The theory of leading edge cavitation on lifting surfaces with thickness

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(I)

ARCHIEF

k.

7

fl

Lab.

y.

Scheepsbouwkunde

Technkcbe Hogeschoo

Deift

SYMPOSIUM ON

"HYDRODYNAMICS OF SHIP AND OFFSHORE PROPULSION SYSTEMS" HØVIK OUTSIDE OSLO, MARCH 20. - 25,1977

"THE THEORY OF LEADING EDGE CAVITATION ON LIFTING SURFACES WITH THICKNESS"

By

M. P. Tulin and C. C. Hsu Hydronautics, Incorporated

Laurel, Maryland, USA

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CONTENTS

Page

SUMMARY i

INTRODUCTION 3

LEADING EDGE CAVITATION 5

Present Status 5

SHORT CAVITY THEORY 11

General 11

Thin Bodies 16

APPLICATIONS

19

Flat Plate

19

Bi-Convex 20

Flat Plate with Thickness 20

High Speed Foils: NACA 16 Series 21 Elliptic Wings with 16 Series Sections 22

REFERENCES 26

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NONCLATURE

a Parameter related. to cavity length

b Parameter related to distance between leading edge and

cavity

c Pressure coefficient p

Pressure coefficient due to thickness

Non-dimensional pressure coefficient, c*/r/

f Thickness of wake behind cavity

k Camber parameter (design lift coefficient)

¿ Foil chord

q Local flow speed

r Nose radius of foil n

t Parameter of integration

u Velocity component in x-direction

y Velocity component in y-direction

x Horizontal ordinate in physical plane

y Vertical ordinate in physical plane

z Complex variable, x + iy

CD Foil drag coefficient

E. Jonest correction factor, wing semi-perimeter

Wing span J

H Complex function

Im Denotes imaginary part of

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iii

L Lift on foil

M, N Constants in formula for complex potential

R.. Denotes real part of

U Free stream velocity (in x-direction)

VC Cavity volume a Incidence Ideal incidence p Fluid density a Cavitation number T Foil thickness

e Local flow angle

ç Complex variable in transformed plane

R(ç)

rl Im(ç)

Complex potential

Complex velocity, d?/dz

Scalar potential, RF)

Stream function, Im('')

Hodograph variable, n

= ß

q - i8

Subscripts:

o Denotes fully wetted flow

i Denotes flow due to cavitation (shorbcavities)

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LIST OF FIGURES

Figure

No.

la,b,c Schematics of Short Cavity Flow and Boundary Value

Problems

2a,2b Flat Plate Foil Characteristics with Short Cavity

T/

= O; AR = ; a =

0.510.0°

3 Bi-convex Foil, Characteristics with Short Cavity

T/

= 4% AR = a _2,3,4°

4 Flat Plate with Constant Thickness and Rounded Leading Edge, Characteristics with Short Cavity

AR ;

r/ = 0-1%; a

5 Series 16 Foils, Short Cavity Characteristics

AR=; T/=6%; a=2-6°; k=0

AR = ; 'r/t = 6%; a =

40;

k = 0,.l,.2

AR

= ;

= 0,6,9,12%; a = 4°; k = O

8 Series 16 Elliptic Wing, Short Cavity Characteristics

AR = 2,4,8,;

_T/

= 6%; a = 40; k = O

9 Series 16 Elliptic Wing, Short Cavity Characteristics AR 4; T/ 0,6,9,12%; a = 6°; k = 0.1

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Figure

No.

AR = 4;

T/

= O-12%; a/ = 0.04-0.14; a =

6°;

k = 0.1

AR = 2; = 0-12%; a/o = .08-0.20; a =

6°;

k = 0.1

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SUMMARY

In connection with propeller induced noise and vibration

it would be very desirable to have a theory for the prediction

of non-steady leading edge cavitation on lifting surfaces with

thickness. As a step in that direction we have developed a theory, ignoring viscous effects, which is generally applicable

to steady two-dimensional flows with cavities which are short

in comparison to the chord of the body. The theory assumes the

fully wetted flow is known and perturbs on it. This theory as

it is developed includes higher order effects where necessary

and differs significantly from conventional linearized theory.

It deals accurately with short cavities on bodies with rounded

leading edges and, in a further approximation has been applied

here as a two-dimensional perturbation to a known

three-dimensional flow, thus allowing treatment of finite span effects.

The theory is presented in its general form, and then in a

form especially appropriate for foils. It is applied to thin

flat plates including the effect of leading edge roundness, to

a variety of NACA 16 Series foils, and to wings with the latter

sections. Results are given both for the increased lift due to cavitation, for the cavity volume, and in some cases for the

cavity length.

The calculations for cavity length in the case of a flat

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linearized theory of Acosta [2] in the limit of vanishing angles

of attack, and are in good agreement with the full non-linear

re-entrant jet theory calculations of Kutznetzov and Terentev

[12]. When small but finite leading edge roundness is allowed

the short cavities are significantly reduced in length and

volume. This result shows the importance of adequately

con-sidering the leading edge roundness in both theory and practice.

The results for the 16 Series foils show the positive

effect of increasing thickness in reducing cavity volume, and

the result for wings of finite span show the marked effect of

aspect ratio, revealing the importance of considering the latter

in both design and estimation and in the interpretation of test

results. For low aspect ratios, too, increases in foil

thick-ness over a certain range reduce cavity volume. For a 60,

those ranges are about 5 to lO percent and 3 to 9 percent for

AR = 4 and 2, respectively.

The results suggest that the use of foils with adequately

large leading edge radii during operation where short leading

edge cavities are likely to occur, could result in very large

reductions in cavity volume.

Finally, this theory applied to round-nosed foils and

wings has not been compared with experimental data, which is

lacking, and it would be highly desirable to do so.

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INTRODUCTION

It is well known now that the unsteady loadings observed on

the stern of ships as well as the radiated noise often increase

dramatically at the same time as cavitation first begins on the

screw(s). The explanation for this event would seem to be in

the high efficiency with which acoustic pressures are radiated

by cavities which grow and collapse, in comparison to the

radi-ation from other sources, i.e., turbulence, the non-cavitating

screw, etc.

The radiated pressures consequent to the growth and collapse

of cavities depend on the maximum volume of these cavities and

their time frequency, and therefore on the unsteady cavitation

patterns which occur as the propeller revolves through the

non-uniform wake at the stern of the ship. The cavity shape and

volume itself depend on the shape of the propeller sections

(foils) and particularly on the foil thickness distribution

near the nose. The effective aspect ratio of the foil is also

important, as we shall see.

The determination of time-varying partial cavities on a

propeller blade is made extremely difficult on account of both

finite span effects and flow unsteadiness. As a step in the

direction toward a more complete theory, we here present a

theory of leading edge cavitation in steady flow around lifting

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short in extent compared with the length and width of the foil,

so that the cavity causes only a small perturbation to the

non-cavitating flow about the given foil. Starting as it does with

the fully wetted flow assumed known, this theory differs markedly

from the usual linearized (slender body) theory which perturbs

on the uniform free stream flow, and would seem to offer the

following advantages: i) it can be applied to thick as well as

slender bodies; ii) it deals accurately with leading edge

cavi-tation in cases where slender body theory does not; iii) in a

further approximation, it can be applied as a two-dimensional

perturbation to a known three-dimensional flow. The latter

possibility arises because the span of the cavity is likely to

be much larger than the cavity length, even though the foil may

not itself possess a high aspect ratio. Given means to calcu-late fully-wetted flows on finite aspect ratio lifting surfaces,

the present theory allows predictions to be made about the

properties of their leading edge cavities; the method can also

be applied to propellers.

The theory developed here is applied to two-dimensional

foils with thickness and comparisons are made with existing

theory and experiments; on the whole, however, data do not exist

for an adequate test of the theory. It is also applied to

pre-dict the effect of wing aspect ratio on the volume of leading

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LEADING EDGE CAVITATION

Present Status

The incipient cavitation number for a hydrofoil depends on

its angle of attack as described graphically by the familiar

bucket diagram. On the flat bottom of the bucket, cavitation

occurs near the maximum thickness of the foil, but on the sides

of the bucket cavitation occurs from the leading edge. The

bubble thus formed grows longer with increasing incidence or

reduction in the cavitation number, , and eventually exceeds

the chord in length. When this happens the flow is

supercavi-tating. We are here concerned with the earlier, or partial

cavitating phase. During this phase the cavity results in an increase in the foil lift, as a consequence of the increased

camber effected by the cavity. The transition between the

partial and supercavitating phases is accompanied by cavity

oscillations and unsteady forces. These facts have been known

for over 25 years. The systematic investigation of the

cavi-tating characteristics of foil sections was begun by the

Japanese researcher F. Numachi during the period 1940-50, see

[i] and other reports in this series.

In

1955,

Alan Acosta [2] presented a theory for partial cavitation on flat plates utilizing linearized, or slender body

theory. This approximate theory provides solutions only for

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6

a long and a short cavity, following on either side of a. length

equal to

₇₅

percent of the chord. In fact, it is the shorter

cavity which is observed experimentally. For values of a lower

than Acosta's critical, experiments show that the cavity at first

approaches the trailing edge, unsteady flow results, and finally,

for sufficiently reduced values of a the cavity extends

suffi-ciently downstream of the foil that the flow is steady over the

latter. Acosta's results were found independently by Geurst

and Timnian

_[1956;3]

utilizing the same theoretical model. At

the same time, Meijer

[l959;4]

reported on the results of

ex-periments on several thin

(4

percent) foils: one a double

circular arc and the other a circular arc on the top and flat

on the bottom (piano-convex). The experimental results for the

cavity lengths on the double circular arc, together with those

for a thin wedge tested by Parkin

[l958;5]

showed good agreement with the theory of Acosta and Geurst-Timman. The results for

the asymmetrical foil differed systematically, however,

approach-ing the flat plate theory only at high angles of attack.

About ten years were to pass before the linearized theory

of partial cavitation was extended to the case of a

two-dimensional hydrofoil of arbitrary shape by Hanaoka

[l967;6].

In the meantime, Geurst

[1959;7]

gave theoretical results for thin cambered foils without thickness and Wade

[l967;8]

gave

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the experiments of Balhan

[1951;9],

Meijer [4], and Wade-Acosta

[1965;10].

The first two of these concerned foils 4 percent thick at angles between 2-6 degrees. The latter concerned a

foil of 7 percent thickness at angles from 2-10 degrees. The

correspondence between predicted and measured cavity lengths

was not very good and worsened at larger angles and for the

thicker foil. It is true that comparisons are made difficult

because of possible discrepancies between vapor pressure and real

cavity pressure and in the determination of cavity length.

Hanaoka's theory is also based on conventional linearizing

assumptions, but he relaxed the condition of cavity closure and

introduced an arbitrary cavity thickness at termination. In

this way he was able to obtain a theory bridging the gap between

partial and supercavitating flow. Hanaoka found empirically that his theory when applied to the flat plate gave

good.agree-ment with thin symmetrical foils for short cavities when he

assumed cavity closure. In that case his results correspond

to Acosta, Geurst-Timman and his conclusions to those of Wade

[8]. Again, assuming cavity closure, Hanaoka compared Meijer's

datafor the asymmetrical foil with theory and found rough

agreement; that is, trends were reproduced but actual values of

cavity length tended to be longer than predicted.

In addition to linearized theories, there exist several

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8

plates: Wu

[1962;ll],

Kutznetzov and Terentev

[1967;12],

and

Terentev [1970;l3]. The first of these utilizes a wake model in which the wake is as thick as the cavity at its maximum. The

second utilizes the re-entrant jet model, and the last a vertical

flat plate termination of the cavity. The last two might be

expected not to differ much and to give the closest agreement

for cavity lengths with reality. A comparison is made later herein between results of the re-entrant jet model and the

present theory. A further non-linear theory utilizing the

re-entrant jet model dealt with the flow about a cavitating

wedge at incidence, but was not utilized for the prediction of

cavity length, Cox and Clayden

l958;l4].

Finally it should be mentioned that partial cavitation on

cascade of foils becomes important in connection with cavitating

inducer pumps, and the conventional linearized theory has been

applied to the case of partially cavitating cascades by Wade

[l963;15]

for flat plates and by Hsu

[1969;l6]

for cambered

plates.

The present status seems to leave room for further studies

of leading edge cavitation on lifting foils both theoretical

and experimental. We say this partially because present ex-perimental comparisons are not really satisfying, and partially

because finite span effects seem not yet to have been adequately

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(1)

Tamiya and Kato

[l975;l8]

and partially because there is reason to believe that the prediction of conventional linearized theory

in the case of foils with thickness can be wrong and misleading.

We discuss the latter point below.

The inception and extent of leading edge cavitation on a

foil with a rounded leading edge must depend on the leading edge

radius, for the minimum pressure on the foil at incidence very

much depends on the radius. In fact, experience shows that the

following expression, incorporating the second order correction

of Lighthill

[l95ll9]

in its last term, yields adequate results for typical thin foils

(_c*)

x

c c4

-

, for - - O

p _p

x/

+ r /(2)

n

On a high speed foil such as the NACA 16 series, the

pres-sure coefficient due to thickness, c*, is negative practically

right up to the leading edge and is proportional to the foil

thickness. The leading edge radius is proportional to the

(foil thickness)2. So,

i 2

(a_c*)

p

_Vx/

_{+ r/(2)}

c*

c*= Vr/

p where

(16)

and c is constant for a given foil series. Therefore, for

sufficiently small values of r, i.e.,

r < 23,'2 (a _cz*)

the pressure coefficient at the leading edge actually decreases

with increasing foil thickness as a result of the corresponding

increase in nose radius. This fact would indicate that, for

a range of nose radii dependent on the foil incidence,

increas-ing foil thickness delays cavitation and would result in smaller

cavities when cavitation occurs, an effect very important in

foil selection.

This latter effect can not at all be predicted by the

conventional linearized theory as that theory inherently gives

rise to unbounded pressure coefficients at the leading edge,

due to incidence. This fact accounts, no doubt for the curious

finding of Wade

_[81,

that increasing the thickness of a plano-convex section (cavitation number and incidence fixed) results

in an increased cavity length. In fact, the data of Meijer [24] for a 4 percent foil and of Wade [10] for a 7 percent foil, both

plano-convex, show that the thinner foil had the longer cavity

The theory presented herein can take into account the

actual pressure distribution on the non-cavitating foil and does

not necessarily rely on slender body theory for its prediction.

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In fact, the results depend very much on the proper

approxima-tion of the leading edge pressures at incidence. Calculations

using an approximation of the form, Equation (1), do in fact

predict shorter cavities and smaller cavity volumes for

in-creasing foil thickness in the case of 16 series foils of

practical thickness and incidence angles, as one might expect,

and in accord with the tendencies found experimentally in the

case of the piano-convex foils of Meijer and Wade.

SHORT CAVITY THEORY

General

We consider here inviscid flows with small regions of

cavitation. That is, cavities sufficiently small in comparison to the body causing the flow, so that we may consider the

cavities to cause only a small perturbation to the non-cavitating

flow. Such a flow is shown schematically in Figure la. We

define,

=

+ i

(the complex potential)

= = = (u-iv)

=qee

(the complex velocity)

dz

ie

o

(7)

f,T = q e

(18)

12

=

_q1e'8

_,

represents the effect of cavitation, so that,

1

if no cavitation occurs. In the case of a short cavity,

is of the order of (1 + e) where

« 1.

The

corre-sponding flow in the complex potential plane is shown as

Fig-ure lb.

It is useful to define the function, w = 'F', which can

be decomposed into two parts:

(lo) w

w0 + lUi

= Qnq0 - ie + q1

- 1A1

We may assume that the fully wetted flow has been found, so

that is given. The problem is then reduced to that of

finding with the boundary conditions:

R(w1)

=

q/q

, for

< D (on the cavity)

where q =

(l+)2,

where is the cavitation number and where, to first order, i.e., ignoring terms of order e:

IA <

<

Itn(w1) = e1

o for ¿ < <

(COA<CP<CPE

;

=O-Conditions at infinity:

R(1) = o

(1k) Im(w1) = O

(19)

The closure condition:

Imw1d? = f

The value of f is zero if wake closure at infinity is assumed,

and a constant ( drag) if the effect of finite wake is taken

into account, see Tulin [1964;20].

CD _Drag

f = -- , where CD

-The problem as formulated may be greatly simplified with

the aid of the conformal transformation,

where (18) (17) = -ia or,

a=

= 2/(ç2 + a2) - 1 +

which transformation maps the complex potential plane onto the

half plane as shown in Figure le.

The associated boundary conditions, conditions at infinity,

and the closure condition are given respectively by,

(i)

R(1)

₌

= -1 < < -b

(ii) Im(w1) = 0 -c < -1

-b < <

(20)

(iv) _Im[w1(-ia)] = O

.!.

(y)

14w1

dÇ = f

The general solution of the mixed boundary value problem

is given by:

-b

(24)

(Ji1 =

_-

H()

-1

The first term on the right above is the particular solution

which satisfies the mixed boundary condition on the -axis,

while the second term is the fundamental solution of the

cor-responding homogeneous problem. The functions H(ç) and P(Ç)

provide the proper flow behaviors at edge points and at

infinities.

The fundamental solution for the present problem may be

shown to be of the form,

i(ç+b)

H(c) P(ç) - _i (Mc+N)

ç[(+b) (ç+l) ]

which has the following behavior,

at = O (leading edge singularity; b

o)

(Ç+b)2 at Ç = -b (smooth cavity detachment)

(ci)

2

at _{C =} -1 (cavity termination)

1 is finite at = (Kutta condition)

[(1+a) -Gnq(t)]

H(t) (t-ç) dt + H(ç)

. _P(C)

(21)

However, if the cavity is assumed to detach from the leading

edge, as on a slender airfoil (i.e., b = o), then, i

wi at = O (abrupt leading edge detachment)

In fact, the viscous boundary layer will probably effect the

detachment point, as it does cavitation inception, Huang and

Peterson

[1976;21].

Conditions (iii), (iv), and (y) uniquely determine M, N,

and o. For given values of q and b, then, the function

1()

is completely determined. For the slender airfoils treated herein we assumed cavity detachment at the leading edge and set

b 0.

The lift per unit span, from

Blasius!

formula may be shown

to be:

L = L + L1 =

-pU2R {(?)2dz)

= -pU2R

(()2

. .

d)

and the additional lift due to the effect of cavitation is

given simply by:

L1 = L-L

The cavity boundary, z, is given by,

Z

(22)

ç

()

=

f (

)'

+

dç

and from these the cavity volume, V, may be found by taking,

D

(36)

y

=

f

(y-y0)dx

B

where y0 refers to the foil surface without cavitation.

Thin Bodies

The theory and results presented above may be applied to

the calculation of short cavities, regardless of the shape of

the body. Here we are particulary interested in short cavities

on thin bodies like foils. In such cases, the calculations may

be simplified by taking advantage of appropriate approximations.

These must be carefully applied, as second order effects may be

important.

The leading edge on foils used in practice, even though

rounded, are of such sharp curvature that a leading edge cavity

may be assumed to start immediately at the edge. At the same

time, when leading edge cavitation occurs, the distance between

the edge and the stagnation point is zero to second order in

the angle of attack, Tulin

[1969;20]

so that we take the points

16

or,

_-ç

(3h)

x

= R

f (y')-'

dÇ _{+ XB}

(23)

A and B as coincident to second order, i.e., b O.

Then, (21!) takes the form,

whe re, O

(38)

I

= -

I

[t(t+l) dt

I

_I

(t-ç)

-1

(39)

n(1+)+

+ +

and where (26)-(30) for determining the constants M, N, and

can be shown to take the form,

Im(I(-ia))

(l+a) = - Im(I(-ia)) Im(I1(-ia)) R (11(_ia)) - R(I11(_ia)) N = Im(I(-ia))

-[Im(I(-ia))

(42) M

-[Im(I(-ia))

Im(11(-ia)) - Im(Iii(_ia))]

where cavity closure has been assumed; i.e., f = O.

The approximation, already utilized that the cavity

per-turbation is small compared to the non-cavitating flow,

sim-plifies the calculation of the lift due to cavitation and of

the cavity volume from the potential, (37). The small perturbation

(37)

i

[ç (ç+l)

]2

i

[t(t+1)]2

q(t)

(24)

approximation is that:

( 3) W1 Ui

-

i

=

Then the lift force due to cavitation is given by,

L1

-

-2

R[wi()

(k!) dÇ J = .14 R. = + R

[(_ia)]

and, the shape of the cavity by,

x Ç(x) i d

()

_yc =

f

dx -

JIm[wi]

-J

dx o (x) 2 (i+o)

firn[()]

[(2+a2)3l

L

(2+a2)2 j

d -1

The volume of the cavity is thus,

6)

_VC

= j

y(x)dx

, where ¿ is the cavity length

o

-(i±o) f

dcp Im[w1] dcp

-i

(c){i

(2+a2) -

(2+a2)2 j

(25)

i (l+o)

[wi

d]

'm [

1)

2

-(l+o) ' (Ç2+a2)

(2+a2) - (2+a2)2J

'r a2

- R [u' (_ia)]

-

8(l+o)

APPLI CATIONS

These results, (37)-(46) are sufficient to calculate the

properties of short cavities for a given foil, i.e., for q0(t)

given. The present theory contains within itself the

conven-tional linearized theory and it can be shown that the results

of Acosta, [2], for short cavities on a flat plate at incidence

are recovered exactly from this theory by setting q(t) = 1, so

that

'H

vanishes.

Flat Plate

The appropriate representation of q(x) in the case of a

fiat plate of zero thickness is the exact theory result,

-(L7)

q(x/) =

Calculations have been made based on this representation

of q0 and the results are shown as Figures 2a and 2b. The

results agree with those of Acosta, [2], in the limit of small

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2G

Figure 2b; however, the non-linear effects on cavity length are

in good agreement with the non-linear calculations of Kutznetzov

and Terentev, [12], based on the re-entrant jet model, Figure 2b.

Bi-Convex

Calculations have also been made in the case of a symmetric

foil with circular-arc thickness and sharp leading and trailing

edges; exact theory was used for estimating

q(t).

The results

are shown as Figure 3 where a comparison is made with the

ex-perimental data of Meijer [k] for a bi-convex foil of four

per-cent thickness and for angles of incidence of 2, 3, and i-I-

de-grees. The agreement for the added lift is excellent (upper

figure) and is good for the cavity lengths; there is a tendency

for the measured lengths to be somewhat less than predicted.

A comparison of the cavity lengths calculated for a flat plate

of zero thickness and of the bi-convex foil of four percent

thickness shows only a slight effect, the cavity lengths being

slightly shorter in the latter case.

Flat Plate with Thickness

The profound effect of the fully wetted flow immediately

at the leading edge is demonstrated in the case of a flat plate

with small constant thickness which is rounded off at the nose

to a radius based on chord of r . In this case the velocity n

distribution due to incidence depends near the leading edge

very much on r . The correct result must take into account

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the interaction between thickness and incidence and was given

by Lighthill, [19]. It is,

(!8)

q0(x/) = a

1 - x/.t

Vx/

+ r /2

n

The effect of leading edge radius due to plate thickness

has been calculated using this representation of q(x/.t) and

the results are shown as Figure . The results show that

lead-ing edge radius very much ameliorates the effect of incidence;

a radius of only 0.5 percent reduces the cavity volume by

some-thing like an order of magnitude. This effect demonstrates:

the central importance of leading edge radius in determining

short cavity effects, as already discussed in the Introduction;

the advantage of the present theory, which does deal

ade-quately with the rounded leading edge, while conventional

linearized theory does not at all.

High Speed Foils: NACA 16 Series

These foils involve camber and thickness distributions

providing good cavitation characteristics when operated at or

near their design incidence (on the bottom of the cavitation

bucket) and we have calculated their characteristics with short

cavities in view of their popularity.

For a given foil section with incidence a, camber index

(28)

±q

(249)

q =1+q

±q

O O,T o,k

where the terms on the right are the components due to thickness,

camber, and incidence, respectively. The [±) signs refer to

values on the [upper) surfaces of the foil. These components

lowe r

are tabulated in Appendices I and II of Abbott and von Doenhoff,

[221,

the values being based on conformal mapping.

We have carried out calculations for two-dimensional

(Aspect Ratio = ) 16 Series (modified camber line 0.8) with

the following characteristics:

= 6% ; k = O

; a = 2,3,4,6°

= 6% ; k

= 0,0.1,0.2 ;

a =

T/

= 0,6,9,12% ;

k =

O _;

a =

and the results are shown as Figures

5-7.

We concentrate on the cavity volume and note that increasing design camber

in-creased cavity volume, while increasing thickness decreases

volume, especially for thicknesses beyond nine percent.

Elliptic Wings with 16 Series Sections

The present method, assuming as it does the fully wetted

flow, would seem applicable to wings of finite aspect ratio,

at least in the limit of short cavities. The idea is to treat

the perturbed flow at a given spanwise section as two-dimensional

in view of the high aspect ratio of the cavity; in doing so, of

course, it is essential to take into account the effect of

(29)

finite span on the fully-wetted velocity distribution. It is

only in this way that the effects of finite span enter this

approximate strip-like theory. We treat elliptic wings here,

using the appropriate but simple corrections of R. T. Jones for

the estimation of the fully wetted flows.

This strip method could conceivably be applied to propellers,

using pressure distributions from lifting surface theory,

cor-recting for leading edge roundness, rather than the Jones'

correction as used here for wings.

According to Jones [1941;23], the ratio of edge velocities

between an endless and a finite elliptic plate is E,, where,

(50) E

J

semi-perimeter

- span

and q may be

It follows that the values of 0;

0;

approximated by,

(5')

EO,T,2D

1

(52)

EOk2D

+

()

_EOa2D

+ q.)

where q. and q. are induced downwash due to camber and

i,k i,a

angle of attack respectively.

For a first approximation,

i,k - E.AR+2

(30)

2n

o,a,2D

/ i,a E.AR+2

3

where AR = Aspect ratio = (span)2/area

In the Jones approximation, then, the three dimensional

value of q may therefore be expressed approximately as,

(56)

q

+ E E.AR+2 +

and these expressions are assumed to be approximately valid for

rectangular and other regular pla.nforms; they have proven

re-liable for predictions of lift curve slope.

Calculations have been carried out for wings with the

following characteristics:

TR = 6% ; k = O ;

AR = 2,,8,

= 0,6,9,12% ; k = 0.1 ; =

6°

; AR = 2,

The cavity volumes, Figure

8,

are seen to be markedly

affected by aspect ratio, especially for the smaller values,

which are of a magnitude appropriate to propellers. This result,

approximate as it is, does suggest that two-dimensional

esti-mates of short cavities, whether based on theory or experiment,

are likely significantly to over-predict both the occurrence

and severity of leading edge cavitation.

The importance of foil thickness (leading edge radius) in

the case of finite span wings is shown by the calculations,

F1.gures ₉ and 10, especially for larger values of a/a, where 24

(31)

in the case of a = 60, increases in thickness from 5 to 10

per-cent cause a substantial reduction in cavity volume for AR 11,

and from

3 to 9

percent for AR = 2. The extreme importance of thickness (leading edge radius) in controlling cavity volume

is shown even more graphically in Figures 11 and 12, which are

cross-plots of the previous two figures.

These results suggest that foils selected for minimization

of short cavity volume should have larger values of leading edge

radius than are normally associated with foils designed to

optimize inception characteristics at design incidence (i.e.,

(32)

RE JI'ERENCE S

NUMACH]I, F., TSLTNODA, K. and CHIDA, I. Cavitation Test on

Hydrofoil of Simple Form (Report 1). Rep. Inst. High Sp.

Mech., Japan, Vol. 8, (1957) pp 67-88.

ACOSTA, A. J. A Note on Partial Cavitation of Flat Plate

Hydrofoils. Calif. Inst. Tech. Hydrodynamics Lab. Report

No. E-19.9, 1955.

GEURST, J. A. and TIMMAN, R. Linearized Theory of

Two-Dimensional Cavitational Flow Around a Wing Section. IX

Inter. Congress of Applied Mechanics, 1956.

!f _{IJER, M.} _C. _{Some Experiments on Partly Cavitating}

Hydrofoils. Inter. Shipbuilding Progress, Vol. 6, No. 60,

(1959).

PARKIN, B. R. Experiments on Circular Arc and Flat Plate

Hydrofoils. J. Ship Res., Voi. 1, No. , _(1958).

HANAOKA, T. Linearized Theory of Cavity Flow Past a

Hydrofoil of Arbitrary Shape. Ship Research Institute,

Japan, 1967.

GEURST, J. A. Linearized Theory for Partially Cavitated

Hydrofoils. Inter. Shipbuilding Progress, Vol. 6, No. 60,

(1959).

WADE, R. B. Linearized Theory of a Partially Cavitating

Plano-Convex Hydrofoil Including the Effects of Camber and

Thickness. J. Ship Res., Vol. il, No. 1, (1967) pp 20-27.

(33)

BALHAN, J. Metingen aan Enige bij Scheepschroenen

Gebruikeijke Profielen in Viokke Stroming met en Zonder

Cavitie. Ned. Scheepsbouwkundig Proefstation te

Wagenigen,

1951.

WADE, R. B. and ACOSTA, A. J. Experimental Observation

on the Flow Past a Piano-Convex Hydrofoil. Trans ASME

J. Basic Engineering, Vol.

88,

No. 1,

₍₁₉₆₆₎

_pp

273-283.

WU, T. Y. A Wake Model for Free-Streamline Flow Theory,

Part 1, Fully and Partially Developed Wake Flows and

Ca.vity Flows Past an Oblique Flat Plate. J. Fluid Mechanics,

Vol. 13, Part

2, (1962) pp 161-181.

KTJTZNETZOV, A. V. and EREN'IEV, A. C. On Analysis of

Partially Cavitating Flow Around Flat Plate. Izvestia

Vysshikh Uchebnykh Zavedenii Mathematika No. 11 (66),

_1967.

TERENThV, A. C. Partially Cavitating Flow Around Plate.

Izvestia Vysshikh Uchebnykh Zavedenii Mathematika No. 6

(97), 1970.

COX, A. D. and CLAYDEN, W. A. Cavitating Flow About a

Wedge at Incidence. J. Fluid Mechanics, Vol.

_3,

Part

6,

(1958).

WADE, R. B. Flow Past a Partially Cavitating Casca.de of

Flat Plate Hydrofoils. Calif. Inst. Tech. Engineering

(34)

28

16.

HSU, C. C. Flow Past a Cascade of Partially Cavitating

Cambered Blade. HYDRONAUTICS, Inc. Tech. Report

703-6,

1969.

17'. HANAOKA, T. Three-Dimensional Theory of Partially Cavitated

Hydrofoil. No.

123

J. Soc. Naval Arch. of Japan

(1969)

pp

22-30

in Japanese.

TJKON, Y., TAMIYA, S. and KATO, H. Pressure Distribution

and Cavity Model on a Partially Cavitating Hydrofoil of

Finite Span. Proc. lLth Intern. Towing Tank Conference,

1975.

LIGHTHILL, M. J. A New Approach to Thin Aerofoil Theory.

Aero. Quart., Vol.

3, No. 3 (1951)

pp

193-210.

TULIN, M. P. Supercavitating Flows - Small Perturbation

Theory. J. Ship Res., Vol.

3, No. 3 (19611').

HUANG, T. T. and PETERSON, F. B. Influence of Viscous

Effects on Model/Full-Scale Cavitation Scaling. J. Ship

Res., Vol. 20, No.

(1976) pp 215-223.

ABBOTT, I. H. and von DOENHOFF, A. E. Theory of Wing

Sections. Dover Publications, Inc.,

1959.

JONES, R. T. Correction of the Lifting Line Theory for

(35)

FOIL; e1 = FIGURE la -CAVITY; ¿nq1

,,-FOIL; e1

= E FOIL;

=0

FiGURE lb FIGURE ic

-FIGURE la,b,c, - SCHEMATICS OF SHORT CAVITY FLOW AND BOUNDARY

(36)

t:

-j

w

o

>

4. * 2° i 5o

7 t:

+

0.3 _a=lOo

I 7

o

K I +

0.2 _/

/ I

2 Ui

/

X X

0.0

0.02

0.04

0.06

0.08

0.10

0.12

0.14 a /, INCIDENCE

/CAVITATION NUMBER

0.5

0.4

-w

h

0.3-

0.2-(

01-I o

¡

a = 10 Ny

X

j

/

X

/

2°

/

I

X

/

X

/

O.5° X +

-.----

fl-0.0

0.02

0.04

0.06

0.08

0.10

0.12

0.14 a /o, INCIDENCE

/CAVITATION NUMBER

FIGURE 2a - FLAT PLATE FOIL CHARACTERISTICS WITH SHORTCAVITY

r/.t=O;AR=; a =0.5-

10.00

0.5

0.4 o

(37)

-i

o

LIi

o

0.4

0.3

0.2 Z

0.0

0.10

0.20

0.30

0.40

0.50

0.60 0.?0

CAVITY LENGTH /CHORD

0.8

0.6

0.4 -0.2

020

50 NON-LINEAR THEORY [12J 7 100

//

_/ _-? X ti LINEARIZED THEORY [2 ] LINEARIZED THEORY [2

0.0

0.02

0.04

0.06

0.08

0.10

0.12

0.14 /

, INCIDENCE /CAvITATI0N NUMBER

FIGURE 2b - FLAT PLATE FOiL CHARACTERISTICS WITH SHORT CAVITY

(38)

O EXPERIMENTS (MEIJER, 1959)

3° o

2°

0.04

0.06

0.08

0.10

0.12

0.14 /a

, INCIDENCE /CAVITATION NUMBER

FIGURE 3 - BI-CON VEX FOIL, CHARACTERISTICS WITH SHORT CAVITY

T/4;AR=x;a =2,3,4°

-0<

0

o

z

o

z

O

o

z

u

:i:

t-o

z

-j

>-

t->

0.5

0.4

0.3

0.2

0.1

0.0

1.0

0.8 -

06-

04-0.2

0.0

OJO Ç

0.02

o o

--o 0 I i I

0.20

0.30

0.40

0.50

0.0

0.20

(39)

0.4

-o _j __j LU D -J O

>

>-

J->

4

L) X

>

0.3

-J

o

z

I-

p )

4 J->

4

U

0.1 z

2

0.4

0.3

-(Q

r /t=0

n

ï

0.0004 0.0016

D7

/

- p -, + ì

-0.0004 . 0.0036 0.0084

/

4

/ :

/

0.0100

/D,/

i Iv

//+//

_//'

D __- ç' + X I I I

0.0

0.02

0.04

0.05

0.08

0.10

0.12

0.14

a /a, INCIDENCE /CAVITATION NUMBER

0.5

-I 0.0016 r ) / _i + I 0.0036

/

_i

0.0084

1/

/1

/

0.0100 1 X - »

---T

T

0.08

0.10

0.12

0.14 o /

, INCIDENCE /CAVITAT ION NUMBER

FIGURE 4 - FLAT PLATE WITH CONSTANT THICKNESS AND ROUNDED LEADING EDGE, CHARACTERISTICS WITH SHORT CAVITY

AR=;r/=0_1;a4°

n

o

-U

0.]

-0.0

D

/2;

77

f X * -f ₄

0.02

I

0,04

0.05 r /0 D

_n

(40)

J

o

-j

(D

z

I

I. >

Y

z

o

z

(Q

0.5

0.4 -0.3

0.2 -0.1

0.0

0.5 -

0.40.2 -0.1

a = 60

/

. p ¡

/

---+

-I i

0.02 Ö.04

0.06

0.08

0.10

0.12

0.14 a /a ,

INCIDENCE /CAVITATION NUMBER

_0

a-6

J40

/1x

¡

/

X + '

0.0

0.02

0.04 0.0e

20

a /a ,

FIGURE 5 - SERIES 16 FOILS, SHORT CAVITY CHARACTERISTICS

AR=; r/6; a2-6°; k=O

0.08

0.10

0.12 0,14

. n -It

i

2°

/

30

(41)

t)

>

Ql

.1 0.2 n n

k=0

Q i

/

A1

0.04,

0.06

0.08

0.10

0.12

0.14

0.1

k=0

0.2 J

_J

I!

/

1/1

4. ₀

///

a O

0.04

0.06

0.08

0.10

0.12

0.14 /

FIGURE 6- SERIES 16 FOILS, SHORT CAVITY CHARACTERISTICS

AR; r/6;a 4°;k =0, 0.1, 0.2

0.4 -

0.3-

0.2-0.0

0.02 a /a

0.5

-X

0.4-c'

o

0.3

-O L)

0.2-0.0

0.02

(42)

E

<2'

n __ft

I

o

(j

0.04

0.06

0.10

0.12

0.14

n 6% 12%

-n

-_

b-I

T

0.0

0.02

0.04

0.06 0.0e

0.10

0.[2

a /a ,

FIGURE 7 - SERIES 16 FOILS, SHORT CAVITY CHARACTERISTICS

AR=; r/O, 6,9, 12%;

a

4°;k =0

z

--i

0.0

0.02 a /

0.5 o

w

0.4

-X -J

o

0.3

o

0.51 0.4 -

n -J -J n

(43)

C) o

J

o

=

o

0.1

0.3 -0.2

i

X

ï

f n _J 8 L 4 / J o

/

x

j

/

j

/t

4 +

0.04

0.08

0.12

AR = 2 AR = 2

y

r

i

-0.16

0.20

0.24

0. f ¿ I I

y

f J

//

/

___X - o f

-I I

0.0

0.04

0.08

0.12

0.16

0.20

0.24

0.28 a /a ,

0.5

0.4 -a /ci

FIGURE 8 - SERIES 16 ELLIPTIC WING, SHORT CAVITY CHARACTERISTICS

AR = 2, 4, 8, ; r/

= 6; a = 4°; k = O 8 4

ï

T

0.1 -0.0

(44)

J

o

0.15 _O

0.L1

--;

0.04 0.0E

0.08

0.10

0.12

0. a /

rl

-

4 +

.- + -

_X -rl cl

/

z

/

/=0 r

i

/;

9% cl

-

-_n__ .

O

* - 4

+ X --I I I

0.04

0.06

0.08

0.10

0.12

0.

a /o , INCIDENCE /CAVITATION NUMBER

FIGURE 9-SERIES 16 ELLIPTIC WING, SHORT CAVITY CHARACTERISTICS

AR=4; T/0, 6,9, 12%; a

6°;k =0.1

2% I 2%

0.0

0.02

0.15

0.4-o

0.3 -=

L)

0.2

-0.1

0.0 0,02

(45)

0.4 -o

z

_I

I

>

Y

z

O

z

0,1

c

0.3 o

O

I

L)

0.2

0.1

-n n n _6%

!

i 9% J, -

_/

I

/

= r] =1 -=-.--- _=-

/

12%

0.0

0.04

008

0.12

0.16

0.20

0.24

0.28 a /a ,

INCIDENCE /CAVITATION NUMBER 0»5

o

X

0.4-n /

/

n (-r

/J

64 9% n

T-0.0

0.04

0.06

0.12

0.16

0.20

0.24 0.2e

/

AR =2; r/t=O, 6,9, 12%; o

6; k

0.1

(46)

0.5

0.4 -0.0

0.02

0.OLI

0.06

0.08

0.10

0.12

0.14 i/i, THICKNESS /CHORD

0.5 -

0.4--0.1

0.08

a/a

0.04 0.12 0.10

\

-0.0

0.02

0.04

0.06

0.08

0.10

0.12 D.H

T/, THICKNESS CHORD

AR=4; T/»0- 12%;a/cr =0.04- O.14;a

6°;k

0.1

L:

0.3--

0.14 -J

o

z

02--

I->

0.12 L)

0.1

0.10

o

z

0.08 a lcr = 0.04

o

\

C',

0.3 o

t-)

0.2

(47)

Q

>

0.4-n " U.3

-o

CG L) 0.08

J

0.0

0.02

0.04

0.05 0,08

0.10

0.12

0.14 T/i, THICKNESS /cNORD

u 0.12 0.16

= 0.20

\

0.0

0.02

0.04

0.05

0.08

0.10

0.12 0,14

1'/.i',, THICKNESS /CHORD

FIGURE 12 - SERIES 16 - ELLIPTIC WING, SHORT CAVITY CHARACTERISTICS

AR2; r/0- 12%; a/a =0.08-0.20; a =6°; k =0.1

o _..J 1

u,'-:

is

-

a/a

0.20

-

( .-) \