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