = -42509 sec-1
96° 8 o
000-'P-- _- "_T
R-r (cms:, I : . - " / .I _Z__._"-_x4 _, \ , , . I//.-_.
.... _" , , , _ f z_'_,_ _ , , , ' I_/ r'_
_: , , , ,_3'_--,/M"_, , , ' / _ \"_
d) span = 1.02 cms, circulation = -0.315 m2lsec, peak vorticity = -35415 sec-1
0.59-
i ,-" 7. ,4_ --'_'- "_ "-,-,
R-r (OTIs ; ) _
e) span = 1.52 cms, circulation = -0.429 m2/sec, peak vorticity = -37397 sec-1
,.4.
($ec-1)
Figure 14 - Velocity and streamwise vorticity results for variation in airfoil span,with chord-length = 1.36 cms.
NASA/CR--2001-211144 21
I064 I ..- .-_.---_--__ _8'_Ii I
I: A _..f_,, - I -_N\_/I_ o94
#1 t t S _ v V t I I t i normalizedaxialvelocity, l
yiiTi!i:i:';_-500 (see-l)
ova"°°
streamwise vo riiclty
f) span = 1.63 cms, circulation = -0.419 m2/sec, peak vorticity = -52295 sec-1
97 ° .83 °
R-rlc t I _\ I
g) span = 2.03 cms, circulation = .0.485 m2/sec, peak vorticity = -50676 sec-1
,, F t .,,,._,/_,/'7_'__.._-_'_-_ _.- " ", / 3450
' ' ' : _ _ft:t_,l_l,J-'J J I , , /
, , +llt)_(____i-_'j, , , __t t ? t t tt%_. "_-'_V//il I
flll%l%%%t__##'/ltt l
h) span = 2.54 cms, circulation = -0.428 m2/sec, peak vor_icity= -44003 sec-I
_883 °
,t:\ " 9!!\
97 °
R-,(cm_,Iil_!_:
3,56 _ X_,.,_.._,X_ "- _ .... #
i) span = 3.05 cms, circulation = -0.476 m2/sec, peak vorticity = -60375 sec-1
R'r(cms}! _ [; i it £ till _ \Izi_-- x.l_ #
,......... :: .... :3, __
_.___-,-_ ............ :: _
j) span = 3.56 cms, circulation = -0.446 m2/sec, peak vorticity = -63832 sec-1
-800 {sec-l),
Figure 14 (continued) - Velocity and streamwise vorticity results for variation inairfoil span, with chord-length = 1.36 cms.
NASAJCR--200 I-Z11144 22
0.00
R-r (cms:
to 30 m/sec
....... _ , q
...... o _ " -
1.27 .......
nonllaNzed a_ial velocity,
vz !vz¢ streamwise vortlctty
_c-1)
300 -3300
0.00" J
t/tZ
t t _ ?.'l'._ ' _\\.%_, _ " " " ", t t T'_.,C.W" 2_',_ i , , ,
152 _ _ _ ". ,. -. _ _ _ _ _ _ , _ _
a) span = 0,51 cms, circulation = -0.322 m2/sec, peak vorticity = -27772 sec-1
I
92 ° b) span = 1.02 cms, circulation = -0.435 m2/sec, peak vorticity = -43956 sec-1
;I• d
025 Jl
,, i z z_/fl _
1
"",,-,L
0
Figure 15 -Velocity and streamwise vorticity results for variation in airfoil span,with chord-length = 2.03 cms.
NASA/CR--200I-2I 1144 23
normal|zed axial velocity,
90 ° 30 m/see -----* I Vz / Vzc
1.40 Ip,_ _ _ _ 7E
?'2 aN'_
streamwlse vorticlty
f) span = 2.44 cms, circulation = -0.667 m2/sec, peak vorticity = -44619 sec-1
Figure 15 (continued) - Velocity and streamwise vorticity results for variation inairfoil span, with chord-length = 2.03 cms.
92 °
. _ _ _- _ _ -/" • I _ -3200
)
- - " {._"_ r%_- , I/_ kkk\_
...... :- "/,, .................................................1.i4 _ ..... " _ _ 0.o4_
a) span = 0.51 cms, circulation = -0.450 m2/sec, peak vorticity = -22914 sec-1
0.00
R-r (cms)
J
1.27 '_'-'t'-_ " - . . . 0 94
b) span = 0.64 cms, circulation = '-0.421 m21sec, peak vorticity = -31387 sec-1
.3,1_,-_-,,
94 °
c) span = 1,02 cms, circulation = -0,525 m21suc, peak vo_ci_ = -43913 sec-1
Figure 16 - Velocity and streamwise vorticity results for variation in airfoilspan, with chord-length = 2.54 cms.
NASAJCR--2001-21 t 144 24
93 °
0.51--_- I "_"_''_
/ .-" ..',7,.¢J_
/ I I 2 ,,,/",_t_(7.]/"---_
7_----_i_-_..... -_4
¢ 4
L
d) span = 1.52 cms, circulation = -0.626 m21sec, peak vorticity = -57104 sec-1
91°° l1.02 /_,/'__.__._...__ ,,--c_.,o,,
Z Z.dtg_.
2.54 i_i_-, v.._,.._._ , ,
e) span = 2.03 cms, circulation = -0.719 m2/se¢, peak vorticity = -73333 sec-1
1.40 912° 78" _ . .
f) span = 2.54 cms, circulation = -0.781 m2lsec, peak vorticity = 58750 sec-1
90' I I
_ _, _ .3,o0
g) span = 3.05 cms, circulation = -0.825 m21sec, peak vorticity = -42748 sec-I
R-r (cm
h) span = 3.56 cms, circulation : -0.862 m2/sec, peak vorticity = -46882
-4..
Figure 16 (continued) - Velocity and streamwise vorticity results for variationin airfoil span, with chord-length = 2.54 cms.
NASA/CR--2001-211144 25
911° 3o m/sec-_-*
R-r _(¢ms) \
normalized axial velocity,
Vz / Vzc I I streamwise vorticity
i) span = 4.57 cms, circulation = -0.866 m2/sec, peak vorticity = -70289 sec-1
Figure 16 (continued) - Velocity and streamwise vorticity results for variationin airfoil span, with chord-length = 2.54 cms.
a) span = 0.51 cms, circulation = -0.399 m21sec, peak vorticity = -23345 sec-1
c) span = 1.02 cms, circulation = -0.591 m2lsec, peak vorticity = -40099 sec-1
Figure 17 - Velocity and streamwise vorticity results for variation in airfoil span,with chord-length = 3.05 cms.
NAS A/CR--2001-211144 26
91 _0.36
I/£
I;
216_ \ x
30mlse_
I
normallzKI axial viloc try,
...,,
e) span = 1.83 cn _1 _ ion : :
1.02"_ _ 7 _
,_,<,r,,>lffIl_.._?_ ,,_
/
d) span = 1.52 cms, circulation = -0.704 m2/sec, peak vorticity = -53251 sec-1
93°
0,761 _ .,...,_ 5 _1
R-r (cms //
l
e) span = 1.83 cms, circulation = -0.776 m2/sec, peak vorticity = -59852 sec-1
R-r (cms)
f) span = 2.03 cms, circulation = -0.817 m2/sec, peak vorticity = -58695 sec-1
_treamwlse vortldt_
g) span = 3.66 cms, circulation = -0.985 m2/sec, peak vorticity = -35505 sec-1
.6o0 (it¢-1)
Figure 17 (continued) - Velocity and streamwise vorticity results for variationin airfoil span, with chord-length = 3.05 cms.
NASA/CR--2001-211144 27
1o norma zad ax a ve o_ity9 30rrd_---_ v _ - I _Ir_amwlsevc._t_ooot I _ t _ _ I ......
127.......... :-,,' .. _ _ -- "_,
a) span = 0.51 cms, circulation = -0.451 m2/sec, peak vorticity = -21114 sec-1
_ i_
'/,/ _ -
# i
b) span = 0.89 cms, circulation = -0.662 mllsec, peak vorticity = -36897 sec-1
mr(cms)l , _ ,"," [ ,si_,L_.V-,x;?_I, ,ttt7'L_'_._'-'-A _,i7_i_)'_ __I _ t t t 1' T _ <'qi_:_Z'_'sl_I/AI//_ _!i i I iI i_,
i, , ' t __ _K_"_'_--_5 _ / / / ,
9"P
000| " I ....
I, _ ,',,'ZZ,_f_-_-'_R-r fcms_/ , _ # .,* t F,.L l_ ,- , \k--_
l..10.:_t._l_L, I _ ""_ w
1 7 _/¢ J_wJtqY'_:l/*
,, ftT \N
__.._ _,.-._:.
c) span = 1.02 cms, circulation = -0.648 m2/sec, peak vorticity = -40085 sec-1
_ _,,_-_....
//1 /
d) span = 1.52 cms, circulation = -0.780 m2/sec, peak vorticlty -- -54079 sec-I
o _l_i ° I I
e) span = 2.03 cms, circulation = -0.944 m2/sec, peak vorticity = -5_139 sec-1
Figure 18 - Velocity and streamwise vorticity results for variation in airfoil span,with chord-length = 3.56 cms.
NASA/CR--200I-211144 28
92 _
O 89 t I ,_, ._.._.._
/ ._/_7-d,_
normalized axial velocity,
30 m/sec _ v z ! Vzc
_,___ ,¢I
f) span = 2.13 cms, circulation = -0.941 m2/sec, peak vorticity = -67331 sec-11 o
g) span = 4.27 cms, circulation = -1.170 m2/sec, peak vorticity = -45710 sec-1
2.92:
R-r (cms)
4 70 w
streamwlse vortlcity
/700
Figure 18 (continued) - Velocity and streamwise vorticity results for variation in
airfoil span, with chord-length = 3.56 cms.
89°
o.oo_ _ I' :-:::-_
_.2r'--_-', ::. " - - : ' ,, " _ "_--_-Z_T --'---- _ °°" - . , , , , 0.97%
a) span = 0.51 cms, clrculation = -0.429 m2/sec, peak vorticity = -19956 sec-1
ooo_ ----_ zso
-400 (sec.1)
_., (cr,_) : ¢ g_ 7 _-'-'v:_'.a".,'., ._ b " I //Jll_ )'_'_Vlll';h _ I It- _l_"h_, t t_-_ x/,_h&.,%_'??, i ,,\<,.,,,v---_////,,__..._ i _' _0
b) span = 1.02 cms, circulation = -0.713 m2/sec, peak vorticity = -39161 sec-1
Figure 19 - Velocity and streamwise vorticity results for variation in airfoilspan, with chord-length = 4.06 cms.
NASA/CR--2001-211144 29
89o025' _-J"-'L'-'----"---_ 30 mtsec ------_
t I
normalized axial velocity,
c) span = 1.52 cms, circulation = -0.824 m2lsec, peak vorticity = -50476 sec-1
8?°076 ..-"7_'--"----_
R-r (cms} _i _
2 _7 \ _--c_..._..._
d) span = 2.03 cms, circulation = -0.961 m2/sec, peak vorticity = -60834 sec-1
t271 910°
streamwise vorticity
0.97
I
e) span = 2.44 cms, circulation = -1.044 m2/sec, peak vorticity = -63493 sec-1
i
f) span = 2.54 cms, circulation = -1.045 m2/sec, peak vorticity = -66113 sec-1
Figure 19 (continued) - Velocity and streamwise vorticity results for variation inairfoil span, with chord-length = 4.06 cms.
NASA/CR--2001-211144 30
normalized axial velocity,
I 65_ "-'--_ , Vz ! Vzc _ity
i _--_-_----_. ._o _ -_ __ _,
!
g) span = 3.05 cms, circulation = -1.210 m21sec, peak vorticity = -66124 sec-1
911o I
19o___/__-_'-_----_ 764' \_---_,, I _ -__ _'-'-_
'7 (_4Jc-1)
h) span = 3.56 cms, circulation = -1.243 m2/sec, peak vorticity = -69638 sec-1
Figure 19 (continued) - Velocity and streamwise vorticity results for variation inairfoil span, with chord-length = 4.06 cms.
30 mtsec _ normalized axial velocity,
....._1 ° v z / Vzc
0.25 j "/
_ctL.:ngn_O:_:2_m2,sec_'--
peak vorticity = -35786 sec-l_
\/
-300 (_
streamwlse vorUclty
Right Hand Core:
circulation = -0.305 m21sec
peak vorticity = -32902 sec-1
a) double airfoil with chord-length = 1.36 cms, span = 1.02 cms.
Figure 20 - Velocity and streamwise vorticity results for airfoil pairs sheddingco-rotating vortices.
NASA/CR--2001-211144 31
30 m/sec
100 ° , _'_.__
?TZ" ,_
R-r (cms)
2,41""
Left Hand Core: _ __
circulation = -0.398 m2/sec _'_
peak vorticity = -35994 sec-1
/ streamwlse vortlclty
normalized axial velocity,
...1
.... sec.1)
Right Hand Core:
/% \ circulation = -0.437 m2/sec
_'_" peak vorticity = -55409 sec-,
b) double airfoil with chordlength = 1.36 cms, span = 2.03 cms.
Left Hand Core: Right Hand Core:circulation = -0.462 m2/sec
circulation = .0,397 m2/sec
peak vorticity = -17349 sec-1 peak vorticity = -18203 sec-1
c) chord-length = 4.06 cms, span = 0.51 cms.
Figure 20 (continued) - Velocity and streamwise vorticity results for airfoil
pairs shedding co-rotating vortices.
NASA/CR--2001-211144 32
104 ° _--)_-_ ._o _ i_"_ __ 30 mlsec -_> _ _...___//_ ..... ltzed axla[ veloc ty_
,_ii _ _-uhx,\\\'_
Left Hand Core:circulation = -0.667 m2/sec
peak vorticity = -40248 sec-1
streamwtse vorticity
Right Hand Core:circulation = -0,742 m2/sec
peak vorticity = -42782 sec-1
d) chord-length = 4.06 cms, span = 1.52 cms.
,-.,.,.,..,,.j
0.97
(
Left Hand Core:
circulation = -0.879 m2/sec
peak vorticity = -48733 sec-1
e) chord-length = 4.06 cms, span = 2.54 cms.
Right Hand Core:circulation = -0.954 m2/sec
peak vorticity = -55171 sec-1
Figure 20 (continued) - Velocity and streamwise vorticity results for airfoil
pairs shedding co-rotating vortices.
NASA/CR--2001-211144 33
normalized axial velocity,
89° 30 m/sec _ v z t Vzc 10"00_ _" "_ I streamwlse vortlcity
_ _ . . ; , , , , . _ 20o
f) right hand core, chord-length = 4.06 cms, span = 0.51 cms, circulation = -0.482 m2/sec, peak vorticity = -19853 sec-1
g) left hand core, chord-length = 4.06 cms, span = 0.51 cms, circulation = -0.480 m2/sec, peak vorticity = -19411 sec-1
90 °
051 %_.._..._._._ I I
R-r (crns) ;_? t _[_'i F_",_tt_'_,_ ! _'i'"k_ L " i_
2.29 _', -, .,. _,_, , / --
f/
h) right hand core, chord-length = 4.06 cms, span = 1.52 cms, circulation = -0.794 m21sec, peak vorticity = -43568 sec-1
'o_. °
2.16 "_ / /.._I-
i) left hand core, chord-length = 4.06 cms, span = 1.52 cms, circulation = -0.794 m2/sec, peak vorticity = 49131 sec-1
Figure 20 (continued) Velocity and streamwise vorticity results for airfoil pairsshedding co-rotating vortices.
NASA/CR--2001-211144 34
normalized axial velocity,
10o 30 m/sec _ ' I Vz ! vzc I I streamwlse vorttclty
140-_ z6o -- -- .I-coo(....,)_
j) right hand core, chord-length = 4.06 cms, span = 2.54 cms, circulation = -1.012 m2/sec, peak vorticity = -56314 sec-1
_')'_----> " -600 (sec-1)
l o2_" /#3Y_ L_-_<_ _"-\ "7-:d'_.9,'t '7 \ _ --_.,,00/7 o
__ ,,', _.__/./ .......
k) left hand core, chord-length = 4.06 cms, span = 2.54 eros, circulation = -1.021 ml/sec, peak vorticity = -62701 sec-1
Figure 20 (continued) - Velocity and streamwise vorticity results for airfoilpairs shedding co-rotating vortices.
99 ° _ _ 8_
o.0o_'m'--. • " _ _ I-_._ ;.._-_., , " "T'-
\ . : i i ; ,, ,,, \\ ' ',', ,,_'_,5.-_',, : ' \_..,, ,,-_ _ ! _---, .... .
R.r(cms)\ ,, ] L"" "] "] _ '0 : : _" "" : : .' "1_ _O't'---'---
_ . , , , ....... . . .
T- .o.,--cor.Left Hand Core" \ -3oo( ],3)o_0_ _':13circulation = -0.269 m2/sec "¢ oo circulation = 0.275 m2/sec
peak vorUcity = -28385 sec-1 / " peak vorticity = 28276 see-1
a) double airfoil with chord-length = 1.36 cms, span = 0.51 cms.
Figure 21 - Velocity and streamwise vorticity results for airfoil pairs sheddingcounter-rotating up vortices.
NASA/CR--200I-211144 35
Left Hand Core:circulation = -0.402 m2/sec
peak vorticity = -36723 sec-1
30 m/sec -------> 84o
\,, , , , ,-._--_ _.._I, ,
190 .-;''_" _"J
'_$ec-1__ / Right Hand Core:
3400-._(_\/t(]/__ [ circulation = 0.403 m2/sec
__._ peak vorticity = 40026 sac-1
</
b) double airfoil with chord-length = 1.36 cms, span = 1.02 cms.
98
1 27_
R-r (cms) .... /
3 0S..,..,_..--."-
84
J
c) double airfoil with chord-length = 1.36 cms, span = 2.03 cms,
left hand core: circulation = -0.567 m21sec, peak vorticity = -39639 sec-1,
right hand core: circulation = 0.545 m2/sec, peak vorticity = 45039 sec-1.
89 °
0.00 I i _ / i _ i
d) chord-length = 4.06 cms, span = 0.51 cms,
left hand core: circulation = -0.580 m2/sec, peak vorticity = -21634 sec-1
right hand core: circulation > 0.422 m2/sec, peak vorticity = 19828 sec-l.
Figure 21 (continued) - Velocity and streamwise vorticity results for airfoilpairs shedding counter-rotating up vortices.
NASA/CR--2001-211144 36
92°30 mlsec ---_
°64'/-'-"--_C'._/_" 1 / 0"9'0'9'_
non'nalized axial veloclty,
e) chord-length = 4.06 cms, span = 1.52 cms, circulation = -1.'120 m2/sec, peak vorticity = -56758 sec-1.
R-r (eros ( ._700 _"
3.56 _ " -700 (_c-) _-
f) chord-length = 4.06 cms, span = 2.54 cms, circulation = -1.340 m2/sec, peak vorticity = -66147 sec-l.
0 (30 89°
1.27 _ , . _ _ _ _ _ ,, _ _ 1_ 200
--..:::,',,g) chord-length = 4.06 cms, span = 0.51'cms circulation = -0.499 m2/sec, l leak vorticity = -18690 sec-l.
0.64 910° _ I I
_;l "IL
h) 91 span = -0.916, peak vorticity -49315 sec-1.chord-length = 4.06 cms, = 1.52 cms, circulation =
343 _"__ -- --
i) chord-length = 4,06 cms, span = 2,54 cms, circulation = -1.201 m2/sec, peak vorticity = -69724 sec-1.
Figure 21 (continued) - Velocity and streamwise vorticity results for airfoilpairs shedding counter-rotating up vortices.
NASA/CR--2001-211144 37
30 mlsec98 °
R-r(c'ms} - " " . . t t l- _ . • # t t
" " " . normalized axial velocity,
I 40 -'_'-"_ _ v z / Vzc
1"
0 q
Left Hand Core: __circulation = 0.280 m2/sec -300 {_._)
peak vorticity = 29840 sec-f 3300 streamwise vorticity .3300
/
a) double airfoil with chord-length = 1.36 cms, span = 0.51 cms.
96 °
R-r (cmsi
1.40_¢ / # f
Right Hand Core:
circulation = -0.290 m2/sec
peak vorticity = -31244 sec-1
_- r ¢ 1 _ iF /_ _ 0.91_
r , , , _ 0,97
Left Hand Core:
circulation = 0.401 m2/sec
peak vorticity = 30432 sec-1
Right Hand Core:circulation = -0.371 m2/secpeak vorticity = -45129 sec-1
b) double airfoil with chord-length = 1.36 cms, span = 1.02 cms.
94 °
, , , 11._._._t,.--2--_'---"_
'-'-"- W llb =°°
L
c) double airfoil with chord-length = 1.36 cms, span = 2.03 cms,right hand core: circulation = -0.494 m2/sec, peak vorticity = -29692 sec-1
Figure 22 - Velocity and streamwise vorticity results for airfoil pairs sheddingcounter-rotating down vortices.
NASA/CR--2001-211144 38
89 o normalized axial velocity,
O,O0.i.-_J_,_ 30 mlsec _ J_.--...._Vz ! Vzc I streamwlse vortlclty
d) right hand core, chord-length = 4.06 cms, span = 0.51 cms, circulation = -0.516 m2/sec, peak vorticity = -21336 sec-1
000
R-r(cmsl
t90-
92 °
S_
I
e) right hand core, chord-length = 4.06 cms, span = 1.52 cms, circulation = -1.064 m2/sec, peak vorticity = -56439 sec-1
9I °
1.02
R-r (cms)
2.
"-_90t _ 1.000.97
1.03 ,_
I
-7 0 (see-l)
f) right hand core, chord-length = 4.06 cms, span = 2.54 cms, circulation = -1.326 m2/sec, peak vorticity = -68872 sec-1
000_5 /
_-'<_'1_ _ ,_,' ' ,' '- , : / _--C -'--_ _ I _-_"--_-_ /I( "_i:,':,'/ ' ",' ' ,' /_-_-___----_ - I _ =°°#/" " _ - - . ; " " , , ' ' , " T _ _ • _32o0
g) right hand core, chord-length = 4.06 cms, span = 0.51 ¢ms, circulation = -0.465 m2lsec, peak vorticity = -19842 sec-1
Figure 22 (continued) - Velocity and streamwise vorticity results for airfoil pairs
shedding counter-rotating down vortices.
NASA/CR--2001-211144 39
90 °
0.13 I30 mlse¢
Jl
normalized axial velocity,
I Vz !Vzc streamwlse vorticity
-500 tsec-1)
h) right hand core, chord-length = 4.06 cms, span = 1.52 cms, circulation = -0.917 m2lsec, peak vortlclty = -53157 sec-1
1 1 °"-
R-r (cms)2.9 _ __ir_f
i) right hand core, chord-length = 4.06 cms, span = 2.54 cms, circulation = -1.215 m2/sec, peak vorttclly = -67317 sec-1
Figure 22 (continued) - Velocity and streamwise vorticity results for airfoil pairsshedding counter-rotating down vortices.
Circulation (absolute value)1.0 ......... , ......... , ......... , ......... , ..... •,.. •
pipe core vetocII3, = 85 mJsec 0 ,."
airfoil aspect ratio = 0.64 .
span-to-boundary layer thickness = 0.57 • •
0.8 •,"
m2 •.,•
se¢ 0 • •
0.6 "_
o• •
0.4 ,"
0.2 ""•," _ Equation (6)
4"
e "O. . ...... I ......... i ......... i ......... z ........
0.00 0.10 0.20 0.30 0.40
alpha (radlans)
Peak Vorticity (absolute value)8000(_ ......... , ......... , ......... , ......... , • _ ......
pipe core velocity = 85 mlsec ,"
1 airfoil aspect ratio = 0.64 . •
se_ span-to-boundary layer thickness = 0 57 ."
Equation (22) _ • •
• @ 0
• " 0
0.00 0.10 0.20 0.30 0.40alpha (radlans)
6000(
4000C
20000
00.50
Figure 23 - The variation of circulation and peak vorticity versus airfoil angle-of-attack alpha,with span-to-boundary layer thickness and airfoil aspect ratio held constant.
L50
NASAJCR--2001-211144 40
Circulation Peak Vorticity
2.0, , , , 100000 , , _ . • •
alpha = I6 degrees alpha = t6 degrees •"
airfoil aspect ratio = 064 0 airfoil aspect ratio = 064 •span-to-boundaq,' layer thickness = 06 • • 0
span-to-boundar? layer thickness = 0.6 80000 •
1._ 1 ••
m_2 . ",e-_ , •" Equation (22) _• •"
sec o .. -" 60000
o o o O O !t O1.C ,•
s ••
." 40000 , •
0 ., s •• 0
0,_ . Equation (6) . •s
20000 ••o
• s
s
0,0 '- ............. 0 ' .... ' i .... , ....
50 100 150 200 0 50 100 150 200
Vcz meters/sec Vcz meters/sec
Figure 24 - The variation of circulation and peak vorticity versus pipe core velocity, with airfoilaspect ratio held constant.
Circulation Peak Vorticity
' ' ' i 80000 , , ,1,0 I, alpha = 16 degrees I- alpha = 16 degrees
t pipe core velocity = 85 m/see ] -1- L pipe core velooty = 85 re:seet span-to-boundary layer thickness = 057 sec span-lo-boundar/layer thickness = 057J
0,8 _
ttt 60000 1 "'''''--
0.6 • " ""
• • o ,, o o "''-..,1Equation(22)• O
• 40000 , oo o
• %_ o o -..__
0.4 ". / ¢
_ o
Equation (6) _"" " -.... ¢ 200001-
0,2 "''''- _"" ...... e
! "l
! .;t0.0 ......... ' ......... L ......... , ......... , .......... , ......... , ' .........
1 2 3 4 1 2 3
Airfoil Aspect Ratio Airfoil Aspect Ratio
Figure 25 - The variation of circulation and peak vorticity versus airfoil aspect ratio, withspan-to-boundary layer thickness held constant.
Circulation Peak Vorticity
1.0 , , . ., . . , , 80000, , , ,
alpha = t6 degrees *f> -I _.= 85 nVsec • _ ..I_ i
airfoilpipecOreaspectVelocitYratio= 1 53 / 4 sec I" " " " " "" "_
0.8 _ -I Equation (22)_,." .<_.
m2 ,, _ 600001" ," ¢ " "
sec ¢ _ , o -..0.6 ,, 1 *,"
0," 4 400001" '
I • -I i #
• ,-4 ii"4
0.4 o ,''"" Equation (6) _,'
o • ] 20000P
,/ alpha = 16 degrees• --i0.2 _," t pipe Core velocity = 85 m,'sec
,s• ] / airfoil aspecl ratio = 1.53
• -I a,
0"00 1 2 1 2
Span-to-Boundary Layer Thickness Span.to-Boundary Layer Thickness
Figure 26 - The variation of circulation and peak vorticity versus span-to-boundary layerthickness ratio, with airfoil aspect ratio held constant.
NASA/CR--2001-211144 41
Circulation
1.0
0.8
sec
0.6
0.4
0.2
0.0
Circulation
1.0
0.8
m.__E5ec
0.6
o.4!
0.2
0.0
0.0
Circulation
1.0
0,8
sec
0.6
0,4
0,2
0.0 T0
alpha = 16 degrees
pipe core velocity = 85 m/sec
span-to-boundary Fayer thickness = 029
oo oo
eC,
Equation (6) "_ ........................
I . , . , ..j.. . . . . i ....
1 Airfoil Aspect Ratio
Peak Vorticlty
80000=
.1_sec
600001
40000
20000
02
0
atpha = 16 degrees
pipe core vetocffy = 8.5 mlsec
span-to-boundary layer thickness = 0.29
¢
O ¢ o ¢
oo •
1Airfoil Aspect Ratio
Figure 27 - The variation of circulation and peak vorticity versus airfoil aspect ratio, with
span-to-boundary layer thickness held constant.
Peak Vorticity
' ' ' ' -S' 80000 n , _ , ,
alpha 16 degrees • i 31pha = 16 degreespipe core wlocity = 85 m;sec • --1-- pipe core velocff_,, = 85 mtsec ....
alrfoil aspect ratio = 06,1 ,,'* sec I airfoil aspect ratio = 0.64 _ " " "
,, 60000 I"o "" _ Equation (22)
o • ,e ¢
,'_ Equation •¢ (6) 40000 I" ,, • o• • O
¢ •t •
,• 0 O •Os" OO • •
• 20000 }- 0 "
o •# ,•4
¢ •." ,,_ s
-_ s. _• . i , i i I i i u I , n i I . . .
0.2 0.4 0.6 0.8 1.0 0.0 0,2 0.4 0.6 0.8 1.0
Span-to-Boundary Layer Thickness Span-to-B0undary Layer Thickness
Figure 28 - The variation of circulation and peak vorticity versus span-to-boundary layerthickness ratio, with airfoil aspect ratio held constant.
Peak Vortictty
i i n n
_ 80000alpha = 16 degrees
t alpha = 16 degrees .j=. s - _ _ pipe Core velocity = 85 m/sec,_ pipe core velocity = 85 m!sec • _,
span-to-boundary layer [hickness = 0.86 SSC _ • span-to-boundary layer thickness = 0.86
8ooooi "-¢ 0 "*,*
o "-. _ Equation (22)
40000;" o " " " "
¢
Equation ( _'" ,, o..... 20000p
.... 0; . . . i ' ' ' I,2 4 6 6
Airfoil Aspect Ratio Airfoil Aspect Ratio
Figure 29 - The variation of circulation and peak vorticity versus airfoil aspect ratio, withspan-to-boundary layer thickness held constant.
NASA/CR--2001-211144 42
L
! -
Circulation
1.5 ,
2sec
1.0
0.5
0.0
0.0
• . . , .... 1
alpha = I6 degre4_s
pipe core velocity = 85 rn:sec
airfoil aspect ratio = 3.06
Circulation
1.0
0.8
m____sec
0.6
0,4
0.2
0.00
!. e"
J
• "" _ Equation (6)
.o S •
o . •
s S
Peak Vorticity
80000 .... , , , ,
alpha = 16 degrees
-L pipe core ,,el•tit>, = 85 re:seesec airfoil aspe_ ratio = 3 06
60000
O
40000 •,, " -.,. ,_
i I
20000 /"""_ Equation (22)
/
_. f*._ ..... 0 _ _• ...., ........ , ........ ....
0.5 1.0 1.5 2.0 2.5 0,0 0.5 1.0 1.5 2.0 2.5
Span-to-Boundary Layer Thickness Span-to-Boundary Layer Thickness
Figure 30 - The variation of circulation and peak vorticity versus span4o-boundary layerthickness ratio, with airfoil aspect ratio held constant.
Peak Vorticity
. , , , 80000 , , ' ' ' ' ,
O_ alpha = 16 degrees O alpha = 16 degrees
I pipe core velocity, = 85 m/sec _1_ " " _ pipe core, velOCity = 85 mfsE_
t span-to-boundar_ layer thickness = I.14 sec ;_ _ span-to-boundary- layer thickness = 1 14
60000 _ o o "•
• i •_ o
_-. o 40000 - _ ""
• - ' Equation (22) _ o
Equation (65 "/'""" " -. o
" "" 20000
I i , 0 _ I I2 4 6 - 4 6 8
Airfoil Aspect Ratio Airfoil Aspect Ratio
Figure 31 - The variation of circulation and peak vorticity versus airfoil aspect ratio, withspan-to-boundary layer thickness held constant.
40
v 0
30 - vg _a_- _o \meters/second
20--'L,
10- i_'--'
circulation = 0.700 meters;second 2
peak vorticl b' = 36000 second" 1
V0(rl ) vg(r2)
V0max V0max
0rl r2
r, meters-10 = [ I-0.010 0.000 0.010 0.020 0.030 0.040
Figure 32 - A plot of angular velocity versus radial location for a Lamb vortex withgiven descriptors (dashed line). The velocity ratio _' is realized at both rl and r2.
i :
NASA/CR--2001-211144 43
REPORT DOCUMENTATION PAGE FormApprovedOMB No. 0704-0188
Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing dala sources,
I gathering and maintaining the data needed, and completing and reviewing the collection of information, Send commenls regarding this burden estimate or any other aspect of thisi collection of information, Including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson
Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0t88), Washington, DC 20503,
1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORTTYPE AND DATES COVERED
August 2001 Final Contractor Report
4. TITLE AND SUBTITLE
Initial Circulation and Peak Vorticity Behavior of Vortices Shed
From Airfoil Vortex Generators
6. AUTHOR(S)
Bruce J. Wendt
7. PERFORMING ORGAI_IZATION NAME(S) AND ADDRESS(ES)
Modern Technologies Corporation
7530 Lucerne Drive
Islander Two, Suite 206
Middleburg Heights, Ohio 44130
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)
National Aeronautics and Space Administration
Washington, DC 20546-0001
5. FUNDING NUMBERS
WU-708-73-10-00
NAS3-99138
8. PERFORMING ORGANIZATIONREPORT NUMBER
E-12996
10. SPONSORING/MONITORINGAGENCY REPORTNUMBER
NASA CR--2001-211144
11. SUPPLEMENTARY NOTES
Project Manager, Tom Biesiadny, Turbomachinery and Propulsion Systems Division, NASA Glenn Research Center,
organization code 5850, 2 t 6--433-3967.
12a. DISTRIBUTION/AVAILABILITY STATEMENT
Unclassified - Unlimited
Subject Category: 02 Distribution: Nonstandard
Available electronically at hnp://gltrs,_rc.n_a._ov/GLTRS
This publication is available from the NASA Center for AeroSpace Information, 301_521-0390.
12b. DISTRIBUTION CODE
13. ABSTRACT (Maximum 200 words)
An extensive parametric study of vortices shed from airfoil vortex generators has been conducted to determine the dependence of initial vortex circulation
and peak vorticity on elements of the airfoil geometry and impinging flow conditions. These elements include the airfoil angle-of-attack, chord-length,
span, aspect ratio, local boundar}j layer thickness, and free-stream Much number. In addition, the influence of airfoil-to-airfoil spacing on the circulation
and peak vorticity has been examined for pairs of co-rotating and counter-rotating vortices. The vortex generators were symmetric airfoils having a
NACA-0012 cross-sectional profile. These airfoils were mounted either in isolation, or in pairs, on the surface of a straight pipe. The turbulent boundary
layer thickness to pipe radius ratio was about 17 percent. The circulation and peak vorticity data were derived from cross-plane velocity measurements
acquired with a seven-hole probe at one chord-length downstream of the airfoil trailing edge location. The circulation is observed to be proportional tothe free-stream Much number, the angle-of-attack, and the span-to-boundary layer thickness ratio. With these parameters held constant, the circulation is
observed to fall off in monotonic fashion with increasing airfoil aspect ratio. The peak vorticity is also observed to be proportional to the free-stream
Mach number, the akrfoll angle-of-attack, and the span-to-boundary layer thickness ratio. Unlike circulation, however, the peak vorticity is observed to
increase with increasing aspect ratio, reaching a peak value at an aspect ratio of about 2.0 before falling off again at higher values of aspect ratio. Co-
rotating vortices shed from closely-spaced pairs of airfoils have values of circulation and peak vorticity under those values found for vortices shed from
isolated airfoils of the same geometry. Conversely, counter-rotating vortices show enhanced values of circulation and peak vorticity when compared to
values obtained in isolation. The circulation may be accurately modeled with an expression based on Prandtl's relationship between finite alrfoiI
circulation and airfoil geometry. A correlation for the peak vorticity has been derived from a consereation relationship equating the moment at the airfoil
tip to the rate of angular momentum production of the shed vortex, modeled as a Lamb (ideal viscous) vortex. This technique provides excellent
qualitative agreement to the observed behavior of peak vorticity for low aspect ratio airfoils typically used as vortex generators.
14. SUBJECT TERMS
Vortices; Vortex generators; Three-dimensional boundary layer
17. SECURITY CLASSIFICATIONOF REPORT
Unclassified
NSN 7540-01-280-5500
18. SECURITY CLASSIFICATIONOF THIS PAGE
Unclassified
19. SECURITY CLASSIFICATIONOF ABSTRACT
Unclassified
15. NUMBER OF PAGES
4916. PRICE CODE
20. LIMITATION OF ABSTRACT
Standard Form 298 (Rev. 2-89)
Prescribed by ANSI Std. Z39-1B
298-102
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