Experiments on ventilated vortex cavities

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Report No. 11.6

May ¶966.

LABORATORIUM VOOR

SÇH EEPSBOUWKUNDE

TECHNISCHE HOGESCHQOL DELFT

EXPERJMENTS ON VENTILAPED VQWIEX CAVITIE2.

By N.C. Meijer.

experimente concIliated at the Hydrodynamics

Laboratory of the California In8titute of ech

nology Paeadena, _{a1ifornia, under nubor E 110.6}

Contract Nonr. 220(43) of the Office of Naval eooarcb Department of° the Navy, U.8.A..

L

i

The report haa been prepared at the

Shipbuilding Laboratory of the Technological

Univerity..Delft

Ho1jan

aß a

part of its own acientific program.

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s

I

A description is given of xp.z'isents on the ventilation et trailing vortices of hydrofoils.

Oha.rvetione of genera]. cliaracteri.etice of advancing

oavi-ties are analysed in view of one specific caes. Tb. spproxi mat. theory which is developed gives good agreea.nt with re gerd to the lower liait of the angle of attack of the wing at which a ventilated cavity will aove upstreas.

Rsults et pressure meaeurenient. in the vortex core ars given th.eô indicate that an exa.ss velocity occurs in the vortex

axis, frodi which it is conoluded that the flow is predominant-ly non.sviscoue and laainar

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II

Table, of Contente.

page

List of Tablee _III

Liet of Figuree _IV

Li8t of Sy*bol _VI

Introduction I

Experimental S.t Up Z

ExperjasntI. Procedure 4

Observation, on Bubble Relsaas Observations on Eubble Capture

Observations ori

Advancing Cavities

6

Pressure Measurements in the Voz'tex Core ₇

Air Flow Rate and Cavity Preesur. M.aeurem.nte 8

Spontan.ou. Vortex Ventilation ₉

Introdno tien to Analysis 10

An*]yei., General

Tb. No Ventilated Vortex ₁₅

The VCti1ated Vortex ₁₆

quantitative Analysis, Pitot Tube Measurements ₂₂

Statio Tube Measurements ₂₃

The Cavity _2k Acknowledgement ₂₇ References ₂₈ Tablee Figuree. £pp,ndix I. 29 Appendix 2. 29

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List of Tables,

i _{ßpeoifioatione of the Free Surface Water} _{unne]. and Particulars}

of the Hydrofoil Models.

2 Approx1ate Distance and Time of Travel of Air

_{Bubbles in}

the Core of a Trailing Vortex.

3 Obs.rvstjona .1 Ventilated Vortex Cavities behind Model I - without flap.

k Obe.rvatian. of Ventilated Vortex Cavities behind Model 2 with

15° flap.

5 Air plow Rate and Cavity Pressure Dt* with Model No. 1 without

flap.

6. Air Flow Rate and Cavity Pressure Data with Model No, 2 with

.15e flap.

7 Air Flow Bate and Cavity Pressure Data with Model No. 3 1/8 inch flat plate, sharp L.E.

8 Air Flow Rats and Cavity Px'eaaure Data with Model No. k 1/8 inch flat plate with rounded L.E.

9 Air Plow Rat. and Cavity Pressure Data with Model No. 5 - A..wing,

bevel *t euction side.

IO

_{Air Flow Rt. nd C*vity Pressure Data with Model No. 5 -}

_a-wing;

bevel at pressure side.

5A-1OA R.aarka at Tablee 5-IO.

11 _{Static Pressure}

Meaeur.menta

_{in the Non..Vantilated Vortex.}

12 _{Stagnation Pressure Meaeurement. in the Non-Ventilated Vort.x.} XII

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4.t of Fiura.

Fig. 1; Pz.ba mounting unit.

Wig. 2; Static and stagnation pressure probes.

Fig.

3;

Air supply tubes

and combined tube.

Wig.

1+;

Experimental ..t up.

Pig.

5;

Path of captur. of very small bubbles.

Fig. 6; Sketch of capture, growth and zalease of bubbles.

Wig. 7.1; Saall bubbles captured by a weak vortex, (Table 3, No. 19).

Fig.

7.2; Larger bubbles capturaa by a weak

vortex. (Tabu 3, No. 19).

lig.

8.1; Too litti. air for s. cavity to develop. (Table 3, No. 17).

Fig.

8.2; Inception of a ventilated vortex cavity. (Table

3,

No. 18).

Fig. 8.3; Th. cavity aovad upstream. (Table 3, No. 18).

W

_Fig. _9; _{Influence of the angle of attack on the vortex cavity for-.}

mation with Model 1.

Fig. io;

Influence of the angle of attack on th. vortex cavity for. mation with Model 2.

Pig. lii Influence of the rate of air supply on th. vortex cavity

formation with Model i at 11.5 degrees.

Wig. 12; Influence of the rate of air supply

on the vortex cavity

formation with Model 2 at 11,5 degrees.

Wig. 13.1; Free swimaing vane type cavity. (Table 3, No. 13).

Fig. 13.2; Tre. swimming cavity at s.oet advanced position with Medal I

at 11.3 degrees. (Tabla 3, No. 13).

Fig. 13.3; Van. typecavity in a very persistent position

befor.

at. tach*.itt Nadel I at 12e. (Table 3, No. 1k).

Fig. 13.1+; After

attachnent

th. cavity reduced to a cavity with a re.

entrant jet. (Table 3, No. IL1).

Pig. 13,3; xe.ss air is shed threugh a funnel;

advancing cavity.

(Table 3, No. is).

fig. 13.6; Th. funnel disappeared while the cavity advanced. (Tabla 3

No. 15).

Fig. 13.7; A short expocur. photograph of the cavity of Wig. 13.6.

(Tabla 3, No. 15).

Fig. 13.8; Fr.. swimming re-entrant jet type Cavity øf analysis. (Table 3, N. 16).

FIg. 13.9; 2all cavity, ade4 by a string of tiny bubbles. (Table 3,

No. 16).

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Fig, 13.10; Without the etring of

bubblee the

cavity stays behind. (Table 3, No. 17),

Fig. 13.11; Bottom view with both vortices ventilated. ('iable 3 No. 20).

Fig. 13.12; Behind a flapped wing the cavity is very blunt.

(Table ¿I,

No. 8).

Fig, 13.131 Th. cavity attached to the flap of Model 2. (Ttble k, No. 8).

Fig. 14; Sketch of an interesting case

at low velocity. (Table k,

No, k).

$ig. 15.1;

Uncorrected

tatiø pressure reduction

in

wet

vortex core. Fig. 15.2; Uncorrected stagnation pressure reduction jn wet vortex oar.,

Fig. 16; The influence of air pressure on tip vortex cavities behind

different hydrofoil modele.

Fig. 17; Influence ot' attached tip vortex cavities on the lift coef-fiaient of different hydrofoil modela.

Fig. 18.1; Inflated free swimming vortex cavity, (Table 5, No. 26).

Fig. 18.2; Short exposure photograph of the cavity of Fig. 18.1.

Fig. 18.3; _{Inflated attached vortex cavity. (Table 5, No. 27).}

Fi1. 18.4;

Short exposure photograph of

the cavity of Fig. 18.3.

Fig. 18.5;

Saae as Fig. 18.1; bottom view.

Fig. 18.6;

Same aa Fig. 18.2; bottom view,.

Fig. 18.7;

Sa*e aa Fig. 18.3; bottom view.

Fi1. 18.8; Sam. as Fig. 18.4, bottom view.

71g. 19.1; Inclined aide view of spontaneous vortex ventilation near afree óurfaoe.

Model No. 1; 11.5e; U .44 m/meo; submergence is 0.24e. Fig. 19.21 Short exposure aide view of the flow of Fig. 19.1.

Fig. 19.3; Bottom view of spontaneous vortex ventilation near a free

surface.

Model No. 1$ 11.5°; U 3.12 m/eeo; submergence is 0.27e. Fig. 19.4; Same as Fig. 19.3, but the submergence is 0.20o.

Fig. 20; _{Th. shape of the hypothetical cavity compared with the cavity}

of Fi1. 13.8 and the distribution of sources and sinke.

Fig. 21; Schematic of trailing vortex flow.

Fig. 22; "Rankinø radius" relative to the half span, an determined

from the pitot tuba measurements.

Fig. 23; Relative excess velocity in the vortex core, as determined from the static tuba measurements and from Fig. 22.

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;4qt

of 5Y!bi].5

VI

- span of the hydrofoil o chard of the hydrofoil 0L liftooef fiaient

d characteristic probe diameter

g acceleration of gravity

k

a

half diet*no. between source and sink distributions

i * half length over source and sink distributions

i

a

masa flow rate of air

P local etatio pressure

r a radius relativi to vertex axis time of travel

u

a

basi axial velocity component

Vr iseal radial velocity component

VQ lessi tangential velocity component

U - trie stream velsoity

r - 1). distano. behind the trailing .dg. of the wing

2). absais of cavity model, positive downstream from midpoint of

cavity along the vortex axis

.

a engi. of attack of th. pressure ud. of the hydrofoil

Y - vorti.ity

fi

s eau density of water

s' kinenatic viscosity of water

- angular velocity around axis in Rankines vortex model - stream function

P

_{- circulation}

Q

a

angular coordinate

co indicates positien at

r*co ,

except in r, , which is at

xs..00.

O indioates position at r-O, except in r0, which is at x-O I _{indicates the transition atreamtub.}

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ExDsriSents on 'f.ntllatedovtexCavj.tjes.

Br NfC. Høijer.

Introduction.

During past

_{eperimnte Which dealt with vapour or efr tilled cavi.}

ties in liqutd flows, certain phenomena hve been observed for which. it

«* difficult to find an explanation, Zn the

aiit Anthony Fall; Hydrau

ic Laboratory of the University of ?{inneaota, air

WaS found to enter

the vertical working s.oti.on of the water tunnel

through the trailing

vertiese of a finite Npan hydrofoil mods. which

was b.in

tested

an4

ventilated the cavity on the model LI1,[23.

_{urtng experimente,}

con-ducted in the Free Surface Water Tunnel of the California Institute of

T*chdiogy (.X.T.), the entrainment of aire supplied to the wake of

dico was etudtad. At high air flow rates two trainling vortice. fille

with air, werf found to develop, through which most of the excess ai

disapps*rød

E32.

In on. case air entered, in the other air left the cavity through

trailing vortices. A sim»].. conclusion was given: the traiing vortices

tarnisd

_{path through which air Waa easily transported. The explanation}

ieemed to be obvious: in the first cace the air war transported

Upwards,

in the directien of the buoyancy force arid in the second cace the air

moYid downstr.a

_{and upwards becauce o}

_{drag and buoyancy forcee}

_{The ob}

servation. beome more intere8ting when at O.I.T. 4i'

was dded So*ø die.

tane. down.tr.m ct

_{finit. span hydrofoil to an already ventilatd}

trailing vortex. The air filled vortex core wideried not only downstream

et th. point of supply as could be .xpeøt.d, but also for

_{aoci distano.}

upstr.a., Th. d.taaeter appeared to be conetant, the forward end being

blunt. More interesting still are observations by Rassen at Langley t]

and others. Thee. investigetore tested finite epan hydrofoil.

near a free

surface and observed t)*t under certain. conditonn air filled cajte

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Th.. oavit1e *ove uptet

a&in un&er

ttob.

tbeiee2v,

to tle ZyroZoì1, eosnetii

_{tztgg*riug u ventiat.4}

Ovi.ty whi

»att1y covered the in4e34 un

_{cui*e4 vibration.}

Tbe reort.d wutediites oi the

_{perforsaoe of finit. spun iydra.s}

fpila and a3mo the analogouu probløm of

t}i

uotion of air by ehipa

pròpeiiera at-Ø aufficiant

reaeone Cor a more detailed iVøBtißetion of

vortex v.ntiZtjon to im»rove the know'edge

o

thie ld.nd of flow, wbi

ta the pur.poae of the present etudr,

xDerim.nta). Set ,Üj.

Tki. xpeiente were performed

_{ri th tree 3urtace Water Tunn*j of}

the Rydrodynamicu Laboratory at the

_{California Institute of Technology.}

$eeificat&ons of this tunnel are lieted in Table 1. )1or the purpes. of

genarating vertices several existing hydrofoil models with finite aspect

ratio were used, of which the epacificutione

_{am also listed in 'Tabla 1.}

Theme modale ware chosen simply- becauma

_{they differed from such other and}

were avuijahi..

Ô

With the xo.ption of th, delta

_{wing (Model No. 5) which was etng}

ouht.d, all the aodeTh were mounted

_{on a strut which was conn.e4 to 'é}

Tusk M

_{XX straingage balanos. 14fb forces up to 1}

_{pound. at sers moment}

relative to the balande axis, could be

_{measured, A "intel" dgttal volt.}

acter was used to indicate the vaines of

_{the lift, averaged over a relu}

Uvei)' long period of time. Of the forcee

_{and moment, acting on the}

mo-dais, only the litt

_{Was measured, Si000 it is directly re3.at.d to}

_the

strength of the trailing vortioeo.

Ter the purpose of supplying air and

_{measuring pressures at un}

_arbt,'

trury point in the flow, three simple

_{probe mounting devices weze mud..}

On. was connected to

_{a crobar which wa supported by't1ie side walle at}

tb. Working aeotton

_{th. others were ConneCted to}

_{a I inch diameter tube,,}

mounted above the water surface

_{in a longitudinal orientation.}

_{A sketch}

of these units is given in P'ig.

1. A useful

_{euture of the mounting unit.}

is that they were provided with a aZeden' surraoe pieroingbody with tse

thin eht.lds. Of.

_{W&5 kept submerged to prevent tb. wake of the probe}

_from

becoming ventilated through the free water surface, The other was above

the water and acted

e

_{spray shield.}

_{iur.gi 2 and 3 show the outline.}

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ór

b. measurement of preseurs ¿ocurring in the core of the vorti'

oes, a "stathait" e.eotx'o4o presuve transducer, tipe P251s2, Wa8

used in connection wtK a "Ba].dwiri"

strain indicator. Zn sit

e-daavoui' to reduc, the unoertnties conneoted with this method of *h$i

ui'emant, a few

suremsnts were iade with the aid of

penrecoz'.

dei-a and iiatohing amplifir. This system was more complicated ande

b.-cua. tus ¡'eeu3t8 eeem.d to be ooiipsrabia to those Otalned with the

"ßaldwin" system, it was abandoned, For aVerage pressure readings,

iie

those to obtain tunnel v.iooity and cavity pressure, fluid sanoastera

were used.

Fòr the meaaur.møflt of length along the vortex axis an ordinari

soale with 0,01 feet diviejos was faetened to the exterier of the test

eotion side Window. 8teady valiiea were read With the aid of an &mag..f

th. ey. in a small mirror, which was held flet against the window;

un-steady valu*

were timated. At each position of the hydrofoil relativs

to the .ngl. of attack, the zerp point of the scale was adjusted

.iateh

the trailing edge of the model.

The sise of air

bubbles

and th. tbtck

ness of cavities were etimatød roughly.

The submergence depth et the trailing edge waa not Constant but

varied to som. extent with the angle of attack according to the follewng

forniula.i

d0 -

i.ai

i - co(ik° - .)

in which 41Lo

submergence depth at o

and d

8ame at a0.

This variation o' the submergence depth was considered unimportant for

th

present experimete.

The air flow rate waa mesoured with a conical tube type "Flowrator"

by Fisher Portr using an aluminurn.float.

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Ext,aris.nt1 }'rocdurs.

B.aaueo the

ct tht at the oneet of ths expejm

the

tho'

baA no in8ight wiat8oßver WLth reepeot to the

_{ie3.d which Iad to b.}

Ix-pLowed, the

eXerimGfltS

were oonduote

in

rather random way.

To begin with, the hyrofoi1 odei No. I (Table I )w

inet1ad

and au' wae .uppli.tl trito the tratXtn

_{vox'tx nearest to the obaerv.r.}

A

Qndttiofl

W*

ø*tbiiehed tn wioh

oavttr waa der1op.d nd ht

*.ved aome distance upßtream An endeavour wa

_{nde to meaøure th}

_re

auras in th iain cvitr anò. in the cavities which formd a aonnottng

tub. between the paint of supply

and the main

aavity. Th.s meaauxem.nte

failed, beøaua. the oavttte

failed, to remain in place when the supply

tube entez'e

the vortex regien. It wa

then decided to rake oberYatieri*

under different conditions of angle of attack, rate of air supply and

velocity of the main flow. Conditions in which aiall air bubbles

_entered

r left th. vortex con, were included in order to obtain

_{soma inforuticn}

about the balncø of gravitational and centripeta]. influ.noes.

The obaeryatjona gave rise to the presumption that the chap. of the

hydrofoil model influenced the behaviour at the vortex ventilation.

Th.r.1ore a hydrofoil modal, baeically equal to Model No. 1, but with

30 degree flap wac installed. With this model the

vortices seemed to b.

totally destroyed b eddies and no steady cavity could be obtained. An endeavour to ventilate a trailing vortex of Model

No. 2, which was again

related, but had a 15 degree flap, waa

more successful and indeed

differ-eneas in behaviour were found to exiat

Thee. differences indicated that the actual flow conditions in the

vortices behind the modele differed and an attempt was rade to obtmi

more detailed information b

_{dying streamlines with perangsnat., wtth}

little eucceai,however. As a next step it was decided to measure the

static pressur. distribution along the fully wit

vortex

axia of both

mo-dele under different conditions.

The observations had shown that relatively steady cavities

øou].dba

obtained when the air

was supplied oloae to the trailing edge of the

hydrofoil and new endeavoura were made to

_{measure the pr.asur. in. the}

cavity as a function of th air flow rate. Successful

_{measurements could}

not be obtained until a new prbe was developed (G, sig. 3 )

_{in which the}

pressure prob, and th. air

uppl

tube were combined.

Thies expenim.nts were extended ta inluda the

modela No.

,

_and

(Table 1, in rer to dotermie

whether further differences depending

(12)

upon. hdrstoit shape could be expected. Anaisia ot )Ie dat& 3.e t

the

tiria]. est of experimente in which the total bead in the coreot the _vo

ticee behin4

the Model j: and _{Was meamured.}

This

random wayof experimenting haí resulted

in a now and then

rather unsystematic set of data, from which, however, it je felt that some useful information can be derived.

ba.rvation,

on Thabble Release.

Sital]. aii'.bubblee

with

an estimated diameter of 0.1 inch were in-.

ected ,nto a tratling vortex, generated by Model No 1, operating st

ow free atrfam velocities and model anglas of &ttack of 11 and 12 de. reee.At these canditone the air bubblee followed the vort. axis for

some d±tanoe and diverg.d from it before reaching

the

end of the test

section of the tunnel. The length,òf travel in the vortex as well as the Ufttorc..on the hydrofoil and the velocity of flow were measured. Tb. remultf are listed in Pable No, 2. The time of travel is included,

as this i the important parameter In Prandtl'a viscous vortex th.ory(51, The emtiEate of bubble size is rough. After' release the bubbles

follow-ed a curvfollow-ed path towards the water surface. Obeeryation. r1 Bubble Capture.

In Pigure two ketohe are shown in which the path of oapture of small air bubbles is given. The supply probe was placed in such positions

that only part o the bubbles was trapped by the vortex. An inereaee of the air flow rate caused an increas, of the bubble sia., but dId net change the path of capture appreciably; the bubbles, however, did leav.

the vortex cor. sooner. Beoue. of the low

velocity of only 3 tpe, the vsluee of velocity and force are only approximations. Under thee. Con

dtions no advancing cavity was obtained by further increase of thesir flow rate.

k sketch showing the eltuation at a somewhat larger velocity (.? tps) i. shown in Figure

6.

In this case stroboscopic 3.igbting wa. used

to study the shape of th. bubbles. It was observed that the bubbles grew in sise and merged as soon as they entered the vortex core. Larger bub.'

bleashowed a tendency to *dvance upstream but the shape remained, very

irregulsr They left the vortex slightly upetream of the point of en

trance. The smaller bubbles were tranepòrted downstream in the vortex core, With less air _{»liod the bubbles did not merge and were aUwSpt}

(13)

.6.

in

the

situation of Figu'e 6 the angle of attack waS 1. With

OnJ4 half degree less it appeared to be poecible

to obtain advancing

oavites in the vortex with the same

position o? the eupplr tube z'eìa.

tive to the hydrofoil. Zn this case the irregularly ebapedbubblee

changed upstream of the point of entrance into * smooth ellipioida)

oat

ty, aven at auch lower velocities of the main stream.

The process of capture of air

bubblea,

when these are sucked down

from th. free water surface,

makes one think of the suction of air throu

the discharge of water from a tasin. The

fact that a vortex caused b

the

discharge of water through a hole in the

bottom of a basin, intersects

the water surface, represents e big jfferenoe, Zn thi case air enters

thÒ core of a' vortex directly and the phenomenon reaembles that

of the

advancing cavity rather than that of the oapture of bubbles.

Some short expoeure'photographe of the capture of aix' hubbies are

shown in Figure 7. Figure 8

shows the inception of a cavity in a trailing

V6rtaX.

Observations on advancing cavities.

The air supply tube E (Fig ) was placed some dttance downstrea

of the trailing edge of the hydrofoil, faz' enough for the oavity' to move

fr.. of the tube,. This set up allowèd the observation of the g.n.r] b.

tiaviour and ehape of the cavity. The

vortices were gen.z'at.d by the

N.A.C.L,.16O6 hydrofoil Models No. i and 2 (Table i).

The rate of air supply was measured only to provide some means of'

comparison. Xts absolute magnitude is of little value, because much of

the aft' wa lost directly into the stream or downstream through the

_vor

tex cor.. The values are presented as a dimensionless coefficients

ci _('ryd4)pt,b

Zn this way the mase of air used per unit time is compared with the *a$ flow of water through a dise with a radius equal to half the span of the hydrofoil. The use of mase flow ratio, rather than volule flow rates seems to be reasonable as most of the'air ie transported downatz'.amwjth

the water in a heterogeneous pressure field.

The specified observations

ara presented in

_{tb.TbleNo. 3and 4,}

whex'ea a survey is ßiven in the Figures 9 through 12 The influence ot

(14)

ca-viftee Le shown. £hotographe, wtih are reterr.d toi. the tablea,are

given in Figure 1. Figuri i4 ehow a ketoh ot a epeoia]. cavit7

UQn which W&5 obtaine4 at a 3.ow veiocft'.

Pobab3.y the moet important finding of these experimente ta that

the optimum conditione for free moving oavttiea appeared in oonnetion.

with different lift valuen when difterent b$rofoil modele were ue.d

foz' th. generation o tb

vorttoea. It te

oneidered tc be accidentai

that tu theee conaitiona the veuee ot the nominal angle of attacic wer.

approxiateìy equal. Another COnClUSIOn is that laminar flow oondtUone

t

the vortices seem to support the oavitLe. This te concluded from te

fact that the beat ftee swimming cavities could be obtained with low or

moderate velootties and that bigbl turbulent flow disturbed thrn.

Pressure measureaentn in the vortex core.

In order to get more information about the possible differenoe

in

the vortex flow b.htnd dtfterent hydrofoils, which may account Sor the

observed differences in the cavity behaviour

experiments were performed

to measur. the pressure

_{s near to the vortex as practicable and at vary}

ing dietance from.tbe trailing edge of the hydrofoti.

o begin with the static probe A 'flg. 2) was used in conjunstien

with a '$tatham" electronic pressure transducer and

a

Baldwin" *traic

indioator

Care was taken to have the metering system crittoally damped

S-n order to reduce the danger of reading overshoot

_{r attenuated vale}

due tò the ver)' unsteady transverse position of the vortex ocx's.

At every selected longitudinal positton of the probe a

_eurejs*d

tng wa

_{taken at a large distance from the vortex axie, where the preC}

eure proved to be independent of transverse shift. Next the vorteX cor.

wae made visible by the bleedin.g of very smell air bubblae into the vor

t.x core from e location upstream of the hydrofoil. Into the

now Yietbi.

core the probe was piaced in such a way that the string of tiny air bub.

biss waS scattered by the nose of the probe,.

At the larger djstacee

ehjnd the hydofoil, the position of the

vortex CorC proved to be very unsteady, so a good average poaition had to

be found. Aftir this We done, th. air supply tube was removed and the

r.eeur. probe WaS *ade Cree froa possibl, contamination with air. Now

the pressure was read that was a minimum dux'ing a relatively long

_ertod

(15)

fl

lyiag the m.thod described abov.

the

4ittez'ene

0',p

W5

_{eaeux,ed in the cores QL}

the trailing vorticee behind the Modele

to. i and 2, or different anglea of attack and in sosse oaees with di.t..

*rent free stream v.1octtjee. A orreotjon for the trenevarse diplaceo ment of the vortex lines

by the probe

i developed later in this report.

In one condition with hydrofoil No. 1 a smaller probe (, 2)

Wao used in an attempt to determine the influence of the probe size.

Zn this oase the tBa1.dwinh strain indicator was replaced b)r a

"Brush"

penrecordex' with atahing amplifiers. hie

075t0m

proved to be less

Convenient and gave no reason to distrust the af.mplar set up, which

th.refor was used again in the Airther experimente. As was expected,

the

amUer probe sensed slightly' lower oxtrese

pressures.

In the same way measurements were aonduøted wtt)t a total bead

tube

(D, Fig. ). Xt was aesimed that it the probe was concentric with the vortex axie, it would not sense the rotational component o the flow.

this eumption was supported by thé consideration that the estìated agnttude of the tangential Velocity is so great that any influence

of

it on the measurement would give irrobable results, euch as hays not

be.n obtinsd. Again a correction tor the

dieplaernent of the vortex

lines is zteceaeary and will e estimated later.

An additional measurement was made agi with a mallar prabe (C ,

E&g. 2), in which caae somewhat sigher preseures were measured than with the larger probe.

The results, in dimenejorlese fore, are plotted in

ftguré1. Itis

shown that in general the flapped model produces larger pressure gradt

ente in. the vortex core than the smooth NQcI61 No. 1.

Air ].ow Rate and Cait Pree8ure Measurementè,

Por the purpose of measuring the air flow rate and pressure n

vor-tex ca.vitiee, probe G (Pig. 3) was made. The idea, inøozporated in this

tube was, to keep the pressure probe continuously surrounded by air so

that, whenever the cavity moved away from it, the probe would. not be

fouled by waters

Although the shape of the aix' supply nozzle could be

adueted to ensure constant pressure measurements with different air

tlow rates, the prformance proved to be dependent on whether óx' not the

oszle was wet. For this rSasÖn the

probe was

oli.bx'ated for different

air flow ratee with thé tibe n half inch aboo te Water urfae an4

with water eplahed against

t.

(16)

The po

_tio

_{at the probe and the nle at}

_{ttak of tite aiodel}

Were sele4t,d to form a cavity eurroundin

_{the tip ofthe probe.}

Thte proved to be relatively

_{easy when Nodal No. I *a5 ueed. In aU}

pthei' oases djffjcultj

_{arde. with the effect that either the nose}

of the cavity would cling to the probe

_{or a cavity which was attached}

to the hydrofoil hd to be accoyted. In

some oases even a fully cavi

tating condition of the hydrootl *o4el could

not be prevented,

The data obtained from these experiments in which all the models

from Table 1 wore ueed, are lièted in the Tabiee5

_{through 10. In these}

tables the asvit preosure is given

_{as the cavitation index:}

po. * p0

The air flow rate te nade

_dimensionle

_{in the same way as before.}

Characteristics obtained tres the tablee

_{are shown in Figure 16. In}

them. experiments again the lift

_{oroe was neaoured. A diagram showing}

the influenc, of attached vortex

_{cavitie, or tip cavittes on the}

lift-Coefficient i. given in Figur. 17. It must be noted that the

coeffi-ciente are nöt corrected for taro forces. Photographs of some of the sa

vities are shewn in Figure 18.

With th. air euppli.d ;intó the

_{eian Vortex cavity itself, it proved}

to b. poesible to inflate tbe

_{cavity. Much larger sizes were obtained}

than with th. air supplied through the

_{downstream tail of the vortex core.}

The excess air which Was not. entrained through

_{the core of the trai]tng}

Vbrtex, left the cavity through

a

_{funnel" whtoh developed above the tail}

as ta shown in Figure i8 No.

_{and 2 for a tree cavity and in Figure 18}

No. 5 and 6 for an attached cavity (compare

_{Ief. 3).}

8youtan.ous Vortex Ventilation.

Although it is not tus purpose of this report to give a detailed

study of the vortex ventilation near

_{a free water 8tr!ao. that te,. with}

the air eupplied in an uncontrolled natural way, it wae thought that for

oomplmttneaa 'ose photographe should be included

which show how vortex

ventilation is encountered in practice. The data of

the related Figure

19 are given in Tabel 11, Froi the photographs it can be observed that

both size. and position of the oavjtje

are covered in the experiments,

X)uring the pteparations for tbe photographe and tres

_expert*ente

with Model No. E it w

_{oberved that the.twø cavities which}

_{occur in}

* .io *

(17)

lo

-th. two trailing vortices of the hydrofot]., never had the same si2e or position. If the conditions were favourable for attachment of the cmvi-ties to the hydrofoil. tips, these tipe were never ventilated ¡t the asSe

instant, which resulted in te occurrence of transient rolling and yewing

morsnts acting on the model.. If the conditions were favourable for the

occurrence of auperventilatlon of the mideectione of the wing, it often happened that ene att*ched tip cavity supplied air through the extremity of the leading edge to the suction side of the hydrofoil; ventilation of

the opposite half of the model span was then

prevented by the central strut and an asymmetrical load was maintained until the second half o

the model vu ventilated through

the second trailing

_{vortex. It io}

evi-dent that these phenomena are caused by- some ever existent asymmetry in the flow,

Another important observation was made when an instable cøndition occurred. I* this casa it appeared that

the

entrainment of the air

which

covered the suction side of the model, waateo auch to maintain

th.

con-diti.n. As a result the who]., cavity broke down, after which a new vor-tex cavity developed to start

a

new cycle of attachment, superv*ntilation and breakdown.

Itoduette

to Analysis.

At first sight perhaps it obvious that a thorough knewledge of trailing vortex flow le necessary to understand the phenomena

in-volving these swimming vortex-cavities which

have been observed in the present investigations. Some basic knowledge can be obtained from lit. rature

_[5)(6(7][8),

but too

little

appears to be known about the subject to expect suoceesful quantitative oomparieona between experiment and

theory. Several comparisons of computed and measured axial pressure

distributions in fully wet vortex cores

were attempted,

but all with

little success. In this respect it should be neted, however, that the experimental results h&ve been

obtained

under very difficult small seal. oonditione

As regards vortex cavit&tìon, th. lack of knowledge of theO-tailed flow pattern in the f*Uy wet vortex, precludes quantitative

agreement between experimental results and.

_afly

theory which ma; be

veloped in

this field. It is

found, however, that a reasonable und.r-standing of the observed phenomena can be obtained on the baeia of the simple vortex modelo

(18)

11,

Tha diftjouities which arS encountered when

_4..cri»tj

_{of e}

tr*iling ortz is attupt.d, ax'. Qausad by th. fact th&t this

vortex

is

composed of

sany liùe vortiaes

at Varying strength, whoea

different

origins li. in a nearly fiat pian.. The line

ørtioe

roll up aroun6

Cash ether and each

bas a viacou. oorC. Usually it

is

aisum.d that aU

lins vortice. are rolled up at approximately one chord length behind

the wing and tb*t farther dowrzstre** the i.iquid, rotate. as if ens ing

le

vortex

existe with its own axieyrs.tnioal viscous core. If

flow i aeeu*.d, it s.øm. obvious that the velocity'

distribution

_in,

this core must reflect the nature of the origin of the vortu,eo it *uat b. assumed that laminar trailing vortioe,, Shed trqj 4ift.rert wings muet have diffør.nt radial distribution. of velòcity, and pr.s

sur,. It turbulent flow is aeeumed., 1,t can be expected that the

differ-oncee disappear with the dietano. from the wing. With this assumpto1

a deacripttu ut tho vortex

f1w wtU certainly not be eaeia.

If an air fill.d cavity in & vortex je consiôr.« _{of which the close} sectional area

i8 large as compared to that of the original vortex

_core,

many of the diftjcultje. dieap.ar. The velocity gradient. at th. cavity boundary will be relatively niall because of the relatively large radius

bere, and therefore viscouS effects become negligible. This mean, that

the vjsaou _{decay of the vortex along a cavity aleo become. unimportant.} The theorem about constant vorticity along a atr.amtube has th. effect that although in the origint Vortex core the radial _{distribution of the} tangential

velocity may be important, it becomes negligible in th.

_very

thin skin around the cavity. These coneideratione motivate the choies of the simple "fl&kln" vortex model to be assumed for a baóie of a

theo-retical

approach tewarda the flow around a swimming cavity.

Asauaptien of axial symmetry for the theoretical cavity boundary seems to be'motivsted by the oserv,tjons of thó smaUer'ro-.ntrant

_jt

type cavities; departures occur with the larger sizes, evidently _{a. a} result of the influence of gravity. It is therefore aseumed that this

influence can

b. neglected in the retationally symmetric cae,.

Aitheugh the precisa details of the rotational core of the vortex

are thought to be uñimportant for the present problem, the

_{existence of}

the rotational cor. with _{a finite pressure in ita *xie cannot b., naglsotsd4} for the cavity boundary Vtreamline along which the pressure must 'be as-aumed: to be constant,, is the same as the .str.amline which coineidee wjth

(19)

-Lt L$po$Lb1s t

use

three

meittc,n

atetja

_{t1'w thscir}

wzLch

Lnclud

_{.skigi line voz'tex.}

The cn1.

_{oLb1e wa}

out of the zemsiuLng dftttes

_tQ

he to ¡e5ur tht the tangentLal

_'/.1ooLt

_{cnonsnt cn b.}

_eptat*

iø

round the asvty and

_{n be onadat}

_t

_{cause a con,}

tiuga1 presauzo fio3tl

n1y,

a ftith.r iiPtp iLcntLön. for

_{the »reennt sna1ais Lt wiU}

_bi

aU»poøs

that the a*vity Ca?

bi r*pias4 b

_{convenient dstributi.n,}

et sources atit aftike Which ni1e

_{a boinds1i 1óø}

_{reeebifttg that o}

*fl ba.xyed cavit3rs With UCh a xouh aseuiption Lt is of

_oouse

iii-pesatte to demand' constant

_{ia8urI alöng tho uU boupdar3r of the}

cavity, but a a iet ap

xiatio

it

be adequate to demand eqa7.

Pressure

ifl

at liait two eactioni, fo _{which the fzward 4tatagnati.n} peint" ena the aøet,on

_with

the irgeat radius

_{ari chosin.}

Ob*e vatjóna cf the

v.nt3,t,a tail behind the vortex oavtti.i

_give

reason far doubt *bøt the aesumed

_a*ieymm.try

_{of th. flow in th..vø't.} car., for the partittoa'of the taU. where the diminuons _{are seaU,}

aye the appear.nc. of a twisted band rather than tub.. 'This is diere..

erd.d in the

ariaiysie as tu

_{the tatleavity itself.}

Analy'sia;

_enerl

The equations sriae why such a free swimming cavity can exist in

a trailing vortex and why it ta able to move upstream and what is the nature of the limitations to

it8

_{existence and movement. Before}

an at..

tempt is made to give the _{answers, certain observatione will be} discus-aed which are thought to be of major importance for the present and for future analysis.

In the first place it has been observed that smooth and

atable

cavity with a gentle movement could be obtained _{only if the wing wee not} stalled and if the velocity of the flow in

_{the water channel}

_{wee not too}

high. Ajeo

if the Vortex was not disturbed by

any object which got too

near to its core. These obBervAtiofl$ lead _{to the hypothesis}

_that

turbu-lence is a major enemy of, the _{swimming cavity. ¡t seems}

_obvious

there..

fore to treat the flow _{rnatheniatioall7 as}

_being

_laminar.

A reduction of the velocity affected the cavity behaviour very

little tIthe angle

of incidence of the wing remained _{constant, This} seems to indicate that

the circujjo

_{to velocity ratio te an important} Parameter.

(20)

The moaøurements of tota]. head and static pressura in the non aerated vortex coro should be corrected toi' probo sise. Disregarding

this correction which is as yet unknown, a preliminary conclusion can be drawn as regards the axial velocity in the vortex axiO, Along great length of the trailing

vortex

nearest to the wing,

the greater

values of the total head

differential

measurements, indicate that the axial velocity in the vortex core te in excess of the velocity at

in-finity'. This conclusion is contrary to the findings from windtunnel experiments (6J(fl, which can be explained from the difference Ln

vis-cosity between the fluids ooni4ered.

('!.

8Ox

106

for

the

ex-periments in

air

and

between 1x1O

and 200x10

for the

present

experiments). This indicates that for the present experiments,

in-vi5oid

tow even in

the vortex core

may be considerad.

It the flow te supposed to be fully invisoid, it cannot be con-sidered irrotational, because the

pressure in thavortex axis muet

b.

*

finite -). The only possIble model remaining for consideration i. that

a twisted bunde of line vortices occupy the Cora of the trailing

vor-tex. This model te in accordance with the

present

general

viewpojnt

Many linevorticea are shod from the wing and rol]. up around each stber.

In those oases where the cavity was relatively small and clearly showed the presence of a re.entrant jet at ita rear end, the

influence

of gravitational and other croeawiae affecte seemed to be negligible, because the

cavity shape seemed to

be unaltered with reduc.d velocity and it had the appearance of

a

purely axisyrmtetric body.

In view of the discussed characteristics it seems, that a

mathema-tical description of the

flow can be based on the principias of

contin-uous, stationary inconpresaible and axisymmetric flow which is inviecjd throughout and which is rotational. The following

equationà may be

the

theoretical bacia or a solution of the mathematical

problem.

Equation of radial motion:

by by

(i)

Ecluation of axial motion:

- U

-

₌

(a)

).

Extremely

low pressures have never occurred

during the

experimente,

for there have been no traces of vapour cavitation in the vortex

axle.

(21)

Equation of tangential motion i

Qontiriuity of coetant density flow: (I'V) bu rbÌL

Axial rotation t

x

br'

Radial rotation:.

.

by0

r

_r

Tangential rotation;

y= -

Q x br

Starting fro,t a distribution of vorticity in the origin wizieb je

net

independent of Q, but whiob is stationary au is the case at a wing tip,

it Le olear that the vorticity linee, for being stationary must b.

0o

ioident with the streamlines, so:

!s

U

V,

_V0

This means that even though the flow is rotational, the law of Bernoulli

is Valid throughout the flowt'isl&:

(9)

Around stres,itubes, the circulation høe to be cone tant,

So $

r

where Woe.

udp = oonøtarit

'

Jroo

roe arr

V oostant

while =

(8)

(10)

(22)

ir

I

Hence:

The following boundary conditions muet be eatiafied;

r

U40

and PP40

(13)

and on the cavity:

at r

»

p0

conatant (1k)

For continuity and symmetry reasono it io reasonable to assuma that ta the axis;

vr(r=O> O ₍₁₅₎

v9(rO)

Q

The Nork Ventilate4 Vortex.

In defence of the experimental indication of high velocity in the vortex

axt.,

the flow model is completed by tb. following assumptions:

1. The flow io independent of the x-000rdinate (16)

2, Th. radial distribution of the tangential velootty component conform with that of the

Rankine vortex

nadel:

in Orr1,

v =r in which 4=

constant and

in r1r

, vQ

= j. Lu wb&ob

l

oonetant

₍₁₇₎

3. In the

outer region (r

r) the flow is potential with u=U

and

V2. O.

These assumptions are in accordance with the continuity equation (k).

n the vortex core (Orr1) the croulation var'iee with the

radius and aoneequently there io vorticity, the direction of which has been assumed to be coincident with that of the velocity.

Equation (5) gives: Y = 2c. X O 2 (8) V9

y

u x u

Y(wjth-=O; (16))

(7) u

and integrated between r and r1:

(23)

¿)

u(Z'

-r2).

Becauae u1 U according to (i8), in the ax&e iÓ toundl

r'?

2 .2 2

2

'co

u a co r

-o i

1

Which ehowe that the volooity tn the

axt

te

arg.r than at infinity.

Thie general eøuìt te not really dependent on the coincidence of the velocity and vorticity vootors; it only requiree that

_YQ>O.

An alternativ, explanation e given in Appendix 1.

jentilat.d Voz.

Zn the pz'eeent analyste only the tre. ewilbining cavity, which i. not connected to any fixed bound&'.y will be dieoueeed.

£s$uing the previously described model for the flow

at

infinity

and axial eya.try for the cavity, it is found that the boundary etrene-tube te a4e up et the same fluid particles es the streamlin, at in-finity which ooinoids with the vortex axis and in which the circula..

tien is mero. On this cavity boundary also (11):

r

Oandas rr0O, vO,

(9) and (1k) show that u2

4

constant.

If this result applies to the whole ca'vity,boundary, also for ite intereectiorto with the axis

(r

O), a stagnation pOint in which

_U3Vr

=v

O, cannot exist. The only possibility

seems to be that the stream..

line of fdie axis te separated to form a cusp. With the sharp cusp between

the tipareted streamlines it is certain that surface teneton wi]. cause the nose of the cavity to be rounded and a small deadwater must b. x peot.d in which the pressure is lees than it te in the cavity. A rough analysis which te not repeated here, has indicated that around the sepa

ration point aleo a region of low pressure muet be expected. 'uture

ex-perimeate in a cavitation tunnel may throw more light into this' matter.

The

_OoflO.ptiofl

_{of a stagnation point at the upstream end of the} caVity should not be ej.ctad fully. It requires reducing pressure

along the cavity 'nose

d*flt*'*a trot the stagnation point, whtoh means that in the stagnation region in the ltui4 the pressure

muet be in

ex-cese of that in the cavity. To balanoe this difference, the surtaoe

(24)

-

17

-eton aust act towards the liquid, which is only possible if then, te an indentation in the nose of the cavity. Although this has not been detsoted during the xp.riaents, it asy b. poseibi.. The influence of viscosity on the ondittone &t the nos. of the cavity has

_nOt

bein

tn.etigat.d.

Aleo litti. thoughts hava been given to the oonditione at the

rear end of the

cavity

and to those at the tail cavity with

tts

isr-EoniCelly varying width.

An analytic treatment of the

proposed flow model haa been tri.d,

with litt3.

success, however, Therefore a very rough analysis of a

specific cas, t. gtven, bseid on radically simplified aseumptionee These aeeumpttons are the following;Equation (i) is writt.n me fo2

lows *

+U+V

P br

òx

rèr

p r r

in which s

up. ₌

or

- r ¿r lt is presumed

that:

1A

¿V r r an4: a u , - o P

or'

z'

The two flows A and B are integrated separately end combined to give

the

Bernoulli equation for the cavity:

2

+

jJ0(u2

₊ _lIn2)

4/

dr

pgE.

Tb. fC.øt of this appromob te, that it is assumed th*t

t

otrøul*tin

free potential flow can be superimposed on a centrifugal pressure field.

The vartem model ue.d

is the Rankje model with the core

rotsttn like

(25)

Tb. potential flow component is described by a suitable etre.. function of the for:

m %7 r'2

)(x,r) with

r') * 1 (24)

Trom this i. d.tsrained:

th. axial position of the stagnation point X5;

the boundary radiue

the *xial and radial velocity oomponente along the boundary:

s

.i_JL11t,

r.

br o br * I

Ic,.

; -

r'

The radius of one and the same stresatube at infinity is assumed to be

r and at a section x$ r In both points the value of is the same7 so:

jU0 r

z '

t,0r'

(z,r)

(25)

or:

r

r(X,r')

(26)

At

Z-.±Oo ,

Rai1cin*'s vortex model is aseused, so when:

v..

r» V and when: 18 (2k). (27)

At the body, the vortex coz'. radius is r1; the transformation of the

vortex model arotand th. body follows tram (21):

r'XVQ*

rc»vQ.. ₍₂₈₎

In Or'

r i. accerding te (27), (28) * V

*

z',t

\

(x,r)

(29) 21t . 19

(26)

19

-r .,, 2

T

Co2U

-

_iik2

7(r1) (3k)

This equation i. reduced by th. nominal velocity pressure

pU; the

ters, which

is caused by y,,

is negl.ot.d

bsoauee only thos. points

wi]].

be oonsid.red in which

20

and in

'SIX

'SrOQ

y

*

i;

₍₃₀₎

The centrifugal pressur. integra], in (22) can now b. d.termined:

2

¡ p2

fri

pr

e 2 1d0

Ir

{fr

(x, r)} r

dr + _dr X

pr2

2 d(r2)+ = r1

{Ir {

(x.r)}

2r, , r1,

pr,2

_[j...

j _{{ yV(x,r}

)}2 d(r2)+.

Vxnix)]

4'it r

_{L 2ri}

.1",,,

in which (26) hai been used. Thu can be written s:

Jr S

/e

, a

drG

₂

r)

(31)

a

Th. circulation at rG 00 can be obtained

froEl

the aesiured liftforcs

and the asiuption of an elliptical circulation distribution over the rectangular wing. With U as the nominal tunnel velocitys

r,

_0L u (32)

With (32) aubititut.d in (31), Bernoulli's equation (22) becomes:

2 2

cU

+

p(u2+v)

p1E_PLk

2 (33)

'It r1

(27)

e

_.

2C2a2

ez.

-ru

P(r)

Equation (35) ia ue.d for

_{the analysis ot}

the pitot tube aansuresienta;

tha static tubi

eaaureents and

o. the cavity.

a. With the pitot tubs the difference

_{had been ss&auz'ed between}

_the

stagnation preseuree (finite bole

_{eime neglected) tar from}

_{the vor,.}

tex core and in the vortex axis:

pitot

II.o

(PSPO

*

j t(r.) }

- 1C4r

_pitot

2C2 2

(d)2

2

r4

r

With this equation

_{the radius}

_{of the undisturbed vortex}

care can

be estimated.

b. With the static tube

_{the difference bad been}

_{measured between}

static

vortex

pressures far from the vortex

cora where

axis, where it ie supposed

_{that uU,0.}

po, o

j.

_;2

_static

.;.

₊

2°°

s

(11L)

jpu2

static

2

2 1'

po,P

U

5.j)*tatic +

(i .)

ji

Uso

20 .2

*

° 2

2Oo2

= jpu2

o, 2

(?(ri)

-r

the

uzU

and in the

static

(5)

2

Q2 2

)tF

) XjL (1(r1) )

static

(37)

Having

_{estimated from the}

_{pitot tube maaeureeonts,}

can be eatj...

mated from (37'),

- 21

2'oo

pitot

er:

pitot

(36)

(28)

21

-o.

The case of the cavity is characterized by the ooiistant pressure

along the boundary. or simplicity, again a potential flow model ta ahoaen, which is aymetrioal with respect to the plane z = 0. Only equal pressure in the stagnation pointe and in z O are

as-sumad to be required: In th. stagnation pointe equation

(35)

be-oomes

22

!

_L&...

(r

(ri) pU00 U00 *nd in. z O: * 2C2e2 I + (

..t:

')2

PJCØ f3U00 U00

n{

(1(r1) x*O elimination of p0 gives:

2 2Qo2

' ) -

_i

fr

) -

-

(F(r ) ) 1I: o (40)

2 br

_r1

xz5

I x-Oj 00

In this equation U0 represente the speed of advance of the cavity relative to th. flow and U» the speed of advance of the wing rela.

tive to the flow.

The factor

(f

.+

₌

()2

i. proximately unity for a slender cavity in its plane of symmetry. represente the circulation and depende

ori the wing

configuration;

- (1(r1) x=O} is mainly a characteristic of the

r1

vortex and of the cavity shape.

Equation (37)ehowe that:

I. The speed of advance of the cavity is proportional with the

liftcoef-ficient and therefore with the effective angle of incidence of the wing: U0

o(oc-This is limited by tb5 stalling angle of the hydrofoil.

2. Tb. general aspects of the vortex cavity are indspend.nt of the

veloci-ty; if the cavity is stationary (IT0 sU0), nothing changes with a change

of velocity.

These two aspects are fully confirmed by the experimente.

(29)

The pressure tn the cavity is expressed by equation

<38)3

2a0a

-

L

_{F(r )}

12

_pu0

I

_pu

,2

_ICr1

1+2

,xx

'

a

at r=

ias

2)

- 22

. 23 .

E1iminat1on of' pgH givee

p110.,p

2Co

1 4.

i +

₍₄₁₎

2?°°

'ICr1

B

QUAntitative Analysla.

tqt Tube M.asjirerente.

The flow is approximated by a source in x=O and

a aink f'ar away

downitr.am (x moe) in a homogen.oue pax'aUel flow tt The radius d of

the tube ii aewned to be represented by the radius of the boundary

etreamliúe at x

O. The measured pressure is áswned to

e the pressure

iii the stagnation point.

The reduced stvesi function (23) isa

,d2

e _Ç

)LI

_{= 1}

.o.

I e w

_'r

₍₄₂₎

r2

_i

_IV'+x2U

The

_{stagnation point te found to be at x}

-

_.

_{By numerical calcula.}

tiort it hao been found that the tube was email enough to approximate:

(see (31)

)

(1)

giving with equation (36):

LL 0M

(30)

23

-which was oalculat.d from th. experimental data and plotted in Pi8. 22.

r1. was reducid by the half epan of the wing in order to be able to judge

the probability of the results.

Th. graph ahowa that al]. the obtained radii are possible, because

they do not exceed the half span of the wing. For the reduced position behind the trailing edge of the wingtip waa chosen:

4%)t

4)Z

2 5

c

oU00

which is in acoordance with Prandtl'e decaying vortex theory, but adapted

to the present purpose [5).

The figure shows clèarly that a much nicer flow existed in the cace

of the model without the flap.

Static Tube 14eaeu'ement..

Fer

iNplicity

_{it is assumed that the flow next to the pressure}

holes of the static tub. i. homogeneous, which gives for the reduced stream funotisa:

(4)2

r2

(i46)

Calculation showed that again with adequate accuracy it can be assumed

that:

I

Because it assumed that: u U0, equation (37) gives:

Z

p00- p0

20202 (4)2

L

= '1

+ 12

(12

00

The results, obtained with veluee from Fig. 22 and the measured data,

are given in Fig. 23, in which relative excess velocities in the vortex,,

con, are shown as a funotien of the dimensionless decay time

- 2e

(,7)

(31)

/2

vr

+(x+

in which; i unitr k 0.3 q8

_0,065761rU.

24

-Tb. Cmvfty.

For the reduced stream function is

taken:

I

6{Jr2+(x+kj2+

\/;2+(X

(49)

The stagnation point is at x=-1,016; the extreme radius of th. sepa-ration streamline is at x=O and is: r0 *0.208; here is the local

velo-city: U(O,r)

\/l19 U0.

3y simple numerical computation it baa been toitnd that in equation (37.) cai be used approximately:

- F(ri)o0.70 - r1

(50)

in the rang. r1 z 0.1 through 0.4.

The stream function wae deaignedto fit the cavity of Fig. 10. abbut which the following data are known:

Vortix generator; Model I without flap.

oc. = 11.5° CL

0.3O

UØQ

z

3.41+ s/eec e - 76.4 mm ("3") b ¿4 * . 0

_3.

Cavity Cfroa photograph)

stagnation point at ¿x/o = O187 behind the trailing edge tL'

x3x106

o tT

total length = 1.2 (c/l=i.6).

/r2

_(x

1)2

(32)

25 Using thee. date, equation (37) becomea:

2 U

j 1.119 - 0.0147

-

-i.- '

= o

Jco

r1

or

I r

+ 0.01514 r1 - 0.0092

O

f

L

U2

XI

*

-.

-

o.oi34 + Vo.0001?23

0,0368 _2_

.2.

I. U00

or approxiniteir;

r1

'

{. 0.01314 + 0.192

-

{

0.096 o.a65

J

Because i wae taken

unitr

r. *r/1

in this equation:

"1

r

1

_j0.09

1

0.0062

u

Becaus. the oavitr is

iuch larger than the vortex cors radius and

be-eau.. it is Stationary relative to the wing,

can 1e assumed to be

one, which gives

s:

oo8j8

(53m)

If for cj2 is taken i.,k in accordance with the pressure tube results:

r

b: -ji- = 0.06?

0,0033

0.0633

a

(53b)

¿6

(i)

(52)

(33)

26

-Th. valus obtained from the pitot measurements

u;

r

'pitot 0.2 2

Although the agreement im x'athea' weak, the order of magnitude i.

cor-r.ot. Tor the discrepancy, the following appologies can be made;

The applied theory t. rough and many approximations have been made. The pr.m.ure measuremonta are expected to have given low resulte;

larger preasure differences would hava resulted in smaller core

radii.

r1

Equation (1), baeed on

-r

0.0838 givem for the reference oase:

o.6 i +0.

o.0178

0.0129. = 1.176 - r 2' 0,00702 1.lkk (i.) r1 for

ç-

0.0635 ii found; 2 0.57 1 + 0' _0.0129 o.006o 1.823.

Censidering that the cavitation number muet be positive, both resulte

are possible. Assuming that everything remains unaltered if CL te changed.

.

Tb. following limitations exist for CL with reapect to 0 S

1+0

CL A). L1

/o

/2

i ,

V

L1 I +

t?&

V0239

o.k88

(a' 10.1,0; compare Fig. 9).

2?

-3). 'o = J(o,3)2 '

i.823

x 0.393

(34)

- 27

It can not be expected that 0W0 can be obtained, because

afl in

finitely wide cavity should be the result. It is clearly shown, however, that the theory demande a lower limit of the angle of attack, at which

the formation of a swimming cavity is possible. The correlation with the

.xp.rim.nt is good. Actually, the cavity had been generated farther downstream where the core radius was larger and consequently muet

have been smaller, giving a higher limit to the angle of attack.

ponoaio,..

Ventilated

vortex cavities are

destroyed by turbulent flow.

It i. indicated that the trailing vortex flow in water Le to a lerge extsnt laminar

J. In trailing vortex flow in water, viscosity can be neglected, but

ro-tation must b. assumed.

k. It is indicated that over a large distance behind a hydrofoil, the axial velocity in the trailing vortex axle is in excess of the free stream velocity.

Vantilated trailing

vortex cavities attach to the wingtipe only at

larger angles of attack and ignite full breadth cavitation at th. tip of the leading edge of the wing.

Vortex ventilation characteristics depend mainly on the ratio of oir-aulatiori to velocity.

The teneration of ventilated vortex cavities may b. a means to

deter'-mine the characteristics of vortex tubes.

8. Theoretically and experimentally it is ehown that below a certain

cir-culation to velocity ratio, a vortex cavity oannot move upstream. This limit depends on the characteristics of the

vortex.

Th. author i. indebted to Staff and personnel of the Hydrodynamics Laboratory of the California Inetitut. of Technology for their assistance and hoepitality especially to Carl Eaetvedt who prepared

thephotographe.

o _{A. Goeman of the Deift Shipbuilding Laboratory preparad the drawings.}

The author is indebted to the Technological University of Deift and to the Netherlands Organization for Pure Research who supported hie visit to California.

(35)

Rs s no iuis

t. Song, C.S.

"Pulsation of Ventilated Cavitiea".

St.Anthony Valls Ey'dr. Lab. Univ. of Mimi. Teoh.Paper No. 32-B 1961.

Silb.ruian, E.

and Song, C.3.

"Instability of Ventilated Cavittee".

St.Anthony Falls Uydr. Lab. Univ. of Minn.

Tech. Paper No. 29-B

1959.

Ccx, E.N. and Clayd.n, W.A.

"Air Entrainsnt at the Rear of a Steady Cavity".

N.P.L.

Syap. on Cay, in Rydr.

1953 (eddington, England).

k. Ranise, TA.

"An Experimental Hydrodynamic Inv.etigation of the Inception of Vort.x

Ven tilation".

MACA Tech. Not. 3903 1957.

Thirand,

"Aerodynamic Theory",

Vol. III Div. O. Reprinted Cal. mat. Tech. Pasadena Calif. tian.

19k5.

Nwaan B.G.

"rro

in a Viicous Trailing Vortex". Aeronautical (uarterly 10 (1959) p. 1k9.

Doaanjb, D.5., Gasparek, LP. and Eskinazi, S. "Decay of a Viecous Trailing Vertex".