Cavitation length on a partially cavitated symmetrical hydro-foil

Cavitation-length on

a partially

cavitated symetrica1 ydroZoil.

Joint interim report of the Institute for Applied

Mathematics and

the Sìiipbuilding Laboratory of

the Technical University oi' Deift.

by

Cavitation-.lenth on a

partially cavitated symmetrical hydrofoil.

Joint interim report of the _{Institute for Applied} Mathematics and

the

_{Shipbuilding Laboratory of}

the Technical University of Deift.

by

Dra. J.A.Geurst1nd Ir. M.C.Meijer

Summary:

The motiva for the study of the cavitation-length is summarized. A supplement to a previously publicised theory is given, to match _experimental results. Particulars and results of static

cavitation-length observations are given; ao;e photographic observations are discussed,

Some future developments _{are indicated.}

Dra, J.A.Geurst: Scientific officer of the Institute for Applied Mathematics.

Ir. M.C.UeLjer; Scientific offieer _{of the} Shipbuilding Laboratory.

Introduction,

In 1955 Dr.van Manen of the Netherlands Ship Model Basin (N.S.M.B.) at Wageningen became

interested in a theory which could be used for the determination of the lift coefficient and the

cavitation length

of a partially

cavitated

propellor blade section as a function of the

cavitation number.

Prof.Dr. R.Tiinman and J.AGeurst attacked the problem with a linearized theory, used a few years

ago by Thun

for the cavity flow

along a symmetric

body.

As to the special conditions, making the

theoretical solution of the problem unique, there

was an uncertainty.

Several possible assumptions on the nature of the solution near the leading edge and the rear end of the cavity have been tried. The resulta of two of these possibilities

have been

published in a report of the Institute for Applied Mathematics of the Technical University

at Deift

11]

and communicated at the Congress for Applied Mechanics in Brussels (1956). They were compared with data which could

be

derived from experiments done by Dr.Balhan at the N.8.M.B in

1951.

Only a qualitative agreement was established. Afterwards the attention was directed towards a report by A.J. Acosta on the same subject (2J.This

report

has not been

accessible at Delft up till now. In

₁₉₅₇

experiments were initiated at the Deltt

Shipbuilding Laboratory

in order to arrive at a

direct comparison between the theoretical results

and experimental data.

The experimental resulta were in fairly good quantitative

agreement with one of the above

a weak singularity at the leading edge of the profile is allowed. At the

rear end of the

_cavity

the pressure

i8

_{discontinuous and the cavity has} a vertical tangent. Only for such values of the

cavitation number for which the singularity at

the rear and of the cavitation

sheet

_{is near to the} trailing edge of the profile, the theory fails to

agree with the experiment. This could be expected

(see fig.1). There is

some evidence that the above

assumptions

are not contradictory to the visual

image of the flow field in

the cavitation tunnel.

So far IneaaurewentB have been done on one

hydrofoil; measurements on a slightly difíerent profile are progress.

In this interim

report only the values of the cavitation length are

compared.

Results on lift coefficients will be published in the

near future.

The mathematical theory.

Por the derivation of the expression for

the

cavitation

length as a

function of

the cavitation number a and of the angle of incidence

we refer

to (i].

Aasming a weak singularity of the order (z -

l)

4

at the

leading edge

_{of the profile we}

have to put A

O on page 7, so the

above mentioned expression becomes: tg

l-sin

a

I + sin

with

cavi 1h

-chor

This formula is valid for the idealized _case

of a flat plate.

Por profilez

_{differing slightly in}

-4-shape froni a flat plate minor corrections have to be made.

The observed hydrofoil.

To check the theory, a thin flat plate is useless as it lacks strength and stiffness; in

giving it thickness

one destroys the boundary conditions at the edges. The logical consequence of the demands for sharp edges, enough strength and stiffness and some resemblance to the flat plate,ls to use a section with circular arcs being the

boundaries; i.e. a symmetrical KrmLn-Trefftz

profile.

To get enough strength and stiffness in the

2 : I aspect ratio hydrofoil, the thickness ratio

was decided to be about .04.

Next to

the fabrication of the h.ydrofoil it8

dimensions and form were checked. The following

dimensions and ratio's were

found:

Chord of the profile

L

-

147,6 mm - 5.813 in. Maximum thickness a - 5,83 " - .229

Span - breadth of

tunnel B - 300 " '.11.8

Thickness ratio

.0395

Curvature ratio of

nie anime fo,,5 o

It was found that deviations in symmetry and arc-form were very small (about I % of the maximum thickness) except at the edges, which appeared to be slightly rounded off (designed chord - 150 mm5.9 in.)

Parameters.

The mathematical theory expz'esses the cavity

length as a function of

in

which o is the angle

of incidence of the profile, immersed in water with

boundaries at infinite

distance and is the

-5-cavitation number.

In the cavitation tunnel _{c*. is measured being} the angle between the profile chord and the

center-line of the tunnel; the error is estimuted to be less than .4 degree (.1 degree for reading the nonius and .3

degree being

_{the negative angle for} which the uncorrected measurement gives _zero

lift-torce).

This tunnel-value of c is corrected for infinite flow with the Prandtl correction:

2

¿) .c4

which must be added to oc.

£

- profile chord, h tunnel bight,

liltcoefficient.

The values of Ct bave been calculated from lift-reaction

measurements on the tunnel walls as

the profile was too small to allow direct pressure

measurements on the hydrof oil surface to be

made. No indication was found that the pressure

holes

cavitat ed,

oc.

was varied

between O and 10 degrees,although

thé importance is limited to the smaller angles as

the cavitation

sheet becomes unstable when oc. exceeds

30

The cavitation number a was varied principally by variation of the

pressure

_{head in the tunnel as} Reynolds number had to be a maximum.

This maximum varied between

Re 1.15 . iO6 and

1.Llo.

io6.

(R is defined as: R

v.t

e e

in which

V

tunnel water velocity, measured at the

flow contraction, gauged by pitot-tube in absence and in place of the hydrofoil.

9 rn/sec _{. 30} ft/sec,

4'

profile chord,

V viscosity of the water).

a was limited by the maximum pressure head of

I a (3,3 ft) water plus barometric pressure andat the smaller numbers by cavitation of

tunnelwalls

and a lack of reabsorption of air bubbles in the rather short tunnel circuit.

The minimum a which could practically be reached is about

'

The

cavitation

length has

been expressed in

1) in proportion to the profile chord as

in which is the

distance from

center of chord to the rear end of the cavitation sheet expressed as a part of the half chord. In order

to distinguish

this value from the chord length we will call it

t;

thus

we will express the

cavitation length

ratio as

In the experiment

i + A'

was visually determined

as follows: the ends of the

hydrofoil had been

clamped between

half

circular "Perspex" windows on

which a scale had been drawn

at both ends;

_was

read by keeping the edge of

the cavitation

_{sheet in} line with two equal

scale values.

The tolerance is

estimated to be ± 1.5 mm which means I % chord length; it must be increased by

some millimeters

for oscillations which often took place.

Results.

In figure 1 the cavitation length ratio,being

a function of

as

observed, is compared with the calculated values for a flat plate. The concentric circles show the values derived from Dr.Balhan's

° experiments, mentioned

in

the introduction. In

-7-reality these values give the length of constant minimum pressure on the profile surface which can not exceed the chord length. Actually the cavity

can become much longer.

Curve e, which is derived from the £oruiuls (1a)

and (1') agrees nearly with the observations in whicho( was 2°. Where the profile is fully

cavitated it is clear that

the

theory must fail,

as the cavity can not longer end tangentially to

the profile surface.

Some experimental results have been reanalysed using the cavity pressure as measured by Parkin and Kermeen (3]instead of the vapour pressure for the

calculation of a.

Photographic observati0DB.

As an alternative for checking the theory or to base a new theory upon, some photographs have been taken. The axis of the camera lens was meant to be in line with the leading edge of the hydrofoil. Unfortunately the view directly over the surface of the wing was obscured by wax, necessary for the enclosure of the wing

attachment.

For future work this wax will be replaced by a water lock so as to get a clear view. Consequently a clear photographic crosscut of a thin cavitation sheet at smaller

angles of incidence could not be attained. A reconstruction at ot= 1.50 is shown in fig.2a.

Figures 2b and e show sketches from photographs at

3° and 5 The profile section a reconstructed in the figures is the intersection of the hydrofoil with the near window.

The photographs have been taken with a

3O,,«-sec. flash and a 50 mm camera lens.

Prom observation by eye with

_<

O most of

the cavitation seemed to be coherent when seen in

-8-continuous

light; with the use of stroboscopic

illumination however small bubbles could be detected, although some unstable patches of pure sheet

cavitation were present. There was

no indication that a difference in character would affect the length of the cavitation.

About the form of the cavity the following remarks may be made:

The flow starts vapourizing a very small distance in front of the leading edge of

the hydrofoil.

At about five to ten

percent of

the chord

behind the entrance, the thickness of the cavity approaches

its maximum

value, which la fully attained at about half of the

cavitation length provided that the forni

of the cavity is stable.

Approaching the rear end of the cavitation the curvature of ita boundary increases.

Some evidence seems to be

present that the cavity ends in a stagnation point of the flow when is small.

e)

At

bigger angles of incidence the cavitation

seems to

end clear 0f the

wing

surface.

A single, pure cavity sheet has

not

been

attained,

80

the question arises whether the

condition, assumed in, the theory, that the boundary

of the cavitation is formed by a streamline is

satisfied.

This question can not immediately be

answered, but the probability of the assumption may

be

demonstrated by the fact, that a.n.y restricted

cavitation, which is

fixed to its place muet dis. place the surrounding fluid because of the

increase

in volume of vapourizing water.

e)

Future developments.

At this moment the cavity length experiments

are extended over a K-T profile with the same

ratio (.04) but with a flat pressure side, so fO18

is about

.5.

Better photographs

are expected to be possible with the aid of a water lock now in construction.

Soon after finishing the cavity length

measure-ments, experiments will start

with both profiles oscillating transversally in order to

get

a harmonic

variation

of the angle of incidence «. _{The aim is}

to vary about _{5 degrees. Instantaneous and mean}

lift coefficients will be determined; instantaneous cavitation characteristics will be compared with the

static situations

Simultaneously a linearized theory will be

developed taixg the instationary motion of

cavitated hydrofoils.

Deif t, '18 February

_1958.

'10 lo '10

-Bibliography.

rl]

J.A. Geuret and R.Timman.

"Linearized theory of flow with finite cavities about a wing". Report no.7 of the Institute for Applied

Mathematics; Technical University of Deift,

Netherlands.

(2] Acosta, A.J. "A note on partial cavitation

of flat plate hydrofoils". C.I.T.Kydrodynaniics

Lab.Rep. no.E-199;

oct.1955.

(3) Dr.Blaine R.Parkin and Robert W.Keruieen.

"Water tunnel techniques for force

measure-ments on cavitating hydrofoils".

Journal of Ship Research vol.1 no.1, april

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