ui: - DTIC · HYDRONAUTICS, INCORPORATED LIFT OF FIGUINEO Figure. 1 -Lif't and Drag, Coeot'icionto of the liyaraloil in Wate~r Without InJoetion F"igure 2 -Couipar:Uion BEetwoon Moourotl

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'LiiJ

(~Sbion' s ruri.--! Ž.S for b ". .. . . ..- ] - - . /

HYDRONAUTICS, incorporatedresearch in hydrodynamics

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HYDRONAUTICS, INCORPORATED\~TEC11INICAL EPI;T_\77O3-1

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-ON THE~ CiANGES IN -LIFTOF HiYDRO~F0LS . - TO - uRFCO INJECTIO.Ns -7 POLI .E-R AII

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Technical Report 7703-14. TITLE (and Subftio) S TYPE OF REPORT & PERIOD COVERED

On the Changes in Lift of Hydrofoils Dueto Srfae Ijectonsof~Polyer ddiive Technical Reportto Srfae Ijectonsof olymr Aditve)6. PERFORMING ORG. REPORT NUMB3ER

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'IT R IOU TI STATEMENT io the abstract WO nteM Black 10, it d~ife. t toi Reot

I% %L4LEMEN'Ar-Y NQTES

14Fl *~' ORDS t(IIemitiIPot on%#* revere soio notevrvam wdio't e In,~ bloc flimnw'bot)

PI., ynier Alddtiveo PrL.'%stre 1)iotrlbutionsilt t 1".ffecCts 13ouiidary La yerDvag E~ffects Hydro±'oil

7o-A a ACT (4&,.IIIhwo om. Ne@# 8ld. It "Oveatvs~ *I'i Savfitv~i f blorii uiiImhor

i~~t:tigdttti on the oVf'octs on lif t L)f -ymnnetrie 'hydrofoilsdue to xh in~jection~ olf polymer solutiow, .n to thoir ~u aes

sho tht he i -t uit either decrease or increzise depn i i onthe polymor,' inject'lon vel1ocitys l~ocatiton of injection, side ofin~iootioti and anigle or' attack. A unified, albeit semiempirical,approach is sugg~eoted in the present report for correlating~ andexplaining the apparoiitly contradictory data. It is proposed

DD) 'jwv 1473 EWIflON or tNovo$ Is 0111601AC7C~iIPCIO O cl PED(se

S CURITY CLASSIFICATION OF THIS PAGE(IWhen Date Entered)

that the observed lift changes may be due to the influence of theinje~ctions oni the boundary-layer displacement effect that isknown to reduce the circulation around hydrofoils.

Four separate boundary-layer effects due to injection areidentified and discussed., namely those due to changes in theeffective angle of attack, the effective camber, the circulation(in addition to the changes in circulation associated with theprevious two effects), and the thickness distribution. Classicalmethods of airfoil theory are utilized to calculate the "effectivd'hydrofoil shape that would have produced the observed pressuredistributions. This effective shane is utilized to identify thedifferent boundary layer effects, and these., in turn, are iden-tical with the changes likely to be produced by polymer injections.,

No explicit treatment of the exact nature of the interaction'between the injected polymier flow and the boundary layer o'r týheexternal nressure gradient is given., since this is out-.1de thescope of thIe pres;ent study. Howeve r," it is argued that thisinteraction must necessarily be dependent on thevc-lsiproperties of the. rolymier. For exzvmple., it is shown that whenthe observed changes in the lift force are noxniv~iied ,,ithrespect to the locai velocity at th lotto ofteinetoslit, the data f'or Aiff'erent polymere,, U.t'Tervent injecv.ion

velocitie an iferent angtes of a t'tak cznn all be oovrelateziin terms of thle si n..le tiond~ime ni onal V1,,zuotr o V t ,.c here Vj ris the injec ion velofity, r iv the ticilmei' relaxatllon tncand 0is the hydrofoil ohord. A csimilar oorrelation is also gilvebn forthe changes in draG due to in~jection.

119C4RIlTy CLA51IVPCATION or VNIS PAGIVMIw" *)gge Vot60

HYDRONAUTICS, INCORPORATED

TABLE OF CONTENTS

Page

FOiREWORD * * * * * * * * * * * * * . * * * * * * . viii,

I. INTR.ODUCTION .. .. .. .. . .. . . ... . .. 1

1 ll.. F1ATU1JES OF THE EXPERIME'N'TAL RESULTS ANDl

OBJECTIVES OF THE STUDY . . . . . . . . . . . . . . 3

Ill. PLAUSIBLE M4ECHIANISMS FOR. LIFT CHANGES . . . . . . . 9

IV.* ShMI -W-PIRI CAL CORRIELATIONS FOR THE OBSERVEDEFFECTS . . . . . . . . . . . . . . . . . . . . . . 15

IV.1 Prezoure Diotribution and Lif't Eff~ects . . 15IV.*2 CorrQJlationo of' Lif't and Drag Data . . . . . 38

V. CONCLUDING RJ2MRKK . .. .. .. .. . .... . . 41

M "EENCE ... . . .. 4


LIFT OF FIGUINEO

Figure. 1 -Lif't and Drag, Coeot'icionto of the liyaraloil in

Wate~r Without InJoetion

F"igure 2 -Couipar:Uion BEetwoon Moourotl ACp V~ablos, and the

Results Of a 11boooti'cal Caloulation

F"igure 3 - f±Act 01, 10""", Chord 1I joetion 01, water ýand 1'ulymonr,oln the Driag CoofVtU cleot. 01' the llydriol,

F'igure, 4 - tt o Of 011ord lzij eeoti:or of Wterto andk 4"lymolr.Oil thot Dragjr 'ooC iln ol the ii'.ii'.foll

F ,,t I. r ) o El.iT'et of IU~Chord kni etli of Wa U.' r and PI. yin o

oilk the ifit. CueI'fIl.e out 01' Uthe Ulydvfoii

Figuro L) -Effect, vo t 30:1 ordi 111jet. I on %*..I, 'At.'.r :tnd~ Po Iynionroil toIltLftt Oooffl~eloiti o'. tht IlydiroiXolit

101,,llrcDi11twrince hIn Vro".uvo kX.ff~loLont,. A0 Vei'%Uo

000r tVI tt Itce olt .

1V1ouro 10. - ol' I. I o'.ln 1o t.Lwool Mtnoilu Je aili'C enIadPi~cur.

ond Ol posad -o'1130

Vi('ý,tUko lI In IlA.i 1 ".i' Diotrl'butilomo. f'or Voill AneLX o1'0'ailti

tolco . 1ri o.-ot Mo. tri bit tfiomo. L fore~~r Vol Ai lo I o


Figure 13 -Boundary-La-yer Displacement Effoct on a Symmetric

Hlydirofoil1

Fiou~re 14 -Effect of Polymecr Injection on Boundary-Layer Dis-

placemo.nt

"i rur 1 3 - Tpcs of Boundry-Layor Intoraction

V gui'o IJ~-) Calculatoki Camber Distrtbutionu

Ficurwo 17 -Li't. Chan111ge Vor&ss V iT /C

- ' Libeiang Vorou. V T /C

Pit-ure TO) - 1)rt(Chango Vo~rou, T/c

Vihawvk 2 - Io:n I)1'a4, Cooffiion't Coiiipcnuntot Iluo to ".olyoxI1J00tion at V Ar .1 H~ato


-iv-

LIST OF SYMBOLS3

cHdrofoil Chord

CD Drag Coefficient

CD Pressure Drag Comiponientp

CD Friction Drag Componient.f

CL Lift Coefficient

C Lift Coofficiont Qc -repuiding to Kutta-J-mouwgj½ value of the elrcuaUon

C Prossuro Coofilikn~'it~p

11 Se Equation )

k Re~duction~ ill ruttu duo to boundarly -Uq O.-,iew Ekquation [I]J

11Integr'al dt~tt'lnd b~y* 1EXjn tton )

VLocal Voloeo! ty oit hykh10t'o11 ziwi%'Wko

V 11luje.etion -Voetee Itv.

V voloett~y onl 111,111,014 e o f iy d 1o Co II

Vt V 0. loct' ty kil 4%44~' r.uCao f.t hyd rkilo 11

Duuy Variitblo

Cc~xiakke ii driot01 t4. IJaidwit 1.L.I

Hydofodil the ci' .ý 1 ~drIL~ibu abmbe l~ t ombm'1

I7-HYDRONAUTICS, INCORPORATED

a. Angle of' attack

CLof Effeetive angle of' attack A

I' Cim.,ulation

r' C2irculatiton e01'roxopQ1ding, to Kutta-Joukowsk! Conidition

6* owidary layer Lii.:plaeemoict tlilckne~s

60~ i~ lcn ont ei'ct oil eainbor linle

* ~ole-lotofe ,i lilo.wd;%rbto


-vi-

ABSTRACT

Existing data on the ef fects on lif t of symmetric hydro-

foils due to the injection of polymer solutions on to their

surfaces show that the lift can either decrease or increase

depending on the polymer., injection velocity, location of in-

jection, side of injection and angle of attack. A unified,

albeit serniempirical., approach is su&;es~ted in the tpresent re-

port for correlatiznZ and explaininG the apparently eontradic-

tory data. It is propozed that the observed lift ehangez may

be due to the influence ofi the Injections on the bou~ndary-

layer displacement ef iet tha~t ii~ known to. reduce the cirCu-

la~tion around hydrofol.

Four ~'para to bounda.ry-layer offects. duo to injection are.

identifieid and discussed,, namoly, thcwze due to. chneo 1i1 the

.effective anlgle ozf attack, thei effeoctive cmetecrua

tion (in addition to the chaigai -it% eireuiatlon aseoiated

with tho vroviouz two. -,offoctý;)s an~d thle -tiickne-z dietributlon.

Classica1 nwothodui oV airfo~il thecory are utiliz.ed to I=Iulate

the t"effoctive": hydrofoil j;hapa that would. havo produced tile

observed prozoure diotribu, -011!; Mao~ orl'otivo Alapt i~

utili-,ed to identify the different boundary. layer effoet;s, atdthese, in turn, are idcntibýal Ait the hnslkytob

produced by yolymer Inj ectiois.


-vii-

No explicit treatment of the exact nature of the inter-

action bet-ween the injected polymer flow and the boundary !ay.Žri

or the external pres sure gradient is given, since thiz iz out-

side the scope of the prejezlt study. However., it is argtzed t~at

this interact-lon must necozzarily be dependent on the vi1.co-

elastic proyertle; of the p olyiter, For example, It I.- 4hovlv

that when the okbseriod cha-ngez3 In the lift force ae '~l~~

with ro..ecet to thet loe-il ve-Qcity at the,, location cX -

jaction slit., t*,e data for oiiievnt~ ynrd

Jection velocitlej alld ditfforem nojez of at~tmek can' all10

correlated I.n Otemv af t Aln dm inal

v r/c ulhere V~ 131%!Oc~tloln volcý~:: i h

1-axatiocn ti- -t and c:i thQ hydrofoii ohord . 1 A.0"r

latioii j. a2zo: cio 'on., h ekn,-100 ti i~i "Irag due.t

i " ' " ' . " '. -, : .-•.:- -•!.• .•• ••. t• ,b• -

a m m m•-5. --. . . . . . . . . . . •, . ... .. .,- . ,,. .

'HYDRONAUTICS, INCORPORATED

-viii-

FOREWORD

The present report summarizes the work that has been carried

out over the last four years at HYDRONAUTICS, Incorporated, under

the support of the Office of Naval Research, on the lift and drag

effects that arise due to the injection of polymer additives on

to the surface of hydrofoils. The work has been primarily ex-

perimental and an effort is made herein to offer a unified, albeit- , snmiempirical, interpretation to the considerable body of data

that has been acquired.

Much of the early work on lift and drag effects were carried

out under the general direction of Mr. Marshall P. Tulin, and a

number of the correlations given in Section IV are based on theo-

retical concepts originated by him. Most of the pioneering efforts

at HYDRONAUTICS, Incorporated on lift effects on hydrofoils were

carried out by Dr. Daniel H. Fruman.

Technical monitoring for the program was provided by Messrs.

Stanley Doroff and Ralph Cooper of the Office of Naval Research.

HYDRONAUITICS, INCORPORATED

1. INTRODUMIO1N

It iolw t known 't-hat; iiuti (:llu'to polymeIllr aWddItlvo ar~e

prosont Ui thu exora 1.Clmwo ovu' t~wo-d:uimollolonl h1hydot'o.1t

they producte ('11 Lii both Ul~ dra(g mid 1,14I't. ohaLrtutwi'Lti

of hlykro Collt 1. ll.-owvor. 1111t tho drwg roi'duut;I'oz Itch I eyed~ io

-genera 1ly auu up twt a"' be.lu illodu to0 I t~hickteu114g Of~ t~hu vltwol:31111lay o'r "Wda.-ollnt. to the waiLi * t-11 11Y'totun Vor Utb e Ii V. iiyu

ti I oin.'r Iv 0 ,ii It. I I uAIM I ý' o .1 % al d ou it. I ma c to rY I 1 ('0 10u uxau o

untut.U' o C tho 1 .1. C t. e oe t..

SL~v tue lo e o~ar ly C Iid1. IY oil I ~ik ot ,I I.1 o%0 il.lc~' tion

ofV pulip ) Ullpolo e .e'tirl t. II adid IAAt , A 'oo I II t. lout o 1 01. "Ok AMd d-Loo .w ith p ropo IiIt'o andk hythrofol ao 11 01 hae luu reokte 1%,

Kowa I .%k L' Wol IT Ct' and ~Iitm 1,olumau 'amd Ii i Il k"i rky

Y rkIllu~ta ~I I daa Itd DIL"O'l, dtk I01,111all, 'hi' I'llt alld Al It

Kowa 1. ols Volink Qi %t~ t, pu. lyme r adIttt t Ivo keoodee rea ed Ii thilont. atid

ffClu letty . 011i tbte otite r handi Voralydr ol. It t-io'l w1.01 t- " yInyo 1

~and 0IIII 't Ia t I repo r t.tt (41 vt t I I Vl t .oli' I rlo I n~ ao~ IA 0'.; 01, e v '

dopenld I 1t; ivol I Whon t'hOPe tilt, p*1vloP 1me to j eu ted PIU 1,1 it 00Ole t~ionl

-kilL Iw t t'tn li.. proo on I v t,

01 t. t 41t.1kisk-~,11 Yt 111001', Witie rort' kit, be~'tt eatlo oV.

Natva I'e Rooovoll p ol tt ~dt tarhe optlonih I p 01' tilO~iteo ot t't' I.Vue o

atid~ I t,1vo~to llti, 1.1k I alot die t0 1'o It . t Vot'Ce oa t he%'C po lytue

have ooti t.01 red 41t'ttud Ulk.t.I~,aIltt t he uhant'~o I ti I, I I't andt

d ra'ag that. u nh rot'oot oto Iwheti Io Lytit adtl, t It 1Wý Itt, Q o t Odok~t'LtAal ly Lit to Li~t't'.wtrtva u ut Iooonth t i ur

fac~e o Up to da t~et.t t i~~ . totu 1M~V0 bttt'el prdom killaul. y


experimental, with the tests having been conducted on 10N and20-cm chord, symmuetric hydrof'oils in the HYDRONAUTICS High SpeedChannel. Measurements have included drag and lif't f'orces, 4swell as pressure distributions with and without injections . Theexperimental studieo have yielded a large body of' valuablo datawhich can assist in the understandin ot'theetet 1poye

additives on ex~ternal flown. The objective 0o' the research do-scribed in the present report -is to utiliso. this body oft data,as well as other data that tos available in the literature, to

attemp t to develop a theoretical ftramework which cani be used toanalyze and interpret the oh served offects.

V C In order to set the objectives of1 the present; report In aproper perspee tive, a bri.of discus sioii of the expe r lnent~aL Ahsurivationos, with omphasis on the most; re nt. reen Iv. Qs*: ftirs

given In S~ection 11 , t'o low ed by a detailed dese rip tion oVi oh-

J ectives . A discuosl on of' the var~ous phyviva mechnnisms ILhathave boen pos tul a td i~n the U.1. Le ro uy to vxp 1 a n the observedii xt ofrwo Ina o dicuossed In UAWot III. :lIin th I 1 IgIt o t' Uti,most roeent o xp erlmonetaL W.da. Tho mo nehanp.1 s t~hat i ons.e eto be, the most pilaus.Io no *a wel no a the voeasos or [ on'deinit- t~o be oo, are aWso Lisepibod In Sect Ion III .1 Asei-pr eiIanalysio lo. thon g I.von I n Sec t on ~IV at d Lo utilly1.o d to co rvreta Itho data. Finally , oome oone luding remarks are given In Bee tloV.

MESO


-3-

II. FEATURES OF THE EXPERIMENTAL RESULTS AND) OBJECTIVE8dO F T'HE STUDY

Most of the experimental studies conducted at HY1)RONAUTICS.,

Incorporated on the lift effects of polymer additiveo-were caririedoutwih 1-cm chord NACA 6,: -020, tw-dimensional hydrofoil

placed in the HYDIRONAUTICS' High Speed Chane (HSC). In theearly tests., the- forces on the model were measured., by the use 0ofblock gauges., for the cases withi and without inJection of -PolyoxWSR301las well as water (so as to provide a "baseline" conditionfor assessing the effect~s of the polymer alone). Thlese early

studies 6 '~, inicated that, under certain conditions,poyri-

jection not only leads to a drag, decrease but also to a lift i~n-crease, so that siniicn fnr'~ ith lit/d rag ra tiocould be realized. In general, it was found that a lift inorea-zoaccompanied a drag decrease whenl thet Polyller injection wa Sonl thesuction s ideo of the hydrofoilý, and that bohda dlf e

J., ~creased when the polymer injection wa. Oil the prossure, si:de of

the hydrofil

In an attompt to unders tand the phenomenolo.gy respons Ib:Le

f or theL obsorvod lift behavior. tests:" were undertakenl to measuIre

the pressure dis,:tributivons, Oil the hlydro.Voil .rei', the ses w I t

and withiout rolyiter Injection * These proll iiin, ' et dLnethat, pol-ymelr 1Lnjoection onl ono ourface (""llet~ion or 0r1,ue'~othe hydrof'oil cllmlgos the rssr dI,L tri~butlon onl both urae

tlough tho offout i"s m1ore, pro'nOlAcL'd1 On the '130 0urac Wn1which t,110injection I., made * It was al""o. found that the pressuokre i tr

but~ions, were ounsis tent with t1e li1"t f0Orces 1meas'ured by the1 b'Iokgautges; ta t 1,6, tho forces obt,,inkmed by :ner it;temauepressuure distributionl were the same, With in aWCe etod OXpcrimon101tiftI

error., as, those meoxsurud by tho blockt gaugos.

HYDRONAUTICS,-INCORPORATED

Based on these experimental results, Fruman, Tulin and Liu

concluded that the observed lift increases could not be exla:lned

in terms of changes in the boundary-layer-separation point, sin ce

the pressure distributions revealed that injection did not signifi-

-cantly alter the pressure distributions in the trailing-edge region. '

These authors argued that the. observed lift effect may not be re-

lated directly to the drag-reduction phenomena. Thliere were also

indications from the trends in 'the data* that the observed lfIteffect may be strongly dependent on the viscoelastic beehavior of

the polymer. One direct method of examining whether vi soela,.tic

effects are resconsible for the observed lift chang:es is 0to co-duct tests with different po!rmters of differ:in vis)e-ueast.-e

behavior under otherwise identical experimental conditions the

results of such tests are described in Reference o for tPhree

different polymers, namely, Polyox, Polyac ryl amide and Jtg...,

Since these results formed the basis for the analyses givev ii

the present -report, they are described briefly below.

As mcntlon&ed arl:le, all of 'tIle te.ts were conduct,, ,: th

a 10-cm chord symiiotric hyd.ro:"oil alnd were performeltd i tn te iU-

NAUTICS high Speod Channel (11 ) modl.ieTd to obta:i. a two-dimen-

sional. flow. The hydrofio:l1 ha two .00.l2.-c) wl.do .,U1j ctKo

slits., one located at 10% chord and Uhc other (on t0he ol o"'itIt

side) at 3O,0; chord. Each a of the Vol" onu

taps botween 18 to i'OT percenlt hord longLth. The drag,, a•id .. t 1

curves for the basic hydrofoil (withOut any im.11,1M,1.01 - arc sown

ill Figuroe .1 The slight a1,1ynunoty n1 tho hydrofoll cused by,

the presellee of t;he W0- alnd 30-peroent chord •.Iit onpposi, t,

sides. of the 1'oil, is responsible Vor, the olight aswyimioetrN noted

in the lift behavior,

*Thleuo aspects will be di eussed in more deta•l in Section ill.


-5-

Figure 2 shows a comparison between the measured pressure

distribution and the results of a calculation using potential

theory9 . While there is general agreement between the measured

and calculated pressure distributions, one feature that is note-

worthy is the relatively large discrepancy between the two dis-tributions near the trailing edge of the hydrofoil.

Lift, drag and pressure measurements were performed on the

hydrofoil under the following test conditions:

•a. Injected Fluids:

1. Water

2. 200 ppm Polyox WSR 301 (Union Carbide)

3. 350 ppm Polyacrylamide (Polyscience - Cat. #2806)

4. 1500 ppm Jaguar WPB (Stein, Hall & Co., Control

#23-0548).

b. Injection Velocity Ratios: Vi/V: 0.1 and 0.3, (V

11 meters/sec).

c. Angles of Attack; a: 2 0 35

d. Injection Sides: Suction (upper) and pressure (lower).

e. Injection Positions: 10% and 30% chord.

In all, tests were conducted for eighty different combina-

tions of test conditions. A sunuiiary of' the test results followo:

Drag (Figures 3 and 4)

Water injection results in a drag increase, or at most, a

slight drag reduction; on the other hand, polymer injection always

results in a drag reduction.

Lift (Figures 5 and 6)

Water injection always seems to produce a lift force in a

direction opposite to the injection side, the magnitude of this

*All of the experimental data are given in tabular form in Apprndix B.


-6-

lift change being relatively larger for Vi/V, .3 and 10% chord

injection cases. Polymer injection produces a lift force in

either the same direction as that of the injection or opposite

to it, depending upon the polymer (compare Jaguar and Polyox at

C0 angle of attack and 30% chord injection, Figure 6), the rate

of injection (compare 0.1 and 0.3 rates of injection for Polyox

at 00 angle of attack, Figure 5),the location of injection (com-

pare 10 and 30% chord locations for Polyox, pressure-side injec-

tion at 0.1 rate, Figures 5 and 6), and the angle of attack

(compare 00 and 2.50 with 50 angle of attack for the suction-

side injection of Polyox at Vi/V= .3 rate, Figure 5).

Pressure Distribution

The pressure distribution data were taken for all of the

different test conditions. Typical pressure distribution curves

are shown in Figures 7 and 8. The changre in the chordw.se pres-

sure distribution due to suction-side injections of Polyox and

Jaguar at a foil angle of 2.5° is plotted in Figure 7. The

hydrofoil has pressure taps only between 18 and 86% of its chord

length; nevertheless, the general trend is good enough to make

the following observations.

1. Polyox injection at Vi/VW 0.1 results in a pressure

decrease on most of the suction side and In a pressure increase

on the pressure side; hence, one would expect a lift increase.

On the other hand, Polyox injection at Vi/VA = 0.3 results in a

pressure increase on moat of the suction side and in a pressure

decrease on the pressure side; hence, one would expect a lift de-

crease. These observations are consistent with the results given

in Figure 5.

2. Jaguar injection at Vi/V= 0.1 as well as at 0.3 rate

results in a pressure decrease on the suction side and in a

A' , .A ., "-A3

* HYDRONAUTICS,JINCORPORATED

-7-

pressure increase on the pressure side; hence, a lift increase is

expected in both cases. However, the magnitudes of this pressure

change on both sides are comparatively larger for V =/V= 0.3;hence, a relatively larger increase in the lift is expected for

that case. Again these observations are very consistent with the

results given in Figure 5.A •The occurrence of a sharp negative pressure peak in the

relative pressure distribution on the side of injection (Figures

7 and 8a) is a characteristic feature of most of the cases con.-Ssidered. Water injection in some cases gives rise to a positiveý

. " pressure peak; see Figure 8b. The magnitude and the chordwise

extent of this peak have an important bearing on the net lift

change due to injection.

Even from the brief description given above, it can be zeen

that the lift behavior is quite complex, with the lift change due

to injection being positive or negative for different combinazii')n

of conditions involving the injected polymer, the injection veioc-

ity, the chordwise location and side of the injection, and so on.

Also, quite evidently the exact nature of the changes in the

pressuro di8tribution holds the key to the observed differences

in lift changes for various combinations of conditions, as illus-

trated in Figures 7 and 8. It is clear that any theoretical

explanation for the lift changes will have to account not only for

the rather striking changes in the pressure distribution due to

polymer injection, .ut also for the changes due to water injection.

The development of a comprehensive theory to encompass all of the

factors involved is undoubtedly quite a complex task, and this is

not the objective of the present study. Rather, the objective

here is to develop a semi-empirical, phenomenological approach

" . - useful for correlating the observed features of the data, appeal-

ing where necessary to dimensional reasoning.

Ii

HYDRONAUTICS, INCORPORATEDI -8-

In the next section, the various mechanisms that have been

proposed in the literature as explanations for the observed lift

effects are considered, with emphasis on the one we consider most

plausible based on the trends in the data described herein.

7W . . .. .


-9-

III. PLAUSIBLE MECHANISMS FOR LIFT CHANGES

As mentioned earlier, based on their observations, Frurnan,

Tulin and Liu7 examlined several possible explanations for the ob-

served behavior. A change in boundary-layer separation behavior

was discounted as a feasible explanation, since the measured pres-

sure distributions do not display any eviden•ce of this (the major

pressure changes occur somewhat aft of the slit rather than

near the trailing edge). These authors then considered boundary-

layer thinning on the side on which the injection is made as a

possible explanation. The decreased boundary-layer Lhi ckiiess atSthe trailing edge of thle hydrofoil can be directly related to the

drag reduction, since the chan•e in the niomoentum flux itl the wake

must equal the change in drag. In turn, the reduced thickinss ofL

the boundar y layer at the trailing edge o"n the side on Ih,,ich the

injection is !,made can be related to a euinge in the effective

angle of attack. It is relevant to note that, accordin, to the

above explanation, inlectlon on the pressoure ide will lead to a

decrease in the trailing-edge boundary-Thyor thickneso on thls

side and a consequent dc reasoe in the eiff.eetivw an tle oi attack,

whereas injection on the suction oide will le'ad to the opposite

effect-a resuilt which it, co istent with observedi lift behavioa r

il most (but "lot all) of the eases.

Frwual, Tulin and h.u calculated the reductiou in tr'atiling-

edge boundary-layor thiekness usiing the m,,aoured drat,, roduction

and calculated the changoe in lift corroespondint,, to th, ange in

the effective anle of attack by using th•e meaured lift-curve

slope. They found that in tll but a few cas, the meaoured lit

changes were conideorably largor than these that oaL be prediet•ed

using the mechanism described above. As will be seen later, a

change in the effective angle of attack due to boundary-layer tltin-

ning at the trailing edge is only a part of the total effect on the

boundary layer of the polymer Injeotion.


-10-

Frurnan, Tulin and LiU7 then considered visýcoelaontic effects,

as possible explanations for the observed changes. They pointed

out that when the observed lift forces are plotted againt st thelogarithm of the free stream velocit~y, a straighit-line behavior

results inl a manner analogous to the behavior noted in pitot-tube

measuremeonts in flows containing polymer additives. Such a plot

also indicates that there is a threshold velocity below which thelift effect doos not appear, the actual value cf this thresholdvelocity being dependent on the test oond.'itions . However., whien

the results are plotted i~n terms, of the local velocity at the

injection slit, a singl, thres,,hold velocity res:ults. These ob-

servationo lend credence to the conceept, th'at Vizscoelst~io ei'fec t"may be responeible for the observed lift offects * Moroovorý the

fact. thaft under otherwi-,t 1s Idetioul ox morenta 1 con~ditions differ-01

ent polymer" leaý,d to LI'zntfleanuly di-fferoint. .li 1 calges. alsol01ndo credenceo to tho cone e"'A that 0iso Lsi e-Ic t ~ rresponsible, at eatin part, for the obseorved effects.

The amnao~louo Ltift effect wqas alsýo nloted by IIIay~ .Illhi

toests onl two-dimenionl"1al. .1,1111it 10m1e00d 11n di1, II)horn'tgOneu0

'whitioens of 1polyothelene0 OXiLde,ý and he0 concldeldki that Lh it t

to-draeg ratio ot' the hydro"'oi.h.S may e11ther10 0t'es r Pe

depndIn o their st e:ll char Lteris1ics. a 'pkay tIeo10 e1d-

that tho observod oiffect may, beL due,1 to theO difeIet in11htenees

of' the polymer oll the l'ollidar.lae --'Ietsis th o

and bottom suiwiaces of Ulhe hydrol'ofl1 and the ons et chane i

thoe i.reutilt ton0 defoe tL ( tht 010 th ifeenebtween t~ho ac tmuai

circulationl anid thait predicted by the Mutta oondition) known to

Tho hypothesi that. tho observed offec t.. maly 11e due toboundary-layer interactilon beomes1101 4 t~to plausiblo whenl onek 00om-paros the actual inca ured. p roa:.ure di s tribu ti ens under undli stmrt'cd

.-4 ° ' " N -, C . < s , ;::; -• . ... . .•: • .>. , " "• " " ' • " : -< ; ' :-' ',' '':7 • " . W . " ,• :: " ' q : " !


conditions (that is, in the absence of injection) with theoreti-

cal computations based on thln-airfoil theory; see Figure 2. It

can be seen that there are significant differences between the

calculated and measured distributions, especially near the trail-

ing edge of the hydrofoil; these differences are typical of such

comparisons, are well known in the literature (see References 10-13, for example), and are attributed to boundary-layer offects.sT hie potential-flow streamlines around the hydrofoil are dipsplaced

outward not only due to the thickness distribution of the hydro-foil, but also due to thoe boundary layer on the foil surface.Hence, better results would be obtained if, ini thoe coiputationsinstead of using the actual hydrofoil shape, an altered shapein which th•e boundary-layer displ cement thickness is added to the

shape is used.

In general, the boundary layor en> the upper surface of the

hydrofoil will grow fa'ter than1 th'at on the lower ourface becausle

of th•e adverse pressur'e g'adi.nt existilnig on the tol 'urface from

the mininum-p,' ure point onward. Thereofore, the effec tlw, hydro-

foil shape (that is, the actual, shape plu,, the di.placement th-lck-

ni.le,) will have a ligh0tly turned up trailint,, edge iat a reoatlvel~y

qsmall posnitive anl'le oro attack. The ditffoereine betwecn\ the cooll-

Sputatio.l r .Yul' .t' alld tho tL-r iot near tho tLraliLL4, Oki[ Le 01

the feMl ar' dti'eetly att'ibut'ale to the effec't mentioned above

a nd, indeed, excellent atoeemetiL betweeni the two resuilto are ob-tailled , boutndary-layor displacemoent efectt are includod in tLh

computation

"•e relewvn e of the above remarko in the proesnt contoxt io

that, as can be s•en frot Viture "2, the boundary-laye-r di-tpls ementSoffe ts a•l have a oilgii1'ieanut offee t on the pro'eo'ure dts tribut.ions.,

Sthesoo efUfects bollin considerably larg'er than the obooslved lift %cC-f cce ts causod by polymer injoctionm. Thus, even small ehantwes in the


dynamic evolution of the boundary. layer on the surface on whioehthe injection is made can be expected to rrodce change i' h

pressure diotributions of the typt? cberved in, the oxperinientý;.

One other important feature' of the, relative rre.suro Clio-tributiono which also indicates' that the oboonred effeet inay bedue to bouzndary-layer interawt_,m ij the rltvl hrcecrease in t~he oressur'e coesffici,:n~t 'some dlotarnco aft of the in~joo-tion posa ioUn. T11i" rol'ativel, sh1arp) negative v-w-ak in the r-

3 sure dio3tr-Lbution is a oharactori.stio feoature of :nos ol thozases co01,dad ~ eo'r tht: 01"!-" :ht 4

injctin i ~d ( ee re~7 ad S .Ao alrcady notodý andtas can be ee from Fi vl roze 7 and 6., the maniui n ~rw.

extent or thsPeak have a-4 I'm'wt o nt, be i~On the nan1cewell a10 thc iet~ or "'he lit"' lift ollntq t Iat r 0 t~ I u o othe pc21y~m'?w injection'.

III man:'. eatol. no aIn thom ease of 0.l vox '-trtd

ing theo 110e-ativo I'votttro -paks . Also, thet~o eu dCt'tt.!oiva fe e on tlke t'oil' .n e 00 I'"611 ý .0k4

01- locatkton: "'t6 0h ioh in4'-0'11 1.1 ion10 Wsmd oeve~ h-a uIIMI1Iwa". vhih' l 111etooi10: le made " e too -have 0t.i rn

121tV'loneo on thelca *,o o he 11'000%14 urO.k f te at o a1'r-

ontlay blnC, l n~iluonood. lnoo Lty th110 bhas! X. '''-'I

t.ribut-b~ll olOno thydrofoil1.

It Ls." ii.otatk note th14at va'tetl lnjoot..o: aloo loedstoL3pasill tho 1,1,0010111 dl .ýt Im ,, tonI. alo 11i Iut. ttetd .1 1(igute h

ohsii;r hw halleno.s ill the p'. 11surliv l.d.stv ribi I'vonsl10

5Iddo i1?J0tioll at. X%'-' 01iexd. 1"or thi 'reohitl caoe Of Volyox

inj e tizi, here Ito ~at iv e peak omiliieli th botoitt- i~ido mrc eouredistribution (01he oide oll which the: injeot4 tonl. io Ilade) . l lowovovr,

5.is~tim -will be d ue in more qummtitativo. em In'k00tionl IV.

7,. 7777-r.-I


for the case of water injection, while there is a small negative

peak for the smaller rate of injection, the pressure peak. bocomie,positive for the larger rate of injection.

The observed features of the pressure distributions de,-crlbcd

above are compatible with the hypothesis that the lift chanigez I-reproduced by boundary-layer displacement effects. The inlteractini

between the injected flow and the boundary layer can be oxt~oeted

to depend az much on the undisturbed preosure diotributioiv, on L!o

hydrofoil ac; it does on the location of the injection. For' cx-

ample, clearly portions~ of the boundary layer in the favorablt.' andt

adverze pressure Uvdint ru-,iono inl the hydrofoil moy rt .'pol~differently to the perturbation~z3 caused by the injoctionto.

Frurnan, Tulin aad Liu~ have oudgeotod that the li*ft oýf'*W

of polymier injection may bo du-. to thL fact that the ol~enters, the flau, around th hydrof'oil in the foin of a "znvollo en st"

due to tho viscoelautie behavior of polymer' *oiution,. Mi .j*- -t i kplauuible to ou11pje thatt the ob;-orviod po'aks in the o sr

rItr butioii may be due to these "o-wollein Jew", iu I%*c

the oboerved .elfect lo a purely vli coela-.tie 1n, 1-ýJoetioti alio leads ito i pk; in. tho. pro-jour diiýtrlbtie.

in any ea -lith) importanti point to note -hwere ha h

loetn ocianqgo';. III the oiplae moment thbnesofte undar'y tydue to. tý its £ uraotlat vith tho injvetod 'flow -will I ea o ,,oechai~oz in tho o'eet'C'N vambi4e o14 tiw lyh'oi'oli. Us turn., thoeffect oC : o3-e hnlo will be to eauvo local ':t-ehanr'o

action va~ well tu to eaui-o taoro general ohangoo -in tile pro.%,uro.

dit~tributloin over: th netire I 1 uoe de~ a~~ithe circulation. ThsiLeatl ha . seill tile dat4.


As mentioned earlier,$ based on his experiments on hydrolfoil,;5

immersed in dilute, homogeneous polymer solutions, Sarpkaya~ has

s'-uggested that the observed lift increases may be due to the

restoration., by polymer-induced changes, of some of the circula-

tion reduction created by boundary-layer displacement effect'-,

.The present results on the effects of the polymer when it is in-

-jected at the hydrofoil surface also seems to suggest th-at the,

-observed lift changes may be due to a boundary-layer displacement

effect caused by the injection, especially 'n view of th e diff r-i

enqes betwý_een the actual measured pressure distributions under

.undistur'bed conditions (that is., in the absenice of injtection) and9

theoretical computations based on potential-flow theory- ;0 vle iui'e

-2. However, th-e observed effect is also evidenitly a finnutlkon of

the characteristic relaxzaI!n time,1 T . of the polyiier,, or., mo re

appropriately, the nondimensional paramfetdll' VfT/lt. whelre Vi Io t~he

in j-ec tio n velocity and t is a characteristic hydrol'oil dmnin

The data also show that the bouidary-layer dlsplacmn f'ce tu

also be a function of the soide., location anu~ veJlocity of 010h polymor

injection.

The hypothosis L,'Jven above can be vor"Ifid ii'tt.y sne hmeasredp rs sue dl'stributioiis, can bo analy ,,edt u sdtng lasTe1

a 1. r f o i 2. th o o ry, and the o ff ective" hydrofoll ollap that. w t 1.

produce the oboerved pro.?4uro i ±iu~om can bo calovulatokd

Thl.wN offective ]iydrofo~.'U hpecan thoen be violwed i.n Levnel. ot, a'

change, in the evolution of the boundary layer arotuid Lime aetuna V

hydrofoil.. By theseo meanos, Teorlatltoie an bo ,Aouti le wemLm

observed effects, and the toot vaIrIabL1s0 ill tQ11411 OV tmint~m

that exist in thc literatuire on the buhaiv:Voi' oi'-boulidary I'tayors

Undor favo rabl:. and ~Idvoreoo, proi'ei uro rdIem and tnImd or tii e .lm-

fluenco of InJoctIinon and varlouo sutri'Ue per~tIraII ens *OIL% This.- I

* ~the approach that :Isutilized in tho next iooetion.


-15-

IV. SM-II-EMPIRICAL CORRE&LATIONS FOR TH-E OBSERVED EFFECTS

IV.1 Press,,ure Distribution and Lift Effects

It was suggested In the previous section that the observed

lift effects are due to a change, caused by the injection of water

o r a polymer at tesurface of the hydrofoil, in the boundary-

layo rdispaoCement phenomenon which -causes the observed lift in

real flo, s to be leo, than -that predicted by potential theory.

In essence, the faster growth of the. boundary layer on the suc-

t~ion side of the hydrofoil comparod to thamt on the pressure side

leado to 'an offective hydrofoil shape which has a slightly turned-up

trailintg edgo InL turn,ý thIiis leads: to a reductIon in- e'f tive

camber anid hence als~o In elreulatio.n and inl lift.

It is-, wellnwn. that re&onabs ~room mt can be ob-

tatr eI I ba otwoon inca urod pres-sure distributions on a hydrofoil.an t 1 roŽditon -I' po tal thoy -ept neair tho trail-

41ge ode;Q, wholn tho mesuedvlus the lift eeetfftcient,

ra, Lhcr Q Aj theo 'I' orotioal. valiior, are used to normaiize the0 fft o i (nto .11 1hsi tho bdstsL f'or the cuniparlsoii

Ahv I i~v 2* ,mewhat 1)o t tor a weemont botween the m.,a

let ~ clatod prsuedis tributionis can be obtained byreon114 n~ t~ha Vth sial t ha potm ia. lif t that 1. mea

1u1e1.Witt 110;oosaviliy bjmply that the actual %tirculation around

th o bydtrofoi~l JiA omallor'.ha that predicted by the Kutta con-dl ~ ~ - trn ece h istbution of vorticoo tuiod to roprozent

.... .... tho hytirioi'l muott corroopond to the reduced ci rculzition ratherthan to th~at pr,ý-dlitod by tho Kutta eondition.

Lot

r rk (1-k)

oil 1Zif1TIt% equivalnt to C1 - LI (1 -k)~ whore C, andel are resopeetivOIy, tih) moasured all predicted value*" of'

-D l...t cooff' ont


and:h =1- k [ 2]

where I'and Pj. are, 'respect~ively, the actual value of the

circulation an -d that predicted by the Kutta condition.

I[t can be s IIo wnI % 6 thIia t,1 for a-sy~mmetric hydrofoil,3the induced velocity 'dstributio~n dueto,'the mnodificed var-

ticity- distribution is given byý,

1l+(dz/'dx)- [17~-x _ +_x)-X7()J

whcŽ.re V is, the free-stream velocity, a is the angle of

t-tack and the integrals. I3 and 1,j are given by,

1 4

I3U II - 2x -O_

andI

14x [521 x( -. (o-

in the abovoe equationo z (x) r op r -!3oi ts the thickn'ess

distribution' olf the hydrofoil and all the. loengthls have: been

~oial~edthoug dvisonbytho hydrofoil chord, c.. In

-Equation 133 tho .posoitivo -ig applies to tho uipper surface

ou, the hydrofoil Wh11i tho.1wk~ativo pitgn applies to tho lower

"In1quato £3,tepiaeLr 12.+(dL/dX)U rep iesents

the rcatliou botwoen theu (predicted) volocity on thte. ehordlinu:and tho actual voiocit~y oni the foil sourface; Its use. avoidm,for~ a round-nosed hydrofoil., tho occurrence of an infinite

tuc tion volocity at thy edu~eg Since '-quto£3

oIhi OquattLon-TF found-by ropreoenting the hydrofoil ourfacebyaproritedi'otributioin of vortiezs and Sourcestv (zeu Appendix A).


violates the Kutta condition, it predicts anl infinite velocity

at the trailing edge; however., this can be eliminated if, as)10 13suggested by Preston and Spence ,the velocities at the edge.-

of the boundary layer along normals to the upper and lower our-faces at the trailing edge are used,, along with a judicious fa21-

ing procedure, instead of the velocity at -the trailing edgeitself.

~qutin [] -oresents the velocity distribution due t o

the effect of angple of' attack., and, for a smtrchydrofoil,

the velocity due to thickness distribution has. to be added to

this to obtain the total velocity distribution. The-latter voloo-

ity distribution can be written as

V _ _ _ _ _ _. [63

I (~) 1 r(dz./dx)d dx .[

0

Tlio volool~ty due to thicknoov disitr:lbution i~ al-s.o

,tabu:latod I'll B0d' renece 9 for a numbor of' NACA foil. shapooinl goneial anld for te opoelf:1e hy droaol"0 under eonioderat Iohoro In particul'ar.

E'quat.:1on [3 nd[i] n bo uod~ al,'one with the simle

relatioln,

to cal culato tho pro,'luroe oetfic.ionlt, CI onl (dithe' ,:%idoof a s;ymnuntrIxc hydreot'ol Lit a soqciefiod "1111."1 Lit attack anldat a arbitnary valuo cfth circulation (that ia., at. a valuo.

=i - 571.


of' the circulation that does not necessarily corresp'ond to -th~at

given by the idealized Kutta-Joukowvski condition). The calcu-

lated pressure distributions -for the specific hydrofoil shzapc

utilized in the presen -t study are shoiwn in Figures C) and 10 for

an angle of attack of 50 and for a value of' the circulation

which corresponds. to the measured value of' the lift cooffici-ent.

In the approximate, calculation., 'the integrals, given ini. Equationis

[4] and [5] were not calculated; ratherý, the expression withini

the flower brackets in Equ~ation [3 as equatdt niysne

this is sufficiently accurate for the present purposeý. Alsoo,

the tabular values given in 1ef erence 9., rat'her than E nlat-lio

[6,were used tocalcula"Ge tVhe induced velocity due to the

thickness ef'fect.

Acomparison betwe th masured and calculat,-ed values- of'

the diffe-rences, in the pressuIre coefficionts Onl the u11poe.l and

lower surfaces o~f the hydrofoil:! I- hownol in Figureo 9., while th

presSUre co1'f1 .Clea~ts t~hemselves, are shlown inVL11 e 0 Value

corrosponld ing, toL the Kuttl-j oukolskl Value oxt' the c ireula, 1,-on1

aro also showni in thoese figure or, 1 uro1, 01f crya oIt.

czan- be seen that; the calecul at. ionl"o, the1 1-rodcdiru. U~I-on

case gý*ive ver"y good 'ag'reemelnt. with the mesrmnt n . d

in1 the" region01 of the tr:2.!n dge, wIeh1ol 1" no Lewort~ly In V*Iew

of a quite approximatce methodt of calcu~lat~lion

The main point111 tobedeivd ro t1e10 ai ossow ii

Figres0 91and .10 .1.0 1"tha whi .1. thel conis,1d0crab1,y 0ns11,0 1- erIi ' t Ut1h, 1.

is 11i0asured in1 t~he ests' (inl 0he a1b1,10 0eu ' Llof 100t. ec n)my be caused

ill part, by a lack of, tw o -dime1. Il001on all. ty in the fl!ow or by wal I I j ter-

fecue ffct ,bounidary -layer disp a ee Toefctt may a o beof

importal ice. 11leon cc, iisp Thusile that ovenl If polymer linjectilon

causes,ý only smal chages In the boundary-layer th A.cIns X the

Ooosrved magnlitudes of lii' t and p01,ur changes can11es.Lt

*Thi 'ljapp '0Xil1altJ ll Ito o qllIva 1.ot en.,to maR I. 1gtb U th,11i -0 1 ro1 01 a s

Appomilx A.


Estimates of the types of changes that will occur in the

pressure distributions on a symmetric hydrofoil due to the ob-

served lift changes caused by injection-can be made using Equa-

tion [3]. Again, as before, making the approximatior, that the

terms within the flower brackets in Equation [31 are equal tounity, this equation and Equations [13, [2] and [8] yield anexpression for the change, AC in'the pressure-coefficient

p -4due to a change, AC ,in the lift coefficient, as

L

= (VAV1)AC__ __ __ __91

/l+(dz/dx)2-. /X(l-x) 2r [

- where, now, the negative sign applies to the-top surface and

the positive sign to the lower surface.

In Equation [93, V represents the velocity on the upper

* or lower surfaces of the hydrofoil in the absence of injection

and can be computed using the measured pressure distribution

* in the absence of injection; these basic pressure dlstrzl.buttions

for the four angles of attack considered in -the present study_

are shown in Figure ii.

Comparisons between the measured chanlges in pre-ssure dis-1.

tributions due to injection and those calculated using .Equa-

tion [9] are shown in Figure 12 for two cases. The first case

"shown is that for Jaguar injection on the suction side of tho

-V hydrofoil at ton percent chord and at thirty percent of free-stream velocity. with the angle of attack being 2.1) as can

be seen from Figure 5, this case leads to a substanti.0l lift

Sincrease.r The second case shown is that for water injection,

* this case leads to a lift decrease.

ww4 -

` - 77 9 - , -7 ..- -;7


-20-

It can b(, sec-n from Figure 1'.- that there is a very good0,inerai agreement botvlaeen the calculated and, measured distri-

butions, except for the, presence inl the. latter of the pres51ur1et~aks already discussed. It should be noted that in each o*f the

comp~arisons shown three separate sets of measurements are In-volved, -namely, the changes in th e lift force, the basic pre~s su redistribution and the change -in the pressure A itribution due-toinj ection. rjh1ius. th(_ grood agreement di~spiayod inl Figure 12 1is*evidence of the self-consistoecy of thie experimental~ datla. Thefact that. the ere.,sure distributions piredic ted by pote.ta

theory using at chlange inl circulation alone go a logw~ty inl pre-dieting the observed changes; .in pressure distribtitions a apersto *su~pport the preen mthod Lit' approach of Viewing the li11ft.anld -pressurlvoe hanges inl tormls of' changes inl the "OfeIoetive" .sha'c'

of, thoe hydrofoll

-1 onteohr ad hn the, ob s rveod lif t changes ar1 vi:!.ew 0-moroKty as Liue to a chanlge in1. the evf:e 0tiv 0 -ang.Le oV attzwk of the

hy -o)iA tho prcodic ted pressr hwoL r ntA oo ge

vient with those cbs ervod. * This can be seenl by o t~tlug, and k I1 .i~qat;t Q [31eqwl. to_- 111.1i-y and z rorspectjvelyý, so as tuo ob-

t~al the y toe it'uY di stributtonl core pnd to theo ielkol v " .uvIeof~~L th-on ion arndte hydro~tei.A. . Then" the ochange Inl the

velovit~y di stribe I :1.n due to a chunge, syIn h u ~ f tc

.cnul be writtenl a,%ý

where , theo Aamelc approxtmation av, th~at Invo'lvoi Iin quatlou I) uhtasb oon muade.

Us (p~qation's 1)and [1010) theo 011auitg inl pm.essure, 0c-ul±fleiolto. Call tholl bo writtenl ao,"

,.-774 -ý

-HYDRONAUTICS, INCORP.ORATED

-21-

V1+ (dz/d~x~z (ac L/aa)m

where (aCL/60L)ml is the measured lift curve slope in the

absence of injection. Equation [111 is also shown plotted

in Figure 12 (in dotted lines); it can be seen that. even the

* .-. qualitative behavior of the measured and predicted distri-butions are differen~t.. It should be rnoted that bot-h Equat' ions

-[9] and [11] correspond to the samne chan~ge inl circulation

* around the hydrofoil., which., in turn, is deduced f rom the

measured value of the lift change. However, these. two equa-tions account for the changes in circulation in termLs ofdifferent distribution of vortices around the hydrofoil chord;the physical si-gnificance of the differences between the twcases will be- discussed presontly.

As already noted, the single czxpressien1 gi'ven inl Equation

[9] does remarkably woll in predicting many of the obser1vedifeatures of -the presoure distribut.i.ons". sIwv~li nco It o1n1";

accounts for the change'Os inl the pressure itioin u t~oc change in circulation for a symme11tric ye ofll the OqutatLI on

fails., not surprisinlgly, to prodiOt. -the11 obse"rvedj peak1 '1n thek

prossure ditstri~bution onl the slieon Ohich101 the Injection Imadoe This,)1 pressure peak is0 evidently a0ssocia,.ted 1,111,h anM 1n-

4 . Iyemti-10i Change, in the effecti-ve shlape 01, the hlydrof'ol.:L anldcanl be p1ridictod only by expl-elotly accouniting for camber)-NIeffec ts.* Note that bonay-a rcosia mn f-feet s wert'.

not accounted for oxp:licit3.y inl Equation I 9] , thloutgh theo ch1angesoin circulation were asumeto d to be :.nue by cAnei oudr-

layci' thiekneo,,s

Au mentioned earlier, seve-ra U thr aeicue hoi'iects of' boundary-laycir displacemenolt onl Lift oxpllelt:ly . Spoel -

±'ically, 11r(~tn 0 1 ha6 resolved tho displcmn f'e

HYDRONIAUTICS, INCOaRORATED

-22-

into symmetCric and aniv~ti <r2and h troated ec

separately. Pres3tor.'s thaeof shiape t l m cis of01 colider-.

able value in ~ o4 ~uCi;n h ffects of surface n.ecic-'

However,, before c, derln-, thi.3 metý.hod in dealit is r. -1evantil

to briefly di-, Uts. an alternate techni que c'eledby , nc

-since it providt~s so-me insight into the inannac i' in which bounda:ry-

layer de-velopmei-t influenceF the outside oetilfo.I

this method the boundary-layeýr displac et efio sresneat the outset b an ,'nýmzi oredsriu.nadtecr

culation is doteriniined b-- uiilizingo' the c o n d ~n' that oa 1.

amounts of vortlcity, thuhof cenposit-e sicns -, mu1st.u si

thIie w a ke f ro~m the bo,,n ýry layer on the upper a-nd l.ower surCe0

of the hydrofoil.

The eal1culaticnal s: c'2 ill Srlcce 's lll0 Ivolvesz 4

tnfrmti~on, by th~o cs of i conforal ],wapr 0, "~th Yr-

fo1: in~to a circle.e ssuoh, the finial rtcs1IT-- in thL-11" c~

involve 00111).1"X Ooord.1&"tes . Ill paIrticW- rý )f 0 te

palwametoro that appear 1n-h i 'lia rcouil t (>u~ton

i~e~eenco13) zatr %I "I'd XQ wIUhihrerspo .y tedI-

taic o iathe coumolox r'. lane between1 the0 1,va hag n'."edtes Of the eiidr h 0115 on th 0upra.1 eert

whi to the result Its cii' Istoo IUlwxold 'to (Neonid hor.It

rele0vanlt to cite "'Pe tee'o obserowat:1on1s oil the eslt

"Thi1s equat-Lortp e t a tho te-al ciianaio of or -

taitn qualitatI ye bnd y-yrefctenLiU't 10h-.ch are w.. 'I

I cognlizodc oxilorilo.menta:1 1y. Vor all,:l'Vol.l of* fIni. to tralllroa

IaNyel', keep lagthe 01 ewicon'LVt; would be t-o l creaseýL %I. TheIOffect weul d t~hlwt be ----- Utdo 'reas the o e~ o andi 1i. ft,as Oxpoeted. The0 00uiwe 50 0tTOe! should our AfX er o

ereaied10ro ( ".,b n~~~n the amununt.. oIthmU11111"1%I'lew oll ho

* HYDRONAUTICS, INCORPORATED

-23-

upper surface*). The effect of increasing and decreasing X2,,

which represents the lower boundary layer thickness are not clear

from the equation, and would appear to depend on actual ratherthan-relative thicknesses. it is expected that the larger the

*trailing-edge angle, the more marked will be such changes in lift

o btainable via trailing-edge thickness."

While the exact nature of the interaction between the-in-

_jected polymer flow and Uthe boundary layer is not known, it is'clear that many of the observations of Spence quoted above may be

relevant in the context of the present study. For example, if

the effect of polymer injection on the upper surface is litaginedas producing a more laminar (tk.hat is., thinner) boundary-layer

behavior on that surface, then the result of the injection would

be to yield a lift increase. Th-is is what is observed in manycases of polymer injoction studied hercin, especial~ly when t~he

effects of polymer injection are, viewed relative to that of water'injoction. However,, it should be noted that the injcted f'lowwill also interact with the external prossure. gradient,, and 'that.

the effect of the 9tron g, adverse pressure gradient present down-

stream of the point of inject~ion (see 11ig. 11) may be to inureaoethe boundary ltayer thi~cknoss * The net effoet will then be tg'ov-erned by tho balance be:.tween those two counteracting influenceos.

Water injection on the uppor surfacc presiunab:ly luads to athickening of t~he boundlary la:yer, since a 'reduction in 1.1ft .O waysaccompanieol such InJ ootion * Also, sinceo watur injec-tion on thoe"upper Slr±'ce causes a drag inorea so, it is difficult to attritbute

the obsorved results to any loa~dlng-odgeo effoots~ isuch a thsassociatod with boundary-layer control throuegh the use of izgedge slats or blowin 1 7

* binphasla aupplied; not in the original citation.

HYDRONAUTICS, INCORPORATED"

Spence t s results also show that the offects nlf,~changres in the upper and lower boundar Cae thcnses r

considerably different. Speci~fically, change in heloer 2V

face botundary layer cani produce either an increase or ecrease40

in lift dependi.ntg on the actual values of thie boundanry laye

thicknesses onl the uppear and lower suraces (that isý, onsetion,

shape, angle of attack, R~eynolds ntunber, etc, and oni thetrl1i

ing-edge angle.. T he 1. cmnta rsullt-s are qualitatively, c'on-sistent with this observation, sin~ce the c'hal-vee in1 14.-"ttle to.,

lowe sufac inj!.,n dsply conisiderabl y fewor trends ats

omparedo toefcs of upper side injection.

As mont:loucd enrlLcr Spencels re-sults are not in a 0or1

f roum whicho aniavv utc.im teoat be obtatined rca di .1 y; toil

thi's p u po se we ap pea l the tWehn~lquo deole1ope., by e zc st o U .2

C~on'sid c'asykt t~wo- d:imens :1 c.101al hyd rofo:1.1 'tn "n" l~e

of attack a~ to ",lue free s~treaml~l as 01ho1n in-1 1i "U", L 1 t~ "I et

te ore adverse6 '1ra1s 011 t-11 rI~~ct on the nee 11ta I c C, 0 11

surface P. ~TheI,,'- l'K'' 0,11 00nd vy-1nOU1110 bV Z-, Ulpl aect-110 hec

thcnesto Il, ofT'ottve ross et: ola I shar Ie o tt.he hyd ye-

'tilo u4ppo and .1owor oul ae h ~ s~ tey .t 1100-1. Uk oail be ro-.

S olved -ou two pa1ts one syMetc anVh te n ismeti

Ila 1110

Whore 60~vsn~ the ohugetIt inl theo kcfet-I'vo camlber ot' t~ho

hyd ro±'oltI and 6tho hanlgo ill At, thi. eknevs t-i s.ri, bn t- oil.

meIt Vfco L~ aro ohowuilt~hIy a~ at for tUe sakeo oVoirtty.

HYDRONAUTICS, INCORPORAT.ED

-2)-

It is all-propriato to fin-t consider the effective a;.i

distribution alore,c since it,- Influenxce onl the lift izý muc> c,

profound than that of the thickne-ss distribution. The ffýtiv

centerline of the hydrofoil is obtained by addinG b ,' o ~_-eonieri cetrinad the effective cross-section'al ae-'

the hydrofoil can be ropro;ýented as in Figure 1'3(b). Ak s c11b

seen from thac figure, the dli placemen t effect ivill produoe not

Only a reductionl inl the effective angle of attack by a m'n

* equal to,,

btalso a camber about. the off .tive chop

B~oth of theowc ~ct will. wild to reduce the "ift 'eue

*by thoe hydro~fol~l

l~t he ~du of he :I eulatiol aboult. the efe~x ~e

fol:.l. shape wore. ,,ovornod by t-he Kutt.a-J oulow si oMt ,

the abo\'e two of'feot"., wiLl, be 11he onIly oete r'. it

dluo to the pro00ence Of the botndary layer5, the t I.tona'u.the lfctlvehdooli net. tvi'ven by 1tb1" K u t U~ouo

bu". , .1% lwv 1oa, sLe t be 't ii '.i 1Oi ~o o

dit.1on1 that. 0qkla : n o st amounk,01A 1100ts0 %M' vOr't1. Ic I y Utpillto fth- walw. ti',*'n tIho twpor mn 1 '''Pbudal 1aeY r

lint. w)hown thal. thelk ab veond I V.!on los U te t~o a * sa

I'proxi mat. Ion,, when, the vcdoe I t lo. at., the ed1e o,1k the 'bcu1 %da y

lay or, on the ukpperv and lower sra at the tro geg l'tho hy1.01oi are, equal . 'Lwsew v oloe 1 U! e t-hl mseolvyeOca

vvt.Ullated by usling, potelt.I tal. hory. with ait arbit-tvary va1, Xi

t1he 11 U10u atio Intheabseo 01, dUh t1 Ipaeetefe o*adt~cirrcul'at~ioz adjtwtod to makethe-1C v'htto e'.a&1 11' t c

thoŽ rkwuuto eani be Imp roved by ltei'atl.ii.


-26-

The effect of the syxnirtri pat s on the p~ressuredistribution can be calculated by adding 6to the thickiies.distribution of the hydrofoil and by using Equations £6] anc [7].

The effect of boundary-layer di~splacemnent on the lift andpe ssure distributions on a symiraetric, two-dimensionaI hyd roft'oiis thus seen to consist of four separate parts. The first :Is

* . associated with a reduction in the effectivre angle of' attack,while the second is wssoeciated with the ncy~atire cabrof t~heeffective hydrofo.)il shape. 1110 third el.ffect reprosents the re0-duction in oirculat!on aos c011ýOm!rd to that1" gviven1 by thO ~aJoukowski eriterion, zand tlh-, fourt~h ezffect; reprosolnt~s thle "~hangedt~hltknoss Mist~ribution. 1,- th'e eomlparlriso'no olhown 11"in .guieand 10, only t~ho thitrd of the -,,bove effects asIncluded( iithe moasui-Ld U ft ) and t~ho othiers were ileee

The influence ofT voi yr,,r or w\ator :t1,10olion1 Oil tho li ft '11dpro.98ure d~i tribut-Ion." :lli nov: bo Viewed 1n11rm of the boundar*y-

1 ~ -dspl noont. ek~f. ieoibdabove. 00n:dert~ nJe

tioln of1 polymeri or wtor ,14,1111 lonat'lovi .1 oll the PtlrfneeooSsymuunot~rlk , two-dilliolslona)1-. hydrofoll I . Tho VfY tvoe ohrdi nd

onelneof t~hw hydrofe:.A.1I itl "he abseolý- of A ~ . n(Imu In-ciidngbounldavy-1:1ye-0 f1 et, als rehoý-n inllta nV~r 14.L

Ats a ftirs 1. approxtimat~don 1A.t a ppropr~nto t~o ue 4 Y t<DiztlIotion 40totl'5 t~ho twundatry- a yor thl-k l~i~ ny'.e the .d

oil whiolh the 111jec t.1ol 1s u.ý;. t~ onlIy 11,11k Ira k,~ Ih t~ I Itof) !,Noection .The offec tVvo 'e~ntorl;111 oki tho hydro'o 1.*1* wl tthen be di~spiaecd vortio.al11y by an 1 alioutit. eojlua to ~S~*wilveroA6I ~ h ellatgo (.1astimed pes'.live whoti thoe I-% W.11wi illn V og e

tho uppker sulrfceo or -. tillnniltv, on tho L. -,r miuetaco) In the

~ O±~ ~ve"'Ileoly a ica t~h~hml tilU hown IinFin.V1 and 15. A.- mont-10110d eZkiller, thotre outn b a 14,10 mt tile

Lil t~ill (holltit"g of t~ho.11L a layer 1 eca Ily dLapotld ill 6111e ther~iatiyenagn th~of~' tho ez4'00UAt due~ ;o tile -lvnl pro;;J1114e

grakdil (Itb it ad tho polylar.


e27 -

boundary-layer-displacement thickness due to injection. The

effective centerline and chord are shown as solid lines in

Figure 14, with TV being the new effective trailing edge of' th-e

hydrofoil.

It can be seen from Figure 14 that the injection leads t'Uoa change in the effective angle of attack by an amtount equal to-Mt/2c, whiere a6t is the change in boundary-layer-dispilacemeno.,t

thicknesý;, due to injection, at the trailing edge. It clan als.obe readily ohown that the chan~go in' camber is: given by j, (6*

x~t*). Since the camber elffect dependUs on the deotal:- D 'th1e

interaction betweePcn the boundary layer and the injected flwA

ma e iher pozitive or netative local~ly and changeo signl wit1hdownstream diztanee deipending on)1 thle specific nature &tU

interactlon. As alreoady pointed out., thei fact that diffo remTApolytioro load to aign'f-4eant1,y differint results alccta

the eharcteri1t interaction len1"t-1 is- like-ly to be ctmýin

on VhoiV 1'flt wOxv am VF whr ¶vi the po lynnr relaxaAioni 'tý'm.

It may be ossectciU~ that the nondimcftnzional ratio b., r ~A61. i1 the, d~zwlace% int thiek1ichiss at the location " t'.t0 w t'Q j

tio, illbean iAmportan t para-meter.

Of ore the eýXact nahture of th i,11teCria ct.on 01 the tntxtepolyinn' soiutions wit the.Vf bounchrtiy l ay, or or it.1h th ou tsid 1 iewo

is unknwn and it io not the ob Jocie-'w 'of vlv ;pmres 1 stud toattempt to dW-esrlbe the det~ails okf tU-11 Inte~ract-ion. I;, It is our chjoetive;, to morely Mlu.,trte that su~ch lt~at~& ~i b h

plausioble reasons Aw ,the obsrved ch.uig0, 01 >th tiriisum. ditributions., lift -and drzag. E~von Is' tho injeocted t:'ntr h

oultsi 1dea rlow aso a ýI.wolIm CI llot" 1-th0.cQ4omputatiotial preodure, de-

seribed herein can be utiliz~ed proivided that 'Uw d i oplavemen It

effect of' this jet. ia known.

- HYDRONAUTICS, INCORPORATED

-28-

It should be noted that the circulation around the modi-

fied centerline LT" will not be that specified by the Kutta-

Joukowski hypothesis; nor will it be merely the circulation in

the absence of injection (that is, for the centerline LTI) cor-

rected for the effects of the changed angle of attack and camber.

The new value of the circulation can, and should be. uniquely

determined by using the condition that the velocities at the

edge of the (-2hanged) top and bottom boundary layers at the

trailing edge must be the same.

Finally, the effect of the injection on the thickness dis-

* tribution can be obtained by subtracting A6"/2 from the basic

distribution when there is a thinning of the boundary layer and

_ •by adding it when there is a thickening.

The four separate effects can be sunmmarized as follows:

(i) A change in angle of attack due to a change in

the boundary-layer thickness at the trailing

edge; given by Au =-A6t /2c.

(ii) A change in the effective camber of the ihydro-

"foil due to injection.

* (iii) A change in circulation duo to injection, dotcýr-

mined by using, the thoorem stated by G. 1I. Taylorthat equal and opposite amounts of vot-olty iuut

pass into the wake from the top and bottom boundai'y

layers 11

(iv) A chargeo in thickness distribution, duo to Inj•ctlon.

•In passingi�it may be noted that in their aly.i, of

boundary-layer effects, Fruman, Tulin and Liut only conl.tO jorod

the first of the above four elLfectr, while in the eouipa4'l.)u

shown in Figure 12 (that :L,0, in Equat-lon [91) , all of the ob-served lift change was attributed to the, third eci'Vct.

* * . Y

,5 ~*• ,:, ., • • .. ., ,,:• : . ' ." . . . , . . ...' .. ,. : .,:':: , • . •. .U . . . . ..-, ."'T»' ' '.. . .: . . . , , .. . ..L:-. • .• - - ' • : .v . . . .. . . . . • . . .


-29-

Before making quantitative estimates of the four separate

effects of' injection, it is first relev~ant to consider their

qualitative behavior for a specific oxample, sazy, that. of poly-

iner injection on the upper surfac'e 11' the effect of pcly1e

injetio Asto akttheboundary latyer dlow ns, bream o1f the pointof' Inection beh~ave I1n a more lamimna r fa shion , then acc omupany in gthe drag reutothere is also a Muction ini the boudary

laye-r-dilsplacememit thicknezos at the t~raillng odge. Thuti, there

w ill be anl in1esei the effective anlgle of attaeok. Of01.ourso,the converse is true If' polymer01 inljetionl is, made Onl the l0owr

Suri'aco.

Ao, U]TOeady poin1ted 011t, theOsign of' the camber effect. willdepond onl the kdeta :lled n~a vttup of the neacin betweel th e

imij cted zand boundary -layer Vi.'ows * :111en 'a t t-110 thinii:UigLll

oI'fect, at'. teasest' 111 Vt.onut. a 1.1 U), t111 :thnere sing 1w00 brea di.s

p051.t-ve.n %o re cIn camlbev Ot(do to a 1001,0U.11 inl tho llegit"IVt-1a11l1er) as Ahowni * owevk 'll if the Ina x111 tillm Ofg'e o I Hilt it in

00VU1i'1., at A a .hrt. dl stanceo downsot'leam11 ot' t-he pohln. ot' Injeet"i olluks olhown 1,n VllureI ( h) the 010Lt'Abhut-:ton of' 0he 01angeL 1,n ca4,11-bve, mlay tit llog' t, I ye OVer1 ZA 110rg potIOM1 ol' th'' hydri'ot I. ,

that. the neot. e 1'i'ee I ;- a I-oeduct t.oll In Ii Vt. fl t t'ne x 1.11t ill kde-te kit, 011.111110 0001,1cuo a t, a '1ow. ncoeto Wth t~ra.I1:1iy; edge"

buti. aheadout tit', as, showit Iii Vl~tgur 11 1-1(l e ,thnthe m.ixtimiul(~ ~ ~~~- tivt y)chaiei alutbol' wA-1 alo o mouear vl3ooo to Wthe t.ra 1l1-

ingt" Olte. Itnea iwt' uw (oee,ý for et, o,~ic

.1 tvg t~ho vahlooe of t~he vawlbel Pnear Wit e bat1itgee

have n liore 11,0,tnilld 1.1ftt UMnc 1V 0 til fthtM the Wtlo I 1.s41141wAv' rethe roduot~ioll or I I t In flht. en. slkl ~ b us~nL~

'iThoonne btu1~ .Ii etvulail-lio d110 to0 111ylov 1eterI1,oetM ol.. may. a tthe top turace., oati ho kttlhW111ftle .ih111t t~ho condikl.klin that Ult!t


-30-

new value of the circulation should be such that the velocity

at the new edge of the top boundary layer at the trailing edge

should be the same as that at the (unchanged) edge of the bottom

boundary layer at the trailing edge. For a hydrofoil with a

finite trailing-edge angle, the trailing edge is a stagnation

point, so that near the trailing edge the velocity will increase

with increasing distances from it. Thus, if the circulation

remained unchanged after injection, the velocity at.the edge

of the (thinned)top boundary layer will be smaller than that

at the bottom; to increase this velocity to equal that at the

bottom, the circulation will have to increase. in other words,

a thinning of the top boundary layer must necessarily be accom-

"V panied by an increase in the effective circulation around the V

hydrofoil*.

Conversely, a thinni:ng of the lowor boundary layer must

necessarily be accompanied by a der,,eat'e in circulation since

to make the velocity at the new edge of the (th:l.nneo') lower

boundary layer at the trailin , edge equal to thlUt, at the (un-

cluiged) upper edge, the e:i.roulation mu't be decreased.

.. :In sunluary" DI the effect of polyme1r ,njecion on t,1e

boundary layer Ls to thi• it downstream of the point, of inAljec-

.. on., the ngtVle-o0--atltaek e:t''fcot wiLl Ilncr'eae the lift for

upper side Ljoeetiu. n and reduce it for lower slde :I1njectlo:l.

The cainbhr ftffect may o1ther 1.1erA.tie or dores0e the 11:t,t1, do-

pondiig oen the detalsl, of 1,te tinteraetiou between the lnjuetod

"11*,1111,o Che welocl. ty dIit 1. ill the reg,.on of the traill)!.ltnl'e :i," deppeldeoit oll the hyd rolil geol'iometry Illn genea . and the2!: <-I•"• !trat i..Hg edge alt igle I~n pa P lI. twldar, thes,,.e p~arame'ters wil. 'U :1 n-

fikllcoe thte (htt ige1 inl eiroklUa-ion. Thulse features are included'. eory


-31-

liow and the Wnundary, layr. The change in lift due to themrodified traiLling-edge condition (Taylor' theorem) Is posi-

tive for upper-side injection and negative for lower-side, in-

ject-ion. Thus the totail changre in lift may be either positive

or negati1ve depending on the relative iiagn:Ltude of the differ-

ent effects., though, inl general, upper-,side injection can be

exetdto prdc-n nra in lift and lower-side injec-

tion a reduction In I ift..

It. Its now app ropriatea to conisider stomec quantitative as ti-

ma tes. Equations, [31 and [411 represent; the induced velocity

dist~ribution foir at symmeiitri-c hydrofoilU with an arbitrary e ir-

eull tionl P. Yor a camlbereOd hydrofoil1 'the1 Induced velocity du

to tinesdistribut-ion Is stA.:i.1 r1P~epreseited by Equation Uli IHoweve .-t1. 1., i -11n the methlod developed by Webor-l,- o it; c

be 'hw*that the induiced voloc Lt~y due to the Vortex distribu-

tioii for -a caimbered hlydlrofol, A"" givenl by

~*tv~ i I dx'

wh evea." be 'l'o r thle po.,ri~t;Iv-o slin Lpi o LA e uppoet'

su r~u. ild tho nekt~ft;1.Ve i.g tO ti I :i1AAWl en tuirVaco . A.1 s~o

a. IB '01 the atg.A OV ofIt'(tok lit 'a "'lrode( w I t-hepa.t the

of freeI 'lve chord Li n T, 1 :.s% the hei:t't it; of, the kci1tec t1Vkc

Canli eo 1,10 M:Ii icaov ho x axiswl'i.e 'ý,I ti th i oi s ~~

t~vibu LIon o the, hydkro Voil (nldn h odr a nd

p:lacoemei it.tl Lkcs

klooe Appwndi~x A.

IR V _-_' 61101100.1101.

7~ Y ~ .


-32-

Equations [111] and [6] toget-her diescribe the velocity dis -

tribution on the hydrofoil. From these equations, the change,

AV ,iin the velocity distribution due to injection can be ob-tained as,

AV= __CL-A [± ~e Vr7* of of f

211153

I-'~T'~~ dxf dx- ITT 0x- -L 4x

where, a,- b ef ore, the. posi~tive sign applies to the Imp'er qur-face and tho noegati~vo M.",gn to the lower Surface. The four,

termsl,, onl the righlt-hanid side represent The 01t'1'0011 On the

volocilty d:.1stribut~ioei of chaingess respectively., i.n the ffctive nl of attac-Lk. In circulation, In camber and Inl th'Ie

thicknss -i tri-butiLon * Note that; tho :t':r t three terims are

Of' opposite sigs011th two sdsof h hydrof oll , wh.llthe J.Lwat term is or 'the1 same ign onl both uIdmtl; -thus" the- 1,as'tterm w I.llno. lit)1f:"lueco the 11ift ch1ange.

Evenl thoughll Equat.ion [.1153 appeairs, tLo be qu~l~t comple~x.t1e changesO,` i.1 V(1.001t~y caZn be calekulateod :tai~rJy roadiLly if

thoe hallies 111 boundar'y - :lAY'LT ipaoe Vthcns due to111jec oem",n aro known , Isince, simple alge;braicý mlet'.iod s for evalu-t "Ing the inl t*-ergralh. s appearl~n lin"1 it; art) avail.1abLe :.n the 1,0 r

atu~:~),21).:V *Equat:1.o1 [:L",' ] can be ,%imnp-11V~od by uil:~n

the oxpkwi'e55ieý 1YI.Vou earlior rolating th1w change In bouxidary


-33-

layer thickness to changes in the effective angle of attack,

camber and thickness distribution. After some algebra, one ob-

tains,

V - V t 1_TAV=+ u *h 2i___

2 2 s{~[16].dAb dA6~

s - - dx'"0't dx' V -x--.~x x-x'-S~0

where Vu and V6 represent, respectively, the velocities on

the top and bottom surface of the hydrofoil. In the deriva-

"tion of this equation the approximation hFl has been made inthe second term, so as to facilitate physical understanding

of the phenomena involved; the approximation will not signifi-

cantly alter the numerical values of the velocity.

The first term on the right-hand side of Equation [16]

represents an antisyimmetric effect of boundary-layer dis-

placement., while the second term reflects both syimmetric and

antlsyimietric effects. The first term accounts for the change

in circulation due to a changed trailingedge con.di tion, and

produces velocity changes which are equal in magnitude but

opposite in sign on the two sides of the hydrofoill. The lift

effect represented in this term is essentially the same as

that in Equation [9] and, as already noted, ascribing all of

the observed lift changes to this effect results in a pre-

dicted pressure distribution which Is in good agreement with

the measured vwlue, except for the presence in the latter0, 'of

certain peaks located near the point of maxinnmi hydrofoil thick-

nesB and onl tho same sido as the one oil which thL injection is

made.

ý,7


-34-

"While the first term on the right-hand side of Equation

[16] is not of a form which can yield the experimentally observedpressure peaks, the second term is indeed of this form, as can

be seen from the following argument. If we assume that, to theSfirst-order, injection changes the boundary-layer thickness only

on the side on which the injection is made, A6 and A6 * will beA C

equal in magnitude as well as in sign when injection is on thetop surface, while they will be equal in magnitude but of opposite

- sign when injection is on the bottom surface; see Equation [12].

First consider tne case of injection of polymer or water on tothe top surface. For this case it can be seen from Equation

[16] that the value of the integrand in the second term has a* singularity at x x', while it is bounded at all values of x on k- "

the lower surface*. Thus, depending on the specific behavior of

S A6* it is likely that a pressure peak of the type observed in-the exu-eriments can occur on the top surface, but not on the

bottom surface. Also, asswuming that polymer and water injections..produce changes of opposite sign in the displacvement thickness

(an assumption which is consistent with the observed drag behavior),it can be seen that water injection will produce peaks of -sign

opposite to those produced by polymer injection.

For bottom side injection, A6 and A6 are. of opposite sign(though of equal magnitude) so that the integrand in Equation

[16) is bounded everywhero on the top ourfaco but not on thobottom surface. Hence in this case the preosure peaks' can be

expected to occur on the lower surface and not on the upper sur- -

face, The features noted above are exactly -the ones displayed

- by the experimental data.

*.In this and all o;her singu Lar interals occurTing in this report,. it is understood that the Cauchy principal value is the one of

conolcern'l.


-35-

Since the detailed nature of the interaction of the injected

flow with the boundary layer is not known, Equations [15] and

[161 cannot be used to make quantitative predictions of the veloc-

ity changes that will occur due to injection. However, these

equations can be used, along with the measured pressure changes,

to estimate the shape of the effective hydrofoil which corresponds

i* to the observed changes-, as follows:

* The change in the effective angle of attack is first calcu-

lated using Equation [13] and the momentum relation expressing

the change in boundary layer thickness at the trailing edge in

terms of the change in skin-friction drag7. This last quantity

is itself determined by computing the change in the pressure drag

from the measured pressure distributions and by subtracting this

from the measured changes in the total drag. The first term on

the right-hand side of Equation [15] can then be readily calculated.

If the total velocity field around the hydrofoil, including

the boundary-layer-displaceement effect, were known, then the change

in circulation Ah can be calculated using the trailing-edge condi-

tion that the velocities at the edges of the top and bottom bound-

ary layer be the same*. Lacking such infonration, the second term

on the right-hand side of Equation [15], which is the same as the

first term in Equation [1.6), can be estimated for various assumed

values of Ah. The fourth term on the r1ght-hand side of Equation

[15) can be eliminated by taking; the differeonco between the inca-

surod velocity changes on the top and bottom surfaces of the.hydro-

foil. Thus, finally, the third tomn (which represents the camber

"change) on the right-hand side of' Equation [16] can be expressed

in terms of the measured pressure changes. The method of Weber.,16

*The method of Sponcol3 provides an approx:hiiate prcoedure for carry-ineg out the step. Howover, even thIs approximate procedure is toocomplex " be useful here.

."HYDRONAUTICS, INCORPORATED

-36-

can then be used to calculate the camber, since she has tabulated

the influence coefficients necessary for the inversions of the

integral.

Figure 16 shows the results of three sample calculations for

the cases of Jaguar, Polyacrilamide and Polyox injections on the, suction side of the hydrofoil at 10% chord, 2.50 angle of attack

and at an injection velocity equal to thirty percent of free-stream

velocity. Calculations are shown both for Ah = 0 and Ah = 0.015.

As expected, the calculations show that significant changes in

camber occur only downstream of the point of injection. For both

of the- assumed values of Ah in Figure 16, Jaguar injection pro-

duces the maximum positive camber, while Polyacrilamide and Polyox

injections give successively lower values. It is relevant to note

that the change in camber is not confined to a small region close

to the point of injection, but. occurs over the entire downstream

region of the hydrofoil. The changes with downstream distance in

sign of the camber (especially evident in the cases for All 0.015)

may be due to the competing influences of the adverse pressure

gradient and polyner relaxation.

Also shown in Figure 16, for the sake of comparison, are the

results for the cases of 300) chord suction-side and 1O',"1 chord

pressure-side: injections of Jaguar at Vi/VA r .3 rate and

= 3.250. As expected, the changes in camber for the case of

pressure-us.de injection are negative (corresponding to the ob1erved

decrease in lift). it is intere.sting to note that for the case of

thirty-percent-chord injection, substantial changes in camber begin

somewhat ahead of the point of injection.

The sample calculations discussed above illustrate that the

changes, due to injection, in the camber corresponding to the

observed changes in the pressure distribution, lift and drag are


-37- • • •

-.quite complex. Thus, even when the effects of polymer injection

are viewed in terms of the changes in the "effective" shape of the

hydrofoil, no straightforward phenomenological, explanation suggests

itself. However, the observed changes in pressure distribution and

the corresponding calculated changes in camber a-re consistent with-..-those that can be expected from a complex interaction of the in- .Jected flow with the-boundary laye.r and the--e.xternal pressure

gradient.

ii..

• . - .•N

• -V2 • .,

~,HYDRONAUTICS, INCORPORATED

-38-

MV2 Correlations of Lift and Drag Data

As already pointed out, the fact that under otherwise identi-Cal conditions different polymers lead to different lift and drag-effects implies that viscoelastic effects may be responsible., atleast in part, for the observed behavior. Based on similar corre-,lations that exist in Pi~tot tube data for dilute polymer flows,Fruman and Tulin1 have suggested that the lift anid drag effectsshould be correlatable in terms of a nondimensional parameterformed by dividing the relaxation length VT by a characteristicgeometric length scale in the problem., where V is the polymer in-jection velocity anld T is. the relaxation time. Such a corro-la'tionis shown in Figure 17., where all of the lift data are showni plottedagainst the parameter (V T//C),, with c bein," the hydrofoil chord.

Note that the ordinatet; in Figure 17 represent the difference,in the lift changes between polymer and wateor injections; thits is

a logical way of correlating the data since, the rel'axation timecorresponlding to water inijection is -*ero, and since for small valueof the abscissae th rd.inato can bo expected to tend to zero.

WhileA there 1i, considerable data scatter in Figut~re :Io It, lo.nevertholesso elear that the corrolationl 1.s a valid on10 61nco allof the different curves" thownl havo esetily the sam sape.. Itis relovanlt to noto th'at eakch ofI the cur1ves, givenl in1 l"Igure 17 no0tonly correl~atos the. castes for the dilferenit pol'ymer0 Injoetions,ý1.but also the diffeorent injoetion veloc Ltiies * The oorrola"t .oit"nalso ohow that, in kvenoral, %it a fid vale o the abclosa larger

hydrofoil anglots display largea' lIft changes,,t and that the Ife oare somewhat stmallell for the 3Q,,' ehord injoetion than for the 10""'chord injection.

'Tho da.ta ohow thatL ror either injoction positionl the, mlaximum111lift effOects" Occur whenl tho value1 of the bcis is. bot'.woon 0.03,41d oAo, with st'uction-lsido itijoction givint- a lift Iiicremaso and

* HYDRONAUTICS, INCORPORATED

-39-

pressure-side injection a lift decrease. As the value of the

abscissa increases beyond this critical value, the absolute magni-tude of the lift effect decreases, and the sinof the effect

actually reverses beyond a certain value of the abscissa. Thus,

the correlation seems to offer an empirical explanation as to why~and how the exception to the rule that "suction-side injection

increases lift and pressure-side injection reduces lift" arises.it was mentioned earlier that Fruman, ui n i 7 wr

able to successfully correlate the observed changes in lift ford~ifferent free-stream velocities by plotting the observed lift

force against the local velocity at the injection slit. Thus it

is logical to attempt to correlate the data in the present caseby normalizing the observed lift changes not by the free-otrealm

velocity., but by the local velocity at the location of the injec-tion slit. Such a correlation is shown in Fig~ure 18. Again,while there is considerable data scatter., the trends itn the at

are readily apparent. It is noteworthy that all of the fourcurves shown have. similar sha ou. The feature that, iN geal

the 30O,,'-chord inJection yields smaller lift effeets than tho 10"-chord injection., may be related to the fact th~at in the formercase loess of the hydrLfofol, surface iza affected by the iiijoetion.Howover, a further corrolation oi' the da La to accounit f'or thothe inkJ ction pooition is, not roadily apparen-Lt.

A corrolation fur tho relative drag, change duo to in~jootionl,Similar to tat shown inl 1iguro 117 for lift, iz.ý ohown inl Ig vuro19 in terms of' tho paramoter Vi T/e. Tho ,,hapv o± thc eurvoo aroSeenl to be the iowwe a~u those for tho lift effoet., with theo occur-

roncev of the maximum drag, reduction coi~neidintL with tho maxitiuti* lift reduction for pressure-,side injection., and t~ho oecurronce of

the minimum drag roeduction (or zuaxinnut drag, inrax icdn

* with the muaximumn lift increase for suction-.,ido injection.


-4o-

The changes in drag due to injection can be resolved into

*two components,, namely, those due to changes in the pressure and

frictional parts of the total drag of the hydrofoil. The change* ~in the pressure drag-'can be computed from the measured changes in

the pressure distribution and from the known shapt: of the hydrofoil

by taking the component of the local pressuro force in the direction

of the free stream. The change in the friction drag an then bededuced by subtracting the calculatcd pressure drag change f rom

the measured change in the total drag.

A resolution of the drag componento is shown in Figure -0for the ease of Polyox injection at. ton porcont of' froo-,txeani

velocity. It can be soon thal fo te oe ijcin\'i ct* te pes uro daeg aotual.ly :ine roa,,tes, both for sue tion- sido alit

pre~s.:uro-oido lnjoctlionzý at all valuoes Of thet 01'e fatak

The doerotooc in ~kin-friotion drag, ý-ieemo to be near y the taim,

f or injokIL10ons-11 0.11eithVer sde ofl thlt hydrolfo-t. I and Vor a~ll n

Of *1ttafLk. It 1,t1 inltere0ýt-i1n to 110te that tilt 1% e t. that. tllo

total drat" roductt Iol f'or ~uto-Id njeet-Ion at ) J,, omal I(~o Fgur 3 :~ bcaseth 11101.0a.o Inl pressureis drag llearly

o±±sotst thoL roditotien inl zd0n-z'rlotlov drat",

For injection at. tho:~-r looatlonl thoe lal'go 1,n 111os-oluro driat" lo gt~v at oswtla.I1ngLov, Of attaoh t'l'Ijetonr

cithor o-1do of tile roilt,, *~thotli I. t. eoes I lOI-ivo att. I a rtit,

%ftneloo of atttoack * The rokilot:Lon Illn-'i IinL 'll* ticaso' is% sVtoi-what* -tim-1oir thanl that Ill the o -&odijetei

eas, -.n. itet ihte act thit. a sma 16%1vrationl "),t 0 1

hydrofoil -ti rlfa o ea. 'etdIntica.

Figues114. 10 anld %.!X) %aL o llustrato that the kinat alld I it tofeiIetsl arc Vlosetly rolka ted 11 doed, AV t~ho 1110%011, Je dp*i4 00on-

lionont alome U,. eorrelatcd 1ith (V Ta/V), it. is. elear th'at .p

xvlatiowt of' tho type uthowi ill Viue 17 u .3-1. 111w ro.sul I t


-41-

V. CONCLUDING IEMARKS

As is evident from the discussion given in the previous

sections, the effects of polymer injections on lifting surfaces

involve fairly complex phenomena. While the actual magnitude

of the lift changes are in themselves not large, it is neverthe-

less necessary to seek a rational explanation for the observed

behavior of the lift-drag ratio before the use of polymer addi-

tives to enhance the performance of propellers and hydrofoil

craft can become a reality.

In the present study, the observed lift offects were viewed

as arising from two separate steps of interaction of the inJec-

ted flow with the flow about the hydrofoil. In the first stot

the injected flow, was asvtuied to interact with the boundary lay.r

and the external pressure. gradient, in qomi .(ao yet) unpetil,..

manner and change the "effectIve" shape of the hydrofol.Th7-11

step of the interaction is undoubtedly deopendcnt on the viCeo-

elastic characterist~co of the injected polymer •ince under

otherwise idontieal condi tiens, difTorent pol.ymer give e dmJ d.'r- -

ably different result,; a detailed con•iderat-ion of tVi6 oter-

action. wtas outzlde the scope of the preooent ntudy.

In ileto01, of theý izitoraotien., tho ltginiuio the prcond ohn ,h

hydrofoil shatpe wao a~tuned to pr'duce C.tng¢• in th ttl'Oth ti.t"

velocity di.,s tributiu. on the. hydrofol, It a aloa

the la"ter Cananges bo calculated utiiiinl pottutl~ ; thel o'

Without -any further cuonide'ratiolns ofte lteasi ffecIn tenn of the ability to theoretically tn'ediet the exact iatttr•

of the lift and pre;.soure ehantgoo that van occur duo to inIIJ Q tto,aolutioti of the fis tph iwsiayv'1etlct lwutiliatilon of' the sond flp, and. t i was out.sido 1 eojie).o

thie proeent study. flowever, thle moea;ureid proesure dlstrIbtitio•is

r . . . - . . . .- : !


as w,ýll as the observed lift and drag changes can be used along

with potential theory to calculate the "effective" hydrofoilshape that would produce the type of changes mea-sured. In turn.,the effective shape can be examined to see if it is compatible

with what can be expected based on plausible hypotheses of inter-

actions between the injected flow and the boundary layer as well

as the external. pressure gradient, without having, to consider

the cietaili3 of thrp interacttions themselves. This is the approach

that was utilized in the presenit study.

Analysis of the types of Jintractioiis thtcItk lc

between the injected f:low and -the extern-al flow arou.md the hiydro-foil indicates, that four di~fferent type,, of changes,, iin t~he pres-sure distributi on can occur. These are a chang iin the0 effee

tive angle of attack due to a chanige A.1n the bounda'ry-layer dli,placement thickness,- onl the. L.side oin whi-h the neto. ae

a change in the effectivo camiber of' the hydrcvlol.1 duei to a cagin the boutidary-layer. dis"placemlenit tck s ;a ci )ango 1in thoCircull.ati on due, to a chlange" Ill the trail ingt-odge eend:1.tlonl re-quiring' t~hat the Volo('.:1tl~es' at t~he edges o'll thep anid bottom

.4boundary layers. at, thle trafl- I~ii, edgetus be t~he s.alme; atid. n,a

The qmciltitatilve features-0 0V the1 o)bser1vedpcsu' d ir~t

ts wel", a s thet w. oi T an I oa Ia Lg I Ir L I'Z 'I t I"LI1 0"1 I h o Iw .1.t~etypes,' Of iille rae tIJ,0ri desc~ribeld above (a . Cumlha t 10n1. 0"011,ed

Out u:.l;potenit.! al thee ry to do Lot'mllloi tQll "otk'.t'ct.ve" shape1 01V

4. ii hy dr01of l"I that p odees1]i oh served i oha ltge a i' at-4 Coll!mpatlbl. e with the typesof tea t.o ie.rhd. t,'ot

ing a spoec "I:c th o py Vol' the in t-eractMe s helles, theollov, (Ilk 0-

O alarationl otfferedho r l lir--1 tasU to be, Vroga rded as t-ltatIv e e 011Y.

Oil tho otheor hiand ,the, Vol low.l igý pel . i alppk)H 0ti1k. plroe out o1X-

fluslit u1, aily aalterniat-Ivo explanlations .

_71 777777TI.

"* HYDRONAUTICS, INCORPORATED

-43-

(i) Even though in many cases polymer injection on the

suction side of the hydrofoil produces a lift increase and that

on the pressure side a lift decrease, this is not always the

case. In general, for injection on either side of the hydro-

foil the lift can increase or decrease depending on the polymer,

the injection velocity, the location of the injection, and the

angle of attack.

(ii) Both polymer and water injections produce peaks (of

opposite signs) in the pressure distribution on the side on

which the injection is made. These peaks do not occur immiedi-

ately downstream of the injection slit, but occur near thelocation of the maxinum hydrofoll thickness for all cases.

- (iii) Pressure distributions in the absence of injection

indicate the presence of boundary-layer effect on the eircula-

tion around the hydrofoil.

.......................... ".

Al HYDRONAUITICS, INCORPORATED

44 -44-

RIEFERENCES

1. WU., Jinl, "Lift Reduction in Additivo Solutions," Journal of'Ilydronautics., Vol. 3, #r5, PP. 198-200., October 1969.

2. Kowalsky., T., "Effect of Polymeor Additives, on Prope-ýller Per-formiance,ý J. Ilydronautico, Vol. 5, No. 1, January 1971l.

3. Wolff., J 1. I and Cahn., F". ,. "Lifting, Surfaces in PolymuerSolutionsl," NS~RLC Report Mo3 ay, 19'11.

4. Lehman., A. F., and Suessmanti, R . T., trAfl Expriena Sudyof the [A:1't and Drag of a 1Hydrofoil, with Po01rlyer EJ ectien",Report No. 72-94ý, 000anics-,, lneorporated, November 1,-72%.

5. Sarplkayn , T .. , "Ontlhi e l'er:1.o vi-aI1WL 01 o1t iy~d v~jofoL. 11 D i :ute

o. "rmaan, 1). 11.1 . Sul da.rai T. K., Ilind I)5',Ud 01 J. ElP.1e t 0 CP ~g edi i Ii i Polylmer Injloc't; Ion on thie I1.1Y U and P~y w

i~ieis lna1. .0 rol 01,A 1,1L ooedIg o'tenteornlattona . Cont-I'(e1,01e0 On P rw' ifedue akf LoT IFIN T,7i7 ,nier.FnUa f i d\

ElglaM Paper, ,T.Thet .97

'(. 1'uiusn 1). 11. "Pu'.'i 1 M. rvl* .4 mnd 1,111i1I11'1 ft Pn and

~iwt~'~akLed hn'e loesll Repo vt. 7101 ~ Oo Lolwr

A. M.* . mk "u, a

). h o t,, I , I *1 \,k l'oii I nhott'l' A. E'. tn 511 ei 1.. N I\ ,

w*'PthfL'" Rpoirt,1" A(IR Numbo~r 1,1005, Malre) 01

~ * I ve ~'k . 1 .I 'Pt A pprox I a I onCaL I ttI tt' t-ite IJ~ I*L ot'~'miie U ~I Aerot'o I I.e '1nik 1 Iy Aoeomt ki t' -he Bounda ry aei,

%It t h An, Ii I~~ on toe Con t ~otI Pi%+' I ems 2 R. & NO. 1.)9k.01Aerolan I o -oRtsark-h klouitd 1 Ionikioii, '19434

,~ ~ ~ ~ ~~~~"h k1*Ihtn,*I * ~P ~~ mIa11o it Mt I'ta 11.1 -ir Ae otnuit. kitIloundk Mr a 't. No. * 2, eouut Ie I oaroh


-45..

12. Sundar~am, T. R., "The Calculation of the Pressure Distribu-tion over the Surface of a Two-Dimensional Wing Taking Ac-count of the Boundary Layr, Master' s TheisTeIdInst'-itute of Science, B~angalore, India, 19630.

13. Spence., D. A., "Predictions of the Characteristics of Two-DimeninlArol, J. of Aeronautical Sciencus, Vol. 21.,

No. 9, P. 577., Septembeýr 1954.

14I. Abbot, 1I. H. and von IDoenhoff A. E. ,Theory of Wing Soctions,Dover 1lPublications, New York., N.* Y.3 1900.,

15. NeIber, J . "The Calculation of the Pressure Di-stribution overth _urface of Tw,,o-dimensionial and Swept inswith Sryn'-jnetri-cal- Aerofol~l Sections, 13. &- M. No. 2)9 1, Aeronautical Re-00arch Council, Lodn,156.

16. W cb er, J. ,"The Cal1culation of the Pressure D..istribution Onlthe o urfa-ce of Th'Lick Cambercd Wincpsý and thoDsig.oCngwithl Given Preo"Iure Dis-tribution, R. & M. No. -0,26, Aero-nautical 1Res earchi Counc-il odn 97

17. Thmiltes, B13. 111COD1essl.M Ib A~oyai Oxford at the

1, 8,. uehA. 11, ad 80c etZO!. JI. P. , lFmuda it.1on of AoroedIynlamic~s,John ~ ~ o I4o In Sos, IIc . o Nw 0, rkhT 9)

.19. Friwa11, 1). 11 . and1k Tulin,1 N. P.E, -~t'es of' AddI& tive Ejectionon LLfting flydrofoib,39" A~FPaper No. 7(7-PE-27,ý 1977.

1A~


0.040 10"rCHORD SLIT ON

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IIYDRONAUTICS, INCORPORATED

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FIGURE 12 - COMPARZISON Or, MEASURED) CIIANorsI gNHWZ VRFS11tWhIHOSL CALCULAILD 1:011 A SYMMETKIC itWD' OFOIL


I ~EDGE OF BOUNDARY LAYER- -

VELOCITIES HERET MUST BE EQUAL

IL

FREESTREAM DIRECTION

()SCHEMATIC OF EFFECT

NEGATI\VE CAMBER

1

LT

CHANGE IN ANGLE OF ATTACKEFCTV

DUE TO B.L * DISPLACEMENT CENTERLINE

ITL T

BASIC HYDROFOIL

(b) CAMBER EFFECT

FIGURE 13 - BOUNDAIRY-LAYER DISPLACEMENT EFFECT ON A

SYMMETPtC HYDRUOFOIL


CHANGE IN TRAILINGEDGE DUE TO A

INJECTION = T2

jT

CAMBER/WITHOUTTINJECTION/

CHANGE IN ANGLEOF ATTACK DUE

CAMBER WITH INJECTION

FIGURE 14 - EFI'ECT OF POLYMERINJECTION ON BOUNDARYLAYER DISPLACEMENT

HYDRONAUITICS, INCORPOftAED

1

lMXIU (POSITIVE) CAMBER CHANGE

L T

()THINNING INCREASES MONOTON ICALLY WITH DOWNSTREAM DISTANCE

1

K ~MAXIMUM (NEGATIVE)CAMBER CHANGE

(6) MAXIMUM THINNING OCCURS CLOSE TO POINT OF INJECTION

LT

()MAXIMUM THINNING OCCURS CLOSE TO, BUT AHEAD OF, TRAILING EDGE

FIGURE 15 - TYPES Or BOUNDARY LAYER INTERACTION


0.002()Ah .015

a JAGUAR0.0015

e PAM

0.01o POLYOX

a=2.5?or 3.2ý?

0.0005 v /Vc = .3

O -0.00050

P, -0.001 30 'CHORD

kINJECTION-,,0.0025 10 -CHORD

Sb) t-h 0.0INETO0.002

0.0015

0.001

0.0005

0.0-

-0.0005 --

-0.001 *"-1 rwSURE

SIDE INJUCTION

-0.0015 L0.2 0.4 0.6 0.8 1.0

X/c

FIGUIX 16 - CALCULATE D CAMBER DISTIRIBUT IONS

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A-I

APPENDIX A

VELOCITY DI7TRIBUTION INTEGRALS FOR SY1.2:IT•_CAND CA:.':BERED 1VDRCO TT'-_

Weber 1 5 has deduced an integral relation for the velocity

distribution on a flat plate at an angle of attack, a , by using

conformal transformation. In particular, she has utilized the

transformation

to transform the flat elate - -. • _in o a circle f:.o-- tl.e

notation, see Reference 15). The velocity d i•s-4C i, _4 .b u.IV

plate for any arbitrary c..cvatton aroun. 1. s . -he.n e ,•i'.

in terms of the co,.lex vel•o•ty potentia 7

V -N~~- V _ , ,

For the 1Kut a-Joul ,a kowe' :oito o b e s ax z f

edge, given by mus , mut be a ata.nation -01nt, thuZ r .I!.%!'n:

tha t r = it= -7rs . Then., 16a the p'-ysi zsl cooinate

Equation rlA-2 - 2 vaj,

V va

*or an arbitra'y eiktcula:ion r d,,. by .quai."i

[23, Equation [A-]2 yields,h -x

v ,.- va_• . .••; :

Of course, Equation tA-4] reduca3 to Squation CA-31 W1011 h - I.

Foil a camber'ed hydrofoil, Webo,'5 has snown that the velocitydistribution is given by,

M."


A-2

when the circulation is specif'ied by the :-cutta-Jo'.ýowski condition.

When the circulatlion is soecIfied by Equa-ions, [13 and [2], Equa-

tion r-1-51 becomes.,dz 'xe'

-I. T 0 1 -C1 ~

Not tht quations [A-51 and [A-6] eue ~~~iey to

.Equati-.ns :A-33l and A4wher. Alz /d=-

HYDRONAUTICS, Incorporated

B-I

APPENDIX B

LIFT, DRAG AND PRESSURE MEASUREMENT DATA

(DEC•EBER 8-19, 1975) ON THE NACA

634-020 TWO-DIMENSIONAL HYDROFOIL

VF = free stream velocity (V, used in the text)

CD = drag coefficient

CL = lift coefficient

X/C = chordwise pressure-tap position

CPP uressure coefficient on the pressure side

CPS = pressure coefficient on the suction side

Ending N means "no-injection" case

Ending W means "with-injection" case

Ab

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