ad-782 013 dynamic noise produced in compressible … · a resume of the programme of theoretical...

-mm»'»»« iwywni ...,..T»»MT^fi I II.WIJHWI^piPW^^IW^WW^TB ■m.WM4iiniV"".ii->»",™MP«lll.'|IU ."J.J. 1. ' »H^llll».,II!1l|lpimUPil»llUl

AD-782 013

THE FLOW ABOUT AND BEHIND THREE DIMEN- SIONAL LIFTING SURFACES AND THE AERO- DYNAMIC NOISE PRODUCED IN COMPRESSIBLE SUBSONIC FLOWS

Josef Rom

Technion - Israel Institute of Technology

Prepared for:

Air Force Office of Scientific Research

31 July 1973

DISTRIBUTED BY:

KUn National Technical Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road, Springfield Va. 22151

'...:■■ fc-fV**^-«"

..^ ..(t. VL .^r......^■w.j, ..'-■: — --■ -■ ■ ■ ■ ■ ..:.:<.■.. r| „^ ittiia|■|.,^itrtjyMti1MrJ;^^>^^.J.-..^■ --^.. w-'- ■ - ^ ...■.^...^■.^■^^^MLäMMm

-i iiini»! j ■( i !■• i II mmmniumwv"- ■' »tv.ni.Hwwu 'w.mimii—mif.L-* WMI'lW!WlW^'»^Wl»RU,y"''^-'-'.™W'''»-ilMl''-''ll-l-1''''1^

UNCLASSIFIED SECURITY CLASSIFICATION Or" THIS PAC.I- (Wi»n (%!(« f.nletod)

REPORT DOCUMENTATION PAGE nEftORT.^UMBtfJ, -t A .1 rv WUJ,\ • in - ? 4 -1 0 dV

2 GOVT ACCESSION NO.

4. TITLE (mnd Sub««/»)

THE FLOW A30UT A,!D BEHIiJD THREE DIMENSICWAL

LIFTIfiG SURFACES A:iD THE AERODYiiAMIC NOISE PRODUCF IN COMPRFSSIRLF SIIRSHMTC rinus

jh. IZA ö/^ READ INSTRUCTIONS

BEFORE COMPLETING FORM J REClPIENfS CATALOG HUMBEH

5. TYPE OF REPORT ft PERIOD COVERED

INTERIM

7. AUTHORC«;

JOSEF ROM

9. PERFORMING ORGANIZATION NAME AND ADDRESS

TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY DEPT OF AERONAUTICAL ENGINEERING HAIFA, ISRAEL

II. CONTROLLING OFFICE NAME AND ADDRESS

AIR FORCE OFFICE OF SCIENTIFIC RESEARCH/NA 1400 WILSON BOULEVARD ARLINGTON, VIRGINIA 22209

M. MONITORING AGENCY NAME & ADDRESSf//c////«ranl from ConUoMinl OH\ca)

U D. PERFORMING ORG. REPORT NUMBER

6. CONTRACT OR GRANT NUMSERfiJ

AFOSR-71-2145

10. PROGRAM ELEMENT, PROJECT, TASK AREA ft WORK UNIT NUMBERS

681307 9781-01 61102F

12. REPORT DATE

July 1973 13. NUMBER OF PAGES

IS. SECURITY CLASS, (ol Ihlt tcporl)

UNCLASSIFIED

15«. DECLASSIFl CATION/DOWN GRADING SCHEDULE

16. DISTRIBUTION STATEMENT fo/(/ii* Kepor»;

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (ol lh» •hstrecl tnlered In Block 20, II dllltrtnl horn Rtparl)

IB. SUPPLEMENTARY NOTES

19. KEY WORDS fCor.l'nut on reverse tide II neceeamry and IdentUy by block number)

AERODYNAMICS TRAILING VORTICES LIFT DISTRIBUTION WING TIPS WAKES

"NATIONAL TECHNICAL INFORMATION SERVICE

U S Department of Cnrnmcrrc Springfield VA 22151

20. ABSTRACT (Continue on reverse side It necessary and Identity by block number)

The work is proceeding along two separate paths which it is hoped eventually to amalgamate. The first investigation consists of numerical work on the rolling up of the wing trailing vortex system when it is represented by discrete vortice: according to a series of models of increasing complexity starting from the two- dimensional elliptic distribution as originally trrjted by Westwater and proceeding by stages through various lifting-line und lifting-surface models,

including some exhibiting non-linear lift characteristics. The second project is a detailed investigation of the flow field near wing tips and wake edges,

DD FORM I JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE '. UNCLASSIFIED IP?

SECURITY CLASSIFICATION ÜF ^HIS PAGE fttjien Data Entered)

iiiifiiiwmiMttiiliiil[fiilliWlüliüliiaiiii(iiiW'J''' ■ ' ' ' ' I '—^■■'t**** I^MllllMIII ilTlllil "-■-^- -i... ^.■^■.I-...^.^:-W^...L ,,■•,.,.,,,,v-ir«^^

iiUll^pifUiailpî.uiWJiWJilWJ'i»W.w »ji|H«iiu).^.^^!p.w » ^MI?i.nmm>mM*f .îiiy^pwjiByiipffBW!WWH|WH!Pff?wv^Wf.Mj» I"|mgp WmmW'■ ".^!'W»1^JiBWU^HWH'MJH'.lW.iW"iW>,lWWWW.^F-^1 'fflJPWT»'^!11pi ifflMPiflyHWiwiffyf;ff.i|HWi"yJl.iTHl!i^.».,1 .■ "^. »j»ii.Tyw^ywuri. i i .HUM'«"l^lfHIB

UNCLASSIFIED

«CUHITY CLAII'ÎCATION Of THIS F*oer>«^Ji D»<« ".nt.r.d)

including thickness effects. It is felt that the precise flow field near wing tips and the wake edges must be understood if the rolling up and structure of the trailing-vortex cores is to be properly studied.

Id UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS P*GE(Wh»n Dmim Enltrmd)

-•-'■" ■ ,.,-.-. .J-...;.„..'.^.,.-.^ , .^^..^.^w.i...-.-..>-^..^..iâi»^^^. -- ..■. ;... „n ■mtiiilll'-1-^-- ■■ - ;--- '■'-'-■'■■■-'-'-'■.■■■- :--■ ^^■...■.».... ■..■. .......,A^*Z^*lai***lUIAUmViM*a*UÜlSim

g»»T.-,lJ».»iii|'lliiJli|iNill»»ni|. ' i.ii.,. I-.M ...IIIPI.-B

^,„,„,...„,„.,.,,.■...1.1. li»..! r^^m^~^.,^^*.m''>'^>^.. M. ... —

81

1-1-

(a) SUMMARY OF THE RESEARCH PROGRftM

l- Wing Tip Vortices Studies

The method for the calculation of lift distribution and the

near vortex wake described in the previous progress reports have

been generalized for the calculation which includes both low aspect

ratio and high aspect ratio wings. Additional computer programs are

being developed to account for the vortex structure in the trailing

wake. Results for linear lift variation with angle of attack were

obtained. Calculations for the nonlinear lift variation for the case

of straight trailing vortices have been completed for the case of

straight shedded vortices. Good agreement between the calculations

and experimental data is obtained.

Calculations using a finite diameter core in the discreet vortices

has been progranmed and computations of lift distributions and vortex

trail are underway.

The calculation 'f the lift distribution on a circular wing

discussed in PR 6 is progressing. Some of the computer programs are

already working and results are being obtained. The results are

being compared with exact calculations for circular wings.

This program is described in Ref. 3, which is presents : as

Appendix A to this report.

2. Aerodynamic Noise Studies

A new jet experimental set up is in its final stages of con-

struction. Upon completion of the system, the experimental program

of noise measurements will be continued.

„. -.-: - - • -— — '""■■' —

m*i MWWW.HW^wwiip

/-2-

3(a) RESEARCH PLAN FOR NEXT PERIOD

1. Calculation of the lift distribution on a circular wing

planform using the potential equation developed in Ref.

1.

2. Calculation of the rolling up of a finite thickness

vortex layer behind a wing including the nonlinear

lift variation.

3. Completion of the construction of the system for measure-

ments of the noise generated by a subsonic swirling jet.

(b) PUBLICATION UNDER PRESENT GRANT

1. H. Portnoy and J. Rom, "The Flow Near the Tip and

Wake Edges of a Lifting Wing with Trailing-Edge

Separation", T.A.E. Report No. 132, August, 1971.

2. J. Rom and C. Zorea, "The Calculation of the Lift

Distribution and the Near Vortex Wake Behind High

and Low Aspect Ratio Wings in Subsonic Flow", T.A.E.

Report No. 168, December, 1972.

3. J. Rom,, H. Portnoy and C. Zorea, "Investigations

into the Formation of Wing-Tip Vortices"to be

presented at Aeromech 41, England, Sept. 17-21, 1973.

■ ^i^..:^-..^,,^,.^,.^: -.,.... ■..,..^.:.: -■......„^.^■■»^„.^^ - - — -- - -■' -.^«to^tMMI»

i»yinj»wiwji,i.i,a.iiipiiiiii.„ii ii. j iii mi»wiiw!*wiiiwap|pBiiiigwpi u iiij ■■ '■^■ nwwipwiww.i.unn.i iiiiii.iiJiiiJiiiiwwfwpipi

I.

INVESTIGATIONS INTO THE FORMATION OF WING-TIP VORTICES

by

Jo Rom, Ho Portnoy and C. Zorea

SUMMARY

A resume of the programme of theoretical investigations into wing-

tip vortex formation which is now in progress in the Department of

Aeronautical Engineering at the Technion,Haifa, is giveno

The work is proceeding along two separate paths which it is hoped

eventually to amalgamate» The first investigation consists of numerical

work on the rolling up of the wing trailing vortex system when it is

represented by discrete vortices according to a series of models of

increasing complexity starting from the two-dimensional elliptic distri-

bution as originally treated by Westwater and proceeding by stages

through various lifting-line and lifting-surface models, including some

exhibiting non-linear lift characteristics.

The second project is a detailed investigation of the flow field

near wing tips and wake edges, including thickness effects» It is felt

that the precise flow field near wing tips and the wake edges must be

understood If the rolling up and structrue of the trailing-vortex cores

is to be properly studied»

Thia

.i^^>..^-...t..^.-^>;.l,..^^l^-j^ .^^^^^^..vi.^**^^.^..,.-..- ,;-..■■..,....,.-, ,/.-.j.ûiî.ÎJ:A|Vl|-,-,fl|r. -fm |■ ! j |■ |if ii^Miil

mmm,ini:w m.«,im>rw.WV . lllll..llll|»lllWWWil»lppf»lWBII|l|iaiimil|l|l|ll I iII,^iMP»Jiilllill»W»ilP|»IJi.i.lilMIIWITW^«fiqi^||llllljB»BpappBpW^ (W,'"»'"1"«»i.»"m MMWHI

2 -

INTRODUCTION

The present programme of research is limed at resolving some of the

problems existing at the moment in the calculation of the form of the

vortex wake behind a lifting wing and ii s influence on the wing pressure

distribution»

The reasons for thd project are primarily connected with the need

for more accurate knowledge of the wake, because of the increasing

importance of interactions between aircraft in crowded airlanes

and the need to minimize these interactions by accelerating the dissipation

of the trailing vortices (see figure 1)• An important secondary consideration

is the more accurate calcul ';.:n of wing characteristics, especially

non-linear effects, by use of a more realistic trailing-vortex model

than the traditional planar waket

Part 1 - Calculations of the Wake Shape and the Associated Pressure Distribution

Using Discrete Vortices

In this part of the work the aim has been to build up a series of

computer programmes starting from simplified models of the vortex system

which have been dealt with by previous authors and proceeding gradually

to more realistic representations; at each stage examining the various

alternatives and eliminating the practical difficulties, as far as possible,

prior to proceeding further

i-iinrrmniiir'-- r-''iiiiniii-iifnl-i r ,■■"■-■- -■--■.»..--....-(■-,..>—...:.. -r..-- ■-■-^MBiiiiiili r HI •ii'iii if-f I'liiiT-iii'ili'inin iri-fiiiiiiii- tirnnin ■rill'rirrtrMlilliii^liillllliiMMlll^'^"*""^'^'*"*^'*"'

-TMP NM*» ... wmwmmwmmi'MVBWW ii11] ..immi..:iim'i>*mK\>immi.mmmitimtmimmmi''*r'.H''"'-"" mm^"""^

- 3 -

(a) Two-dimensional Wake Model with Elliptic Spanwise Circulation Distribution

(2) This programme reproduces the early work of Westwater in which

the flow in a fixed stationary plane far behind a moving wing with elliptic

circulation distribution is identified with the problem of the unsteady

motion of a two-dimensional array of point vortices moving under their

own mutual influences (see figure 2)„

This basic programme enables us to investigate many of the practical

problems which occur in all discrete vortex models: whether to use equal

strength or equally spaced vortices, the problem of "cross-over" of the

vortex trajectories "escape" of the tip vortex and the development

of irregular shape of the sheet (see figures 5,6 and 7),

These problems are basically due to the use of a concentrated

point vortex model with its associated velocity singularities, plus

accumulative effects of rounding-off errors etc. The solutions of these

difficulties were found to lie in a proper choice of the number and

spacing of the vortices and the size of the time intervals (corresponding

to the donwnstream separation of the planes at which the vorte< pattern

is calculated step by step) „ The correct caüainations were found by

a series of numerical experiments» It is important to note that due to

the causes mentioned, too many vortices or stations can lead to deterioration

of the results rather than convergence on a limiting form.

ifîirinrlir"' .. ,.,,a-.<-i:.^, -i^j.>^^,^.-.in.l..;ua,>î.l:^ i-amVHIfl"1"'''-" - ■-—.-■■^»aMnlftlt'i™. , n < - - "• -'JJ ■■^..-.-■■-■^■^^•IÎJMI/Milto

HUPPHHK« ... •

- 4 -

At a later stage in the work the two-dimensional programme was used

to develop the following idea. The actual wake has thickness and is not

an infinitesimally thin sheet. A better approximation to the wake than an

array of point vortices is, therefore, an array of Rankine vortices. If we

assume, therefore, a core diameter fixed by the conditions of constant

vorticity per unit cross-sectional area of the core (the density being

calculated, say, by using the method of Spreiter and Sacks to estimate

the ultimate diameters of the final pair of cores) , we can impose the

condition that, as soon as two cores touch or intersect, the two vortices

are replaced by a single one at the centre of gravity of the pair, with

core diameter fixed by the same constant density condition. This procedure

overcomes the difficulties caused by too close approach of the vortex

centres, i.e. "intersection" and "escape", it limits the number of vortices

automatically, thus helping to eliminate other irregularities, it preserves

the distance of the centre of gravity of each half of the array from

(6) the wing plane of symmetry (which is required by Betz's theorem for

strictly two-dimensional motions but applies fairly well to the three-

dimensional wakes examined later, also) and it finally ensures that, when

all the vortices cure amalgamated far downstream, we end up with a pair

with the correct diameters as predicted by the ultimate-core-size theory.

- — - ■ i ii «IIIUM i urn«» -

<1WI" "—"

- 5 -

(b) Lifting-line Model with Elliptic Spanwise Circulation Distribution

In the next stage of the work a wing with an elliptic circulation

distribution was dealt with, employing a single bound vortex in the wing

quarter-chord line (fig. 2). The trailing vortices were continued straight

from this line to the traling edge and, from the trailing edge backwards,

their rolling up was taken, as a first approximation, to be given by

the results of section (a). The corrections necessary to this wake shape

to conform with the lifting-line vortex pattern were then calculated by

a series of iterations, assuming the elliptic spanwise circulation

distribution unchanged throughout. The experience gained in stage (a)

was utilized to select the number of vortices etc and at a later

stage the finite-core concept was incorporated here too,

A practical point worth mentioning is as follows» The rolling-up

calculation was only carried as far as ten semi-spans downstream of

the trailing edge. However, a correction for the influence of the portion

of the wake downstream of this station was incorporated. At the ninth

and tenth stations the centre of gravity of the vnrtrx-wake

cross-section (in the half-span domain) was found and a line through these

two points was taken to define the position and direction of a single

replacement trailing vortex for each half wing, starting from the tenth

station extending to infinity and having the appropriate ultimate

"•""•--"■---'"''" -——- ..-.^U.::^.^.:^^*M**~ I - -

■" ' mumm" 1 i"—■■' ppHnmiii

- 6 -

strength (i.e. equal to the value of the mid-span circulation). The

induced velocity due to this pair of send-in finite vortices was

taken as an appropriate correction for the influence of the portion of

wake downstream of the tenth station on the flow upstream of it.

This method was employed in all subsequent stages.

A comparison of results form this programme and that in section

(a) is shown in figure 8.

■

(c) Model Employing a Single Sheet Vortex Lattice

At this stage a lifting-surface theory was introduced to enable

us to deal with general wing shapes with some accuracy (see fig. 3) .

The standard vortex-lattice technique was employed to

start with. The wing was divided into cells by a number of equally spaced

streamwise lines and by either equally-spaced constant-percentage-chord

lines oi by lines perpendicular to the stream, as shown in figure 3a. A

bound vortex element was placed at the "quarter-chord" of each box and

a control point was established at the middle of its "three-quarter-chord"

line. The usual trailing vortices were taken to spring from each bound

vortex element and to continue straight downstream in the wing plane,

and the unknown vortex strengths were found, in the usual way, by evaluating

the downwash at each control point, equatiw^ it to the value given by

the boundary conditions and solving the resultant system of linear equations<

. JV-"i-liBH<l*H!^*BÜ,JHi*ri;5W«^1^ft»j'.n

■ ■ nrniniiiMi ■MJMMT-- tmmm

P|lpi|pPIPpiPPmi|PII.II I ■MWJIHIUIUI iill.JH I I. II IIWI -. . ■ JLiWil-l,^H-. ^■<J'»»MM-W.^. 1.11. -.www

I The wing aerodynamic coefficients were also found from the results, to

a linear approximation.

A first approximation to the rolling up was now found by taking

the actual spanwise distribution of circulation found (which will generally

not be elliptic) and doing a two-dimensional calculation of rolling up,

as was done in section (a) for the elliptic case. This was now taken

as the first approximation to the rolled-up wake shape starting from the

trailing edge. The finite-core model has also been incorporated into

this calculation, using a vortex density based on Spreiter and Sack's

work :.nd utilizing the C calculated from the linear vortex-lattice D.

(12) 1

solution ' . As in section (b) , the first approximation to the wake

shape warf then improved iterativelv using the calculated vortex strengths.

During these iterations the vortex strengths were kept constant. The

next stage involved keeping the wake shape constant, re-calculating

the vortex strengths and the aerodynamic coefficients and adjusting the

vortex core sizes to suit. We then returned to recalculation of the

sheet shape by iterations, keeping the vortex strengths fixod, and

so on. Some wake shapes calculated by this method are shown in figures

18 and 19.

ii HI rrr-iin-tn». B Ml ■"■■'•" ■■:.^.:.-.-.i..^.......^-. . i..—^-.,,.,—.-..- .— im jiii»i Hi i- ■ rrnr i ...iiiirri ■aniiiiiiiiii -1 i i^-imii»!.»

- 8 -

(d) Model Employing a Multi-Layered Vortex Lattice

In order to deal with wings of all types, including those with

leading-edge separation, a multi-layered vortex lattice model was

now introduced as the starting point in place of the planar version

(see figure 4). This employs the assumption utilized by Bollay

(14) and Gersten that the trailing vortices do not leave the wing

parallel to its mean plane, but that they immediately leave at an

angle — a to the plane and then extend in straight lines to

infinity. The system of bound-vortex elements was set up exactly as in

the previous section and the vortex strengths and wing properties

were found by the process described there. In this case, however,

the results have been found to be sensitive to the choice of swept

or unswept bound vortices. Unswept vortices have been found to give

results closer to these found from experiment. In dealing with the rolling

up the first approximation to the wake shape was not found by the earlier

two-dimensional method, but the calculated vortex system with straight trailing

vortices was allowed to act on itself to produce the first distortion,

followed by re-calculation of the vortex strengths usi'g the distorted

wake shape and then re-calculation of the wake shape etc. This last process

is still, in fact, in its trial stages and the difficulties of intersection

and "escape" of the vortices have been encountered very strongly because of

the close vertical separation of the trailing vortices from neighboring

layers of bound vortices. The finite-core method is to be introduced into

this programme to overcome these troubles.

«iwm

U - ■ - -.-.--■ .. ._ , ^^^.„.Mfca^MMJMIMli

- 9 -

It is hoped that the final version will be able to reproduce

automatically an adequate version of the correct vortex configuration

for wings of any shape and aspect ratio.

A number of graphical comparisons of the aerodynamic coefficients

as predicted by the various methods described here and by previous

work, both theoretical and experimental, are shown in figures 9-17 and

in figure 20 the vortex centre position is shown for a rectangular wing.

Part 2 - The Flow Near the Tip and Wake Edge of a Lifting Wing with

Trailing-Edge Separation

In order to be able to calculate the dissipation behaviour of the

trailing vortex cores properly a knowledge of their inner structure

is required. The central part of each core consists of vorticity

emanating from the flow near the w,\ng tip and the neighbouring wake edge.

However, it is precisely in the region of the tip and near the

wake edge that the results of linearised theory are most unsatisfactory

(15) because of the singular behaviour of the solution there, Jordan

has recently elucidated the nature of the complicated singularity at the

tip point whilst at the wake edge we have the familiar singularity due

to incompressible flow round a sharp edge.

ayknii«. - .■>. IWMWI-IVTJ^JH

^^..i.^ ,^L:,^:■.:.â^-i-U^^-^^^t^^L..^^*..^:*.<-..â^f.;^^■....^^^f^D.^^^^.„.....■--^■îUiâ^..i, '„^ ^khrinfrflvtH^Vi,h..;-. ■.■-.^:„v- ... *,—»^„it.-,,;^,>,-■..^^.-^■^»a^-^.-.î r^-v*.:^ül.^,W.-ttf.fek-Uî -' ■ I 1' ■ 'i ^määtillMtliaälÎlk

-10 -

In this section of the work the wake-edge and tip singularities

were removed by taking into account the finite thickness of the wing

and the effective (displacement) thickness of the wing boundary layer and

wake. The method of matched asymptotic expansions was used to match

an inner, two-dimensional solution with the outer linearised result so

as to obtain a uniformly valid expression (see figures 21 and 22) . This

is combined with earlier results doing the same thing for the rounded

leading edge singularity to finally obtain a uniformly valid first

order result for the wing potential in all parts of the flow field-

The final result is shown in Fig. 23.

Work is now in progress on programming this result to obtain the

flew field about the wing and the unrolled wake to a uniform first-order

approximation. At a later stage it is hoped to combine the results of the

two sections to obtain an idea of the effect of wing thickness and boundary

layer development on the rolled up wake structure.

H^., ,■■ ■.||,^,.„,^.-....î.[|l^^.::ü„'^v!i|:Vflf--:^/..^.':^-^^^^î^î' '■■ tit^WWiatftVtte mug^nmni i giîig^ngQggiMin iuMiiwHiiiaul

»■wirwMt

-11-

1., Wetmore, J.W» and Reeder, J.Po, Aircraft vortex wakes in relation

to termnal operations, NASA TN D-1777, April 1963.

2 Westwater, F„Lc, Rolling up of the surface of discontinuity behind an

aerofoil of finite span» British AoR.C, R&M 1692, 1935»

3o Hackett, JoEc and Evans, MoRo Vortex wakes behind high-lift wings,

AIAA Paper No, 69-740» CASI/AIAA Subsonic Aero and Hydro-Dynamics

Meeting, July 1969=

4„ Butter, DoJo and Hancock, GoJo, A numerical method for calculating

the trailing vortex system behind a swept wing at low speed- The

Aeronautical Journal of the Royal Aeronautical Society, August 1971«

5. Spreiter, J.R. and Sacks, AoHo, The rolling up of the trailing vortex

sheet and its effect on the downwash behind wings. Journal of the

Aeronautical Sciences, Vol., 18, No« 1, Jan« 1951, pp, 21-32-

6o Betz, A,,Applied aerofoil theory. Aerodynamic Theory (Vol. IV), WoF»

Durand, Berlin 1935„

7o Falkner, V,Mo, The calculation of aerodynamic loading of surfaces of

any shape, British AoReC. R&M 1910, 1943o

8o Falkner, V.,MoC The solution of lifting plane problems by vortex lattice

theory» British AoRoC. R&M 2591, London, 1947.

- - - —■ ■ -- -■- ■ ^...—~ ^^^,.^J^MMl^,— mmmlmmmmil^^

i

■ 12 -

9. Filkner, V.M., Calculated leadings due to incidence of a numbei of

straight and swept back wings« British A,R„C« R&M 2596, London 1948,

10. Moore, D.Wo, The discrete voitex approximation of a finite vortex

sheet, AFPSR-1804-69, TR 72-0034, Oct„ 1971.

11. Margason, RoJ„ and Lamat, J-E^, Vortex lattice fortran program

for estimating subsonic aerodynamic characteristics of complex

planforms« NASA TN D 6142,, Feb., 1971.

12. Kaiman, T.PU, Giesing, J„Pr and Rodden, W.Po, Spanwise distribution

of induced drag in subsonic flow by the vortex lattice method. Journal

of Aircraft, Vole 7, No. 6, Nov-Dec« 1970, pp. 574-576„

13. Bollay, Wo, A Theory for rectangular wings of small aspect ratio.

Journal of the Aeronautical Sciences, Vol.. 4, Part 1, Jan. 1937,

pp. 294-296„

14. Gersten, K., A non-linear lifting surface theory specially for low

aspect ratio wings, A1AA Journal, Volo 1, No. 4, 1963, pp. 924-925.

15„ Peter F. Jordan, The parabolic wing tip in subsonic flow, AIAA Paper

No,, 71-10, Jan. 1971,

16, M., van Dykef Perturbation methods in fluid dynamics. Academic Press.

17,. Thwaites, B.„ Incompressible Aer'.dynamics, Fig.. VIII, 28A„ Oxford

Press, 1960.

- -■-.-.-.-^ ^......--»■.-.-.».-^...:--....-J....- ■-'•wiiflitBiiMiirrMtf^^

■ i

- 13 -

lü. Jones, R.T., in the Princeton Series on High Speed Aerodynamics,

Vol., 7, Part A. Fig» A7t, Princeton Press, 1960.

19., Tosti, L.P., Low-speed static stability and damping-in-roll characteristics

of some swept and unswept low-aspect ratio wings. NACA TN J468, 1947,

20. Bartlet, G.E., and Vidal, RoJ., Experimental investigation of

influence of tdge shape on the aerodynamic characteristics of

low aspect ratio wings at low speeds. Journal of the Aeronautical

Sciences, Vol. 22, No» 8, Aug. 1955, pp. 517 -533, 588.

21„ Chigier, N.A. and Corsiglia, VcR., Wind-Tunnel Studies of Wing

Wake Turbulence, American Inst« of Aeronautics and Astronautics,

Paper 72-41, IGth Aerospace Sciences Meeting, San Diego, California,

January 17-19, 1972. Also NASA TMX 62, 148, May 1972.

22. Legendre, R., Effect of wing tip shape on vortex sheet rolling up.

ONERA, La Recherche Aerospatiale, Brevds Informatins, July-August 1971,

227-228.

23. Razur, K. and Snyder, M., A ruview of the planform effects on the

low speed aerodynamic character.i sties of triangular and modified

triangular wings, NASA CR 421, April 1966.

..i.^.^:i^..-.^^---^...^~.^^^^..^^ —^,:^^.^^.;..\.^J.;-.T<-;ll^rtV.r.i^7i-»>H1 . ,i - ^ ;-,i r ■^-■^i^.,. -"'—' '■ —-""^

IM

l. ■

C u

e^

x

> Vj

H

a

to

Ü H

I

!</

BflaimAia i at, „ii i'-miiMriMBiaMii ^^■.-..„^...^■^.i—M—^--.^J.-.-.^^..».»^^-. ^-^.^ ^ - .-.-:- '■- ^^,^.^^.<~^*

i, ;

INPUT COMPUTATION SYSTEM

(a)

ELLIPTIC CIRCULATION

DISTRIBUTION

WINO GEOMETRY

(b)

ELLIPTIC CIRCULATION

DISTRIBUTION

OUTPUT

BOUND VORTEX SVSTEM

VORTICES INTERACTION

£1^1

10 ROLLED UP ViU-

DISTHIBUriON Willi :'l 2-T -r

BOUND VORTEX-SIMPL.rsüi.- 3D WARE

FIG 2 -ELLIPTIC LIFi VORTEX WAKE CALCULATIONS

IS

■ I I i-f N --'*' I ' ■ .^.■^„■^■^.^.^^■■^ —^ .^-..-^fi ,. | -^ ...--.f-^.^. i,, ■-. ■■■.■.-■ ^■^.^^^^„^...a.M.

INPUT COMPUTATION SYSTEM OUTPUT

A STEPl

WINO OeOMCTNV

STEP 2 CIRCULATION

DISTRIBUTION ON T.E.

STEP 3

CIRCULATION DISTRIBUTION ON WINO AND

20 ROLLED UP WAKE

STEP*

«0 ROLLED UP WAKE

STEPS Ißp NEW 'LINEAR' CIRCULATION

DISTRIBUTION

2 SiiF PLANAR :'/ WAKE

4/-/,/.S ,','

'syy/^

ID VORTICES INTERACTION

WINO AND fRAILINO WAKE

VORTICES INTERACTION

r^^

CIRCULATION DISTRIBUTION

VLM-20 ROLLED UP WAKE

VLM- JO ROLLED UP WAKE

MODIFIED VLM CALCULATION

10 ROLLED

UP WAKE

NEW 3D VORTICES INTERACTION

'LINEAR' CIRCULATION

DISTRIBUTION MODIFIED BV ROLIED-UP

VORTEX WAKE

MVLM ID ROLLED UP WARE

FIG 3 -VLM LIFT VORTEX WAKE CALCULATIONS

/fe

...„u^r—.-^.--^--^--^ ^„^^■^^^^-^-^^.^W.^^-^-^M»»^^^..-.— .— i . -1, | |imM|||

ll«,«rM.j»».. i....... --..-.^. 1,. ._. .„ -J-,. ..- ..^ ■- - -■■-'-^■~~- :■' ■ ....^^^^^^^^^MM^^ää^^^

STEP1

WINO OCOMETRY VORTICES PLANAR DEFLECTED

STRENGTHS WAKE

HP NON LINEAR LIFT OISTRIBÜTIOIJ WITH PLANAR WAKE

STEP 2 HP NON LINEAR CIRCULATION

DISTRIBUTION WlTH PLANAR

WAKE

STEP 3

3D NON LINEAR ROLLED UP

WAKE

3D NON LINEAR ROLLED UP 4AAE

NEW VORTICES STRENGTHS

CALCULATION CORRESPONDING

TO NON LINEAR ROLLED WAKE

FINAL 30 NON LINE6« ROLLED

UP WAKE

FIG A - NON LINEAR VORTEX WAKE CALCULATIONS

■ iJ.^,.-,...J^»-,.-,t.,„...,.-;1.-,.w^„„.-;..-.J.^^ .J,.-..„-^.,1.^.^^^-...,.^....Jt-J..,>.jJ..^il-l|l|lTllll ■ririniiiiiiBllllimii ii ir r - M mr -iiltiriiirliiiiiiTiifiriii ni i ^»MaiaMlHM^iMtMitlliiiltlii^^

■

0.60

0.40

0.20

0.00 «n x <

i N -0.20

-0.A0

-0.60

-0.80 A-

STATIONS: 3.8 b- 0

4 b - A

«.10-

M>8

N-10

0.00 0.20 O.AO 0.60 0.80 1.00 1.20 Y-AXIS

FIGURE 5 - PROBLEMS IN THE CALCULATIONS OF THE VORTEX SHEET ROJ.L UP "ESCAPE" OF THE "TIP" VORTEX.

/<?

..,^^..^.^—.,- -...n*.^.'-*-■. -^.:-,^.:,.,,.^.. -,....,> ,.^...J.Ji^-^..„.l... ..-......■..■ ...:...,....>^-.^- . .„■: . ^

0.38

0.18

-0.02 -

x <

i IM

-0.22 -

-0.A2 -

-0.62 -

STATIONS: 3.5b- a Ab- a

o( •10*, M-104» N>10

0.00 0.20 0.A0 0.60

Y-AXIS

0.80

FIGURE 6 - PROBLEMS IN THE CALCULATIONS OF THE VORTEX SHEET ROLL-UP

CROSS-OVER OF VORTEX LINES.

ao

"-- "' ■ .,,,i,-..-.....^..-......J.„......... .,..,......,.J,......- ^........^..IJ. .i ^M,.^^..^ ^.,-a.jj,-»aailia^tit»«— ■'■•■i" - '•■■

■ ......i:w..-„---,..,,.> .,.--.,.■:■..„ ^ . ..-,,....„^^.^J^^ittyMiaiM||j||i|||jgMggMh. , ., M

I 0.38

0.18

-0.02

w -0.22 10 x <

i M

-0.A2

STATIONS: 35 b- a

4 b - A

oc •10% M.104, N.AO

0.00 0.20 0.40 0.60 0.80

Y-AXIS

''IGURE 7 - PROBLEMS IN THE CALCULATIONS OF THE VORTEX SHEET ROLL HI TOO MANY SUBDIVISIONS.

21

Ik. -i ni-fitiit irrfHniMfc.nr— -: —--.,^^^ . -.-. ■ ■,.J.,....,.^-^.. lim«'"- -■-■■ -■ ^ ■ . ..—^v.^~~^—^.-^^—M»J

'■■'■>■,■ ...

mm

' !:'

OHJI H W

0-

i

oj »«'I

» *

S o

CT O r>l O C --t ro (N

3 II II II

^ Ö S Z

| (0 4J U

n

61XÜ/

U M

s

«A ■U •H C D C

11] X

>1 x> c 4) >

•H CT

(U

to 0)

SS

M D in a

a p4

w z M O a H

Q

O H

EH iJ

I

co

s D Ü

*■'*'******* «V

T^tm.. LlrfWH-l-niir.irMJ!^!..!! .. —- ^- -■ -- ■ - ■ , ,■ , ,,,.,.a-k-a^^—.-,.. -.., ■......,....:.. ,.:....»J..^^„.iJ.,^. trt- iMiii« ■ miMMMMiiiiiiiiiili».

-^^^^" »lilllM »luiliiwiu, l|lTM.|ll!ll||»JiWMlll]l.lp»IIIIIW^WWWI^^ ■mwm,m,mjE.w<mmm«.rmwm^m

14

1.2

1.0

0.8

0.6

0.A

0.2

CL

PRESENT NLVLH RESULTS

(N, .2, N .10)

(Nc.2 , N .30)

WINTERS EXPERIMENTS

BOLLAY'S THEORY (Ref. 13)

10 20 30 AO 50 of

L'lt.UKE 9 -- TIC'; LIFT COEFFICIENT CALCULATED BY THE NLVLM PK'-OK.'Vi WITH RESULTS PRESENTED IN REF. 13, FOR RECTANGUJ-Ai. ' 11 AR = 1/30.

&

■ M&HMrti MM&m&wfjjJüQjjjQ

MjîiäUJkî. im-,-^rftA^t-^-^-^^^--^ ■ ^^w ^^..^..^^^^^^^îrMiiiärtttiüiriiii-iiiir^-rniiliiiri-'iiflili'iMtiir--^--—^^— -^-.^—-^ wmmmämtmmmmmmti

«2MSU ■i111« "■ vmmm^^m^mmmmmmm „,^™ra>»»^w<^w.««w<»»«'.'»r>*<«»M«1

PRESENT VLM (2x20)

V HOLME

A ZIMMERMAN

0 WINTER

0 JACOBS t CLAY

■ GOODMAN ft BREWER

Q LETKO ft GOODMAN

A JACOBS

• PRANOTL ft BETZ

FIGURE 10 - THE LIFT CURVE SLOPE CALCULATED BY THE VLM PROGRAM. COHPnRlSOU v.Tr; EXPERIMENTAL RESULTS (REF.17, fft ) FOR RECTANGULAR WINGS.

M

tftttfiWlMMMMlteiWi •■"' -■-"■--^'■---'•-—"•-"-— L.^.^****^ MMM^iHtfiMMa

fmmmtmmmmmmm "- "" mmmmmfmmm

<

2' 15

> ^ :-- X o h ö

Lr >> a «*M

jj a > Ü3 LJ f-- »: I

< K3 CJ r^ * , u F J r:'

;'"■»

,.-, >•» A--

tu c, H < v.1 k-l r-' T"! til .,; l'^' U3 »-■'

a

- I-

a', ^

rj

O

w X

^r

..^...■w..,.—».— J„_>^^—.-—......—..—.»^^^— —,^— ^.^^—^^

t/e LE. R*f. o 0.092 round f$ A 0 round 19

—PRESENT NLVLM (Nt.4, N^«20)

tIGURE 12a - THE LIFT COEFFICIENT OF A RECTANGULAR WING OF AR

X

•™»*i*ait*sa,jSmj

i|-Mimiliiillri<i^riii1lili«»^iiMiliiWiiiitnllrt1Mifiia.yiiiiiiiMiliiriilrri-.riiii,.i -rnnurt iitrn^liyiiMilMltMMi.i,.! »ii.ii.ri(hMtMifiirHyifiiiaiii lilliliiir ——^ miiiifflairi

WWPIWWI.IH»I''H ' ii., jiLinpiumi •■Wfi»Bn»W*WjW*'.i»«tri.,v,.,1.,

m»impiiiwi,i)>*^^wmww" i PHHWW wm i.»wiwpiwwwwwff.i^'j|iujipiii.iiiiiij-iwi)ffi'»1111 mm '".".fiiiiin

t/c L.E Rtf. 1 0 0.092 round in 1

1 A 0 round 14 1

PRESENT NLVLM | 1 (NC.A,N5.20) 1

FIGURE 12b -- THE LIFT COEFFICIENT OF A RECTANGULAR WING AR

57

t^,.«.,,.....„,,^,..>it'..;at;.-av^^ ,.....- ...., Ana»*»^....:.,.,... ..,...,...-^.-. ...-.,.....,:...,.„.. .il...|..1..tm[)t^MB|g||M|g|||||||||j||||^^

■> j " !iii"miBwiiminww|^iii|'i'i|i|lll'!»'^''WB»w." n wm i "u»«««»»»»!!!!«»! jf^NVPH ■|^^i.ul!m^WJ.VWmi4muK.)UJII|i|IMimii>WJMRfWRqAlll^wyi>lWIWlili^^^^^ l|.iliillJRIII,«|RH

A «45

1.2

1.0

Q8 -

'VK \Ai

O GERSTEN ( R«f. 14 )

— PRESENT NLVLH

(Nc.2 ; Ns«10)

0.6

i.4

<r i i

' \ LINEAR THEORY

0.2

1 1 1 10 15 20 25

üivE na - THE LIFT COEFFICIENT OF A SWEPT BACK RECTANGULAR WING OF AR = i

n

-

* ■ - «mav^rfrfa

'--"-■•-— '■ " :■■*•■■■*-*• -../:.>,^.i^...J.^-.,~.^-.,-,-i.-,„...,~~—»„.v.^»^..^-.!.,^..-^..^. ri itrHilimiiijlli' - -^ '- —.■-.- >-.-- J.- ...-J^.. i- ■.., ■- ....^--.^..^^«..».v»--....-. .

■www...»- i iwiipiminiiipipppipiip W.iWiJAIilWJ'"!l^!W41iîWJMÎ.!ii^l|»NMWiipiJiliPJ)WU.)]llii|i|)tfl^^

...... .. ■ • ■ ■ ■ ■■ ■-

■"•^

t/c L.E. Ret. 0 Q.092 round H

A 0 sharo 19

— PRESENT NLVLM (NC.*|N...20)

FIZ'JRE 13 b - THE LIFT COEFFICIENT OF SWEPTBACK RECTANGULAR WING OF AR » 2.

J?

.. .,M.„ ,.. ^ ■ - ■■•'"•"- aM«—Mriri i - - imîin» 1—M —Mi

WPHWP m^immmrmm »nwwmwwiwwi^w VW^'^^'.ijiiiiiu.injiiNiiiiiumi

1 I | i

§ g U

C3

< E-i

w Q

« O

o

o —

D • " fa

> s W O

§

< 5

s I

D O

i '

(O'^O »t))*'',D

31 ■

■*■ ' "—'■' ■■ -■^-^■-..-.•■-■■-~.-^.-^—■^^-^^^.:-^^-^^^^..-^--^™^-^^^. ■ , - - ■ - —

•*•••- .,-",, •rynimw i MiiunTmiiniTmiHii .1 JIIII mim iiu —,,: .11 .1... i...i .,i........,„.,, )>ww.i^> Bimpwi '»■ ;'!..«iijiiiji.ii.iLJiiiimi.i

o a

5 5 > z o H

z V» o hi

Ul lil <4 (/) (A in ii < üJ tu

Q. OL z

<

in

0, u o

> w a

cn ö c, t, f- vr,

'.; *■

< w cj a:

u ta M

fa < o c

IX, (.J 1-1 .J l-

« w

u; o

o 7.' K K < C

o Ü

< [.J 0 *■ o

o

o

.^2 IMMM«mMlg|

a^..^.^^.j^^-.^M.f^Jrn ..■■...-..^^.)..J^;i...:.......1,^...J..,.,J. -■-.,,.mr' ^^^^ r..f...;„.. ^.^W^nn^^V.,!,,,.,. ., ■..V^^.^^...^-... .■:...^-_.. -^fmldjijaimn- || ..^^^^ .„..^ ^■.■■;. ^■■■^ .... ; gjgg

iiiliiiÛl.liWIWIIjW.pp.tJJ.ilWMWI'.W'iP'IWW^Wi^^PWM'P' iPMpffiiipiîff^W!^wpf»Mîi.iiff.ffîMWlffiTPwnii-JijM>.ii... 111n n,J.J . .■■ m j..wpiiw iiiwiwjiif.mwwui.mj qpWl«.?'*««»*' "»jr^BlfTM^-f «•nnmin«w^.w*f,,w.n:

0 EXPERIMENT (Ret.20)

PRESENT NLVLM

(Nc«2,N .20)

25 o(

FIGURE 16a - THE LIFT COEFFICIENT OF A DELTA WING OF AR

53

J.....,.,.,.;:;■,.,....,..., ,-,..^.»,l.»,-....-....,v... .. ,.,...--.,.; .- ^ -.. ^. ..^^^,^.1 --w.-.,..—..... , lim -. - . ..„.l -....^..^^J.iJa.»3^»l.M-.i-^J^rfM^äia

mmmmmmmm ■■ ■'■■ ■

,ll■l■,, l,,"l, *" ■■■^.^ ■ A * ...» ^^llllll in .,«,,

■ ■ ■ ■ ■ ■ ■. .

O EXPERIMENT (Ref.Zf )

PRESENT NLVL^

(Nc «2, N .20)

20 25

: GURE .ob - THE LIFT COEFFICIENT OF ;i DELTA WING OF AR = 2-

3y

. v...w^.....;^ ..*.,■ ..■.*.:~^ . ^^ ...■,1J. . ... --.- , — — i i . r —'- ' ■■■: -■ ;..J^Miaa*iciMMJ«^Mamiu^M1.

WW«"^«i*iPPPIIWPPMIW»^WP»ll«WPP™*P^^

1 I

...... . . .,

i

FIGURE 17 - THE LIFT COEFFICIENT OF A CROPPED DELTA WING OF AR = 0.78.

36

^■' ■;■—.->. ■-■■■ --- -.■■^^^,-,.,.. .-.--.- .^..■. .■-.,..-..^m******,.*. ,. ■.^■■. ..■■- ■ ■- —ll<JHJHIMti»illllilll 11

imiimwnm»"! ' ■ll1 rmmni' LJ UMiiiiwiiiuu.iii.iJ H*m*vmwm.*mmw*m". ■'- • '- -. .....M ....... .

l l

g i

H O O 3 ^ CN

I

II

v^

oi

•^v

Vi

t- IX \

-NT

U a £>

■-t

ii

?:

M

I

O <M H

as

O w » 1) g a in w

01

B p> I •■< o t». H~ u • O IM OS fi

s1 H 5K

I

00

i H tu

■ -■-■•-^- ■ ' --■.-.^».—.— — " ■ -

pry. .,". ]J[<l»l,ll)WI-:lll!iy.»AmBti.»l»"l . -T—^-T-? MlrtMlur ^^ ^ ^^ ^^^^ UllliJ Jiill .lWt.ll.WWlilWiff4>' (i»MWi'-S.^-¥pBPiWÄPi>"?^W.'1!lUiMiWVipJl.i!.i HI Ull!iHJlil,ii:Hl!iH'liWWMHpi^9|l|p,.l,il|l)ll!| .•.r.fM**M«M,IIWW*.. **■"" MMIMlllB

g1

no -t r ' (J

li .1 t,

0) er;

- 1^

<^

K

\i

OH L:

\;

,■>

K? ̂

.•\

cN8>

v5 rv

w M H

Ö &5

M 3

oJ en

s

!• 1 WO SIKH •/

nn»! in u uf »

ä

•4-1

c •H

r. > 0>

01

10 0)

4) M

S -iH a b

s ^ II

I a o

EH X

l

s

vO

^^

CO

■ia^t^J..^^.>,...^.-^..t™.,,.;^-^-,....- fljM. - ■ ■ - ■ --- ;-- ■- -■■-"■ ■■.■^■—^^-. •-"-"*■'■*'

mmfmmmfiFmmmm ' J«.! iiujmuiiiiiiiwu '*™*****<mm****mmmmmmim'immmi* MPgpppHlllllim 111 H MMMMMMHMQHNMHIMlh«

FIG.20-SPANWISE AND NORMAL LOCATION OF VORTEX CENTER

FOR RECTANGULAR WING-AR «5.33

iîwM-.l.TM.-r.^<iV.-f,t«^^m-M«M,>^^!...-.^.^..,.l,„.^,au,,.'..,.-.->,i. :.,...^...;-.,—■.,..-~.-.â-...,».......,......ii:,„,1,,„..,,

V'.»".y^JHIiîJJ-,

I" Ir'H MiairH-lllWl ....J.i.^.,ja».J.,..l,tl.-»,».u,i^4.i.^.n,-.J-.mJ.J...,.iJ,Jl..l,.;..,...,., .:..a...! ..,-■:.....^^.^-.i^glgmjûj!

... ........... mm mmmmmmmm* mm*^m*mmmmm*mmmmmmmm

H •■Wrr^nWi mmmmmmgs,: -ifteew,^ .

I 1

i

FiavRtti-

■5..-Y^r

general «frrea*«^

m~. , ■ . ..,-...— ..■■- - --■ ■'. ■■.^_..J -—~.., — .... .. ^■.J-.

ad-782 013 dynamic noise produced in compressible … · a resume of the programme of theoretical...

Documents