Analog study of hydrofoils near the free surface

by Daniel FRUNAN

(under the süspervision

of Pr. R. SIESTRUNC)

February 1965

This research work was supported by the Office

of

Naval

Research

Department of the Navy

(Contractnb 62558-4113)

B.ASR.A.

47. AVENUE VICTOR-CRESSON

ISSY-LES-MOULINEAUX

(SEINE

Office of Naval Ree.earca.

american Fmbassy

Xondon

Lab. v. Sche2psbouwkunde

.Technische Hogeschool

DeLIL

BUREAU D'ANALYSE

ETDE

RECHERCHE APPLIQUEES

ANALOG STUDY OF HYDROFOILS

ANALOG STUDY OF HYDROFOILS

NEAR THE FPEE SURFACE.

by Daniel FRUMAN

(under the suspervision of Pr.

R. SIESTRUNC)

February 1965

This research work was supported by the OffiOe of Naval Research Department of the Navy

Analog design of supercavitating hydrofoils at non zero cavitation number near the free surface,

1, Introduction

The scope of this work, is to give a method of design for super-cavitating hydrofoils near the free surface, if the pressure distribution on the lower surface of the foil is given. We suppose that a cavitation pocket is formed from the leading edge to a certain distance down stream. The length of the cavity is another parameter of the problem. If we suppose that the pertur-bations of the uniform flow due to the hydrofoil remains relatively small it is then possible to solve the problem by means of the bidimensional small per-turbations theory. The resulting mathematical model is easily solved by means of the method of analog computation.

The pressure distribution on the lower surface of the foil is equivalent to a vortex distribution located on the projection of the foil in the direction of the uniform flow. The thickness of the cavity corresponds to a source and sink distribution placed on the projection of the cavitation poc-ket. The strength of the sources and sinks are unknown too. For a given Froude Number the strength.of these elementary effects can be determined by an itera-tive method starting with the solution for a very large Froude Number (Fr = ).

Each step of the approximation needs the computation of' the shape of the free surface due to the elementary effects by means of the process shown in (1)

To illustrate the method proposed, we have calculated two super-cavitating hydrofoils at two different submersions and two different cavity lengths. The results are compared to those found without taking into account the gravity effect.

2.

1 .1 Theory of small perturbations

Let us consider an irrotational and permanent f lOw of a perfect and heavy fluid around a supercavitating hydrofoil (Fig. 1) of chord S and at depth of iinxnersion £. The stream function of the flow is written.

oy

where is the perturbation stream function and V0 is the flow velocity up-stream. To remain in the scope of the small perturbation theory we assu.me the hydrofoil only slightly cambred and the cavity thin. The boundary coziditions

of the function are imposed on the projection of the physical boundaries in the direction of the uniform flow, The linearized Bernoulli equation gives then the pressure in any point of the flow field.

=

-- r gy

Ci)

where u represents the horizontal component of the perturbation velocity,

p the pressure at infinity upstream on the stream].in .the depth of the

point considered and g the gravity constant, ('

/

At the free surface we have p = p = p where p stands for the atmospheric pressure, hence equation

Ci)

simplifies to

Vu=-g

(2)

where gives the shape of the surface wave relative to the static surface. The slip conditions on the foil, cavity and free surface can be written in the linearized form

dy V

-

dx - V

0

where v represents the vertical component of the perturbation velocity and the local slope of the boundary.

b) Usually the cavitation number is defined Po - _PC

o1

2

-

_2'o

where denotes the pressure in the cavity. We shall now change this convention in the following way

- PC

0 - PC

+

2F2

--

= + 2F f ₍₅₎

1

_1pv2

ss

0

2

Io

2 o

From (i) and (5) we have on the boundary bits AL' and FL

-2 2y

u ₌

(1-F5

) (6)

2

v2

0

where F , is the Froude number. On the upper

sur-gs

face of the wing section the function.is

Vt!-1

= 2

(1-F2

2y crs

while on the lower surface we have

-°

(iF

2y

2

's

Cs

c) On the boundary AF, projection of the lower surface of the profile we can write, taking into account (i)

Vo

- /

ii -,

= -

u_

£ (--)

_-

°

(1-F2

2y

s s 2

n

C

We choose the function

£ (--)

₌ so that it satis-fies the following conditions ;

L

=

for x = XA = XF, £ = 0 for all

_{XA< x < XF}

f (1-) > 0

because on the lower surface of the profile

'

Condition assuring the closed cavity

At the down stream end of the cavitation pocket the ordinates of the lower and upper boundary of the pocket are equal, there fore we have

= (9)

At infinity the perturbation velocity is zero, consequently

grad = 0 (1 o)

Lift coefficient From (i) we can write

=

(u

- u) +

(y -

y)

(ii)

taking into account (7) (7') and (8) and integrating (ii) from XA to _XF we find

2r

C "L

Vs

Lo 0 since J

i(---) d(--)

= 1. A

Function

4'

_{l'unction ''}

2

) At the free surface a')

-'i,

=-k

'ln = - 2 b) On AF -Y in c) On FL

V-

2y ,

VC

2y

iln

= - 1-

(i_F; ) =

F2

o L -2_2y1 - 2

5S

d) On AL! d') +' -2

2y,

ln = 2 (i-F5 -s

(°()+ F2

2y2 2n 2 s s C = -F

VCL

2y s 2

Cs

2n 5.,

y1 and _{y2 . being the ordinates of the "profiles" corresponding to the above} stated conditions

1.4. Conditions at the free surface

The condid.ons (1.3.a) for the free surface can be transfor-med as shown in (t). nowing the function)ori the free surface due to a submer-ged source or vortex, it is easy to determine the function41 corresponding to a distribution of sources and vortices.

In L and L' e') + = _{1L =}

_V2L

At in.finity grad = o grad 2 = b') )

Q

The elementary solution for a submersed vortex of unity strength is given by

-.kf

1

Pk

sin m-m cos ml) -mx = +2e sin lcx +

-

1(2 _{+ m2} e dm for x 0 TI

(13)

= +

-.1-.

_j'°°

"

ml cos e' din

-

2 2

"0

k +m

and for a source of unity strength Q, we have

-1(2 1

'(k cos

nil+m sin ml) -mx

e cos

kx

+

-J

e din for

x > 0

m2+k2

(14)

1 (k cos in2 +nisin

e din for

x < 0

7V

m2+k2

1.4.1. The functions

1

and correspond to a distri-bution of vortices sources and sinks on the profile and

the cavity, their strength being

Function

(+

_{u -u) &ix}

-

A

we can write in a general manner the following distri-butions Vortices Sources 1 =

F2(+-J1)

(xA< xxL)

and Q = v+

-

V 6. Q1

=L(

) (15)

7'...

It is evident that the strength of these elementary effects are unknown beforehand, is they depend on the values of the functions and

2 on the boundary

AFLL'A,

with exception of the vortices

/\f'20

Situa-ted between A and F and which are known f'om the beginning. Thus, in order to completely solve our problem we have to make use of successive approxima-tions0 First, we suppose the Froude number to be infinite. The resulting sim-plified system allows us to determine the values of the functions and on the boundaries. From these values we can calculate the expressions (15).

and (16) for any Froude number. Using the procedure described in (i) it is possible to determine the corresponding free surface by means of the

formu-las (13) and (14). With these initial values we determine the new values of and on the profile and the cavity. This process is repeated until the difference between the values 4I, and cf)2 of two successive

calcula-tions is sufficiently small.

1.5. Considerations concerning the effect of gravity upon the pocket and the free surface

We saw in chapter 1.4.1. that the strength of the vortices sources and sinks are functions of

1

and on the boundary AFL'LAU.. It is easily shown that they are connected in such a way, that certain terms in the expres-sions (13) and (14) multiplied by (15) and (16) are equal.

Let us take the values of the functions L) 1

and P2 on the

boun-dary when the Froude number is infinite. It is then possible to calculate a distribution of sources and sinks equivalent to these values. If the distances between the sources and sinks is sufficiently small the free surface is given by the expression = 2 + 2ke sin kx

k

çcc (k sin2mf-m2cos nil) _{dm (17)} D r _-"0

K +m

where represents the strength of a dipole given by

but dQ

_'

= d + -

y) =

dy x dx dx and i = [i :m x dy +

J

x dY] D 0

For the equivalent vortex we have

dV' =_v0(y_y)dx-V0kydx

and

C-

_-Vk

_ydx

°

A

but as y = y(x) is a closed curve

P

5 =

k

By introducing (18) into (17) and adding the expression for the free surface due to a vortex (13) we obtain

f7f

___

('9)

kr

This expression is completely independant of the Froude number which we have verified by numerical calculation. Figure 4 shows the shapes of the free sur-face for the funtiofl in the following cases : reduced i1nmersions -s-- - 1,

reduced length of the cavity pocket -j , Froude number F = 0,0628 and

0,1256. The calculation has been carried out by taking a distribution of 15 8

1.6. Calculation of the drag (Figure 5)

The drag of the profile is given by

2

r

R =

-

.JFAL' p

dy JFA

(p_p)dY =

LFAL' z

But instead of taking the integral on the contour LPAL' we shall calculate it along the boundary defined by

C, where "yrepresents

a

singularity placed at the down stream end of the cavity. Thus

R=+Jf/) dz

( dz

C

9.

dz (20)

The second term is equal to the wave drag, because the integral on the con-tour is zero. There caly remains the integral cnes

BC

2 o o

22

4

dz =

v2dy = ___Jk2vA2e2c

A

(22)

which is equal to the wave drag. fer the foil. The first part of the expression (21) must therefore represents the resistance of the wing section due to the

if

cavity. The function

is

singular at the point z = - where it takes the form

_(z-1" (2),

in such a way, that the integral is reduced to

2

/1ff\

2

-JJ

2

)

dz =-2ir-K

2

with K =

'2

and where the

total

drag is

4(6-x) /

r 2

2

R=-- jkV0 A+4(_

₎ and ₂

kA______

CD = CD + CD = +

8Vs

(-e_ 2

10

1.6.1 .From what we have just sown in § 1.5. the contribution of the wave to the total drag is practically negligible. Therefore the drag depends only on the distribution of

the vortices along AF'. On the other hand the drag due to the cavity will only depend on the values given by the functionS ( - close to the singularity as the values, taken by ( - are negligible

compared to those of QIjs

2 analog calculation

The harmonic functions and are represented. by an elec-tric field in a two dimensional conductor such as a flat bottomed electrolytic tank with its inversion tank (i). The frontier AFLL'A is represented by a plate of plexiglass covered with electrodes 1 cm long except near the leading

edge where, to improve the representationof the functions

and\, we

choose smaller electrodes. In a similar way the free surface is also repre-sented by a plate of plexiglass carrying electrodes.

2.1. Representation of the function

2.1.1. Without the gravity effect on the cavity.

Figure 6.a) shows the analog set up used to represent the boundary conditions of the function (j?j. Each pair of electrodes on the plate is connected across a

resis-tance to the secondary terminals of a transformer T in such a way as to assure the continuity of the normal

The measure bridge PM is connected to a

transformer

having the same ratio as the transformers used to supply

the

electrodes. The potential of the secondary terminals of the transformer is put equal to -100. In order to ma-ke the potential difference on the electrodes relatively small

(between

-5 and 0), we choose the resistances R sufficiently large.

+1 -1 V0c- 100

2 -hIC

n n

where j = width of the electrodes (cm)

h = depth of

water

in the

tank

(cm) -1 -1

-

Cr=

conductivity of the water Cr2. cm ).

From (3) the potentials

4'

measured on the plate give us

2y1 V t-s lOOs

0

2 hX

On the electrodes of the free surface we impose by means of po-tentiometers the values of the potentials of calculated from the expressions

(14). For, the first approximation (Froude number infinite) these electrodes are not supplied.

2.1.2. With gravity effect on the cavity

The normal derivatives of 4., on the plate are now given by the complete expressions of

1.3.b, c and

d, the values of

y

and

being obtained in the same way as in

§ 2.1.1. where "the Froude number is infinite. The

elec-trodes 'of the plate are, no longer connected to transfor-mers, because now '41

-'

.

Instead, the resis-tances R are connected to -100 for the electrodes of the upper surface

'and

to +100 for those of the lower surface. Figure 6b.

with R = K

V0 -

S

=

the circulation will be

0

K

where K is a constant which has to be chosen as to make - 5 and

5 These potentials are made adirnensional by deviding the values

I + -

Vc-S

of and by o given by the relation

2

1 OOs

2

At he free surface the potentials are determined by means of (13) and (14) for the distributions of vortices, sources and sinks as calculated by (15).

2.2.. Simulation of the function.2

2.2.1. Without gravity effect

In this case the electrodes situated between FL and AL' are not supplied while those between AF are connected to large resistances 0

X '_

100 x - S 9

'

hK s. with ER = 2y

_s

c-s

For an infinite Froude number the electrodes located on the free surface are disconnected ; in the other cases, i.e Froude number not infinite; we impose the potentials

2

calculated by means of (13) and (14).

2.2.2. With gravity effect on the pocket

In agreement with, the conditions 1.3.b',

C'

and d' the electrodes of the plate are connected to the potentials + 100 for the upper surface and -100 for the lower sur-face across the resistances

K = _{for XA(}

_X(XF

£ (--)

+F'2 K

F2

XF<

X<XL

Lo

where y and y2 are obtained in § 2.2.1. for a fluid with no gravity effect, The free surface potentials are calculated as above,

2,3. Superposition of the functions

_+'

. '1' The expressions 2*'i / CL \ 2'i2 2y1

V0aS

cr )

VCLC

=

-

s

I-,

(25) and (26) lead to the following relation

which determines the shape of the foil. It is important that the factor

does not exceed a certain value, lower and upper surface intersection for the rcsulting profiles. We may take the factor (_

ax a criterion for choosing a favorable pressure distribution on the lower surface.

, CL \ as 2y2 2y (27)

r )

CLS

= -r

C in magnitude R =

3. Experiments and results

The aim of the experiments was to show that the solution to our problem for an arbitrary Froude number is converging if we Start with the solution for an infinite Froude number.

We studied four different cases for which _{f/s 0,5 and 1.0} and -/s = 2.0 and 3.0. The Froude number was defined by the ratio

gs f s

F2

_{= 2}

=2rr.=ks

v0

A

For these four different cases we have also numerically calculated the values for _{and '.'2 with and without taking into account the gravity effect, on} the Cavity.

3.1. Function

1

The figures 8, 9, 10 and 11 represent the shapes of supercavi-tating profiles of zero lift for different Froude number _{and with arid without} the gravity effect on the cavitation pocket. It is not surprising _{that the} smaller the Froude Number and the larger the pocket is the more the results of the _{two last cases differs For sufficiently large Froude number we can assure} that the hypothesis of the light fluide leads to results not too different from reality.

3.2. Function

14.

The figures 12, 13, 14 and 15 show the function +'2 for _three different cases. The remarks we just made concerning the function '11 _are

al-aie nearly equal for infinite Froude numbe and finite Froude number, but with the gravity acting on the cavity, while theactual shapes of the lower surface of the foil are very different.

In a general manner, we may say that the two hypothesis consi-dered do not affect much the shape of the lower surface, but are important for

the values of (c/-

)maximum in the combinations of the functions and L12

§ 2,3. The fig.ares 16, 17 and 18 show 'sing sectjonS characterized by

f/s=1/2,

?/s=2andFrO.1256.

3.3. Total drag

Seen the comments of § 1.6.1 and taking into account 24 we iay write 2 2 CD kf -2k2 S 2y s e £ 32

ç)(P

)](

(28) CL2 - 2

For the maximum amplitude of the free surface due to a vortex

r

is (i) -kf

=-2e

CD A

For the four given foils we find the C total O085 + 1 .36 0.085 +

_1.615(--\

0.028 + 1.258(t)

0.028 + 1.475(-(29) For S 2 S

=2

CD CL f -e CD =-

and=3

-=

L

and'_2

S S '7 C - _{= 1 and 3}

1

= 5 5 _C L

3.4. Procedure of the calculations

By making use of the figures 8 to 15 we can establish the shape and the hydrodynamic characteristics for a supercavitationg foil, The cavitation number

- is determined by the physical constants of the fluid and the flow ve-locity. By means of (5) which we may write

C-

= + 0.1258 we deduce the

smaller than sio (29).

values of from those of. The value of (

) being chosen

(_CL ') we calculate CL and determine CD by means of the expres-- /rnax

0

p=po

V0

I

p->po

S TI

Fig.1

4=ic4

B

S L

A

'F

L

qr'..I'of(X

Vocr(1_12

2fl

fl - S

₂

_/

_V0o (1_E-2 2Y)

A

=

VOCL

F

+F2

Y2

SCL

FUNCTION

;;'

V

(i

2

=

F

V0o1

_{F2_2Y1}

2

1S

FUNCTION 12

Acx

2fl2

Ba,

tb

F2 VOCL

Y

2

Y2n-S

₂

L

)

L

(p_s

OL.O F2 v; 2

2fl

₂

_SCL

''2L.2L

A

_P1

-.-Sources

and sinks

efFecr

...1OO

V00s

F

2

x

0

I,

Fig.4

plus

vortex

/1

_pIus

_vortex

0

4X

S

100

0

1.-a

_a

a

100

0

0

0

.-100

₀

I

PM

FUNCTION

4,

Analog

represenohon

cR

eR

(b)

Fi96

I

a

idLdãb

ILL

.PMP

PMP

0

N_

0

-cc

'I

-lOOo

eR

K.

CL.

FUNCTION '2

Anà log

represenahon

eR

K

eR

K

F-.f2Y2

-2 Y22

CLS

ij

_CLS

Ii

PMP

+lOOo-0

grovily

eFfect

over

he cavfty and

the F.S.

T' t=L,

FcD

no gravity

effect

over

he

Fig.8

Is

0,25

045

0,75

-=1.#. IL..)¼).

CaviIy

t_.

0

S2'

the

S

'

-efFect over

the

effeci

over the cavity

F.5.

6'

cavity

..-___

no gravity

gravity

and

/

/

/

/.

'I

-

-/

/

I.

/

/

I

0,5

0,75

._.

\

0.25

₀₅

_0,75

a-Fco

no gravfty

e?ecF over the

cavity

---gravity eFfect

over

he cavity and

the ES.

0

41

F0,0628.

no gravity

effect over the

cavity

gravity effect

over

the

cavity

0,5

-0,5

CLS

_CL

Q25

0.5

0,75

-4-=3-,-k-.2. F2O1256

_Fco

ro gravfty eFFect over ihe coviy

grovfty

effeci over

he cavity and

the

_:F.5

Fig.12

0,s

U

£

2Y2/o

CLS/CL

i,

3, F2=ô1256

F5=co

no gravity

eFFect

over

he

cavity

gravity effect

over

the cavity and

he F S

-F

0,5

0

A

-0.5

'7

I,

025

2O16

Fs=oo

no

gravfty

effecr over

fhe

cavfty

graviy

eFfec

over

1he

cavity and

tt%e

_{F. 5.}

0.5

Fig 14

0,75

2Y

CLSI

(0

CL

0.5

0

A

2Y2/cy

(.

Fco

-- -- no graviy

effeci

over

he

cavity

-- gravity

effect. over the

cavfty

and

the F S

A

-

-_47p

O,5

-- -- --

--.- -.

N

N

N

0,5

-Y2

so.

S.-Gravity eFPect

over

the ES.

Fj

s

,

F2-O 1256

--5'

\

--5

\

CL

0

0

2

4

6

2

I

0

jY2

so-

Gravity

ePFect

over

the cavity and

he ES.

s=--, r=0,125.6

0,25

.-.

0,5

Fig.18

N

N

CL

0

0.

4

Design of supercavitating hydrofoils near the free surface for zero cavitation number,

Introduction

In the last few years the design of supercavitating hydrofoi].s cor-responding to a given local pressure distribution has been studied in a large number of technical papers where the profile is located in an infinite fluid and the cavitation number zero o' different from zero. st of these papers are ba-sed upon the linearized theory due to Tulin and Burkart (1).

The same theory has been successfully applied to supercavitating profiles near the free surface for zero cavitation number. Recently, Auslaender

gave an interesting method to solve the same problem. Ha makes uses of a trans-formation leading to a problem which can be treated by means of classical airfoil

theory. The general equation yields the shapes and the hydrodynamic performances f:r given load distributions as well as for different flow angles and thickness distributions. Bymeans of an IBN computer 1620 families of profiles can be cal-culated.

It is interesting to notice that the pressure distribution over the final profile, having a certain thickness and an angle of attack, differs naturally from the initially imposed pressure distribution. We shall see later that the differences between our method and that of Auslaender is not only due to the different calculation method employed (analogue computation instead of numerical computation) ; but also because the comhination of the effects

al-lows us to keep the initial pressure distribution and the center of lift unchan-ged

know-2.

supercavitating profiles in an infinite fluid (4-) it was pGssible to define the boundary conditions and a number of simplifications for ,the analogue represen-tation, as well as to compare the analogue and numerical results.

The first part of this report deals with the analytical solution o.f the problem. The physical plane is mapped into a half plane where the lower surface of the profile corresponds to a half cercie. The complex perturbation potential is determined by an appropriate distribution of singulari.es in the transformed plane.

The second part gives the soLution by the analogue method and com-pares the results obtained to those of Auslaender. Three other pressure distri-butions are then analysed and compared wit' the previous ones taking

into

account

different hydrodynairic and mechanical factors.

Finally we give an application of the method for the 4esigi. of base-vented hydrofoils of zero drag for a given prersLre distribution.

1. Analytical calculation

Let us now consider fig. 1 .1.. show a thin wing section located in a permanent irrotational and incompressible flow bounded by the free surface. The flow velocity at

infinity, v0

is supposed sufficiently high as to make the hypothesis of infinite Froude number valid. The cavitation number

pp

10v2

2'

o

is consequently zero. This is equivalent to the hypothesis that there exists no gravity effect and that in the cavity the .' essure is.p

= p0.

If we adopt this scheme of flow we find it convenient to caracteriSe the position of the profile as Green has shoWn (5) by the thickness

S

of the jet and not by the depth of im-mersion. In fact the free surface shows at infinite a parabolic behaviour

ana-logue to the streamlines of a profile and can therefore not be used as a re2ernce. (i.i.)

We can however suppose, and experimental results confirm it, that the so cal-culated flow patterns represent in the neighbourhood of the profile with a suf-ficient similarity, the physical relity.

We may write the conditions of tangential flow at the free sur-face and on the profile as

x = instead of the more exact form

0 V

yx

=v +u

without introducing significant errors. The linearized

theo-0

rem of Bernoulli is given by

P_Po=

The complex perturbation potential f(z) = + i J , is an

ana-lytical function defined by boundary conditions which, instead of imposing them on the profile cavity and free surface, can be satisfied on the projection of' these boundaries upon lines parallel to the x axis

on = 0 = 0

OB'

=0

B01

(f

=

'f'o

+ (-1

where

p = p0

and consequently u =

= 0

Irepresents

the circulation on the hydrofoil and Lf' is a constant related to the drag due to the jet as we shall see later, It also characterises the ope-ning degree of the cavity. In fact, the fluid has at infinity downstream in the

4.

On the lower surface of the hydrofoil we distribute local pressures

C /"

= f (2E), _{The functions and}

'f

can be written as function of C

P

g()

and

_{g' () =}

£ ()

(1.2)

where

_{g ()}

is a function chosen in such a way that the Joukovsky condition at the trailing edge is satisfied and that over the whole profile a negative de-rivative is assured so that the local pressure is everywhere positif between

A arid F g = 1 - 0

LJ

g' (.) <0

g (i) = 0 ; g' (1) = 0 The expression g () = I - 0

L()J

fixes the load at the leading edge. If

r'1

the flow will be smooth, if n= 1 the load on the leading edge will take a finite value but becomes infinite for n<.1e As. ethers have already me:tioned

Ci)

the

tolerable limit is n = 3/4.

1.2. Analytical solution

The physical plane

z = x +

iy is transformed into a half plane

(A)

by the conformal mapping

z =

(5 -

1og, - i)

and

>-

=

(I +_

\

1

2 4

whereF represents the affixe of F in the plane. In theJ

plane the sec-t-ion OF is represented by a half circle of unity radius. The real axis x

re-presents the limiting jet lines of the cavity and the free surf ace We find the

- boundary conditions which determine the function £(z) in the plane

(>)

The function

f. A ). 1

In particular, on the cercie ¼f = flg(9) _; g (o) is an add Lunc-and satisfies.. the following conditions

9=0

g(o)=

0

g'(o)=

0

g(o)

<0

Q=-ff

_g(-1T)=i

;

f()

I for is lo.g (, j

aa

10gN

P2 -2

-rr

log

If we take into account the mentioned conditions, the function

_{£(>\ )}

is given by

(A

_-B

(..

1)

I

=iiog

.

+1hog

+

-L

n=1 An

Xn

+ i - log

- _{9 +} -r-;-i s-r-;-in n 5. (i .3.) (cos n 9

-A

B representing B, = 0 is

For one cavites tion in (9)

where the coefficients An are given by

An

= 'g( o) +

sin n d

In the same manner we can write the functionW on the half circle

S

log 2

77-

F1

where the An are the same as above. On the real axis is defined by

LQ0 .

A

-?=log

iT

C ,\)

I

Vs

II 0

+1

og 2 - 1 + cos 9 / +--- :n cos n G (.1.5.) / 'I n=1

+ - log

r

+ cos 9) +

-/

t.p n= 1

/

(J

2

(-n=1

(\j

1 An cos n 9 6. if

>\ <1,

and if

_A

1 by _(1.6.) log +

L

_log

₊₎

.11 _7? L n=1

1.3. The shapes of the lower surface of the hydrofoil of the free surface and the jets forming the cavity.

Let us call the ordinates of the streamline forming the profile the free surface or the cavity.

The

Slip

condition is

=

-. (1.7.)

S

Vs

0

but as we shall see later we may replace

Vs

0 therefore

r'Lj

₁

It

Vs

2 2

and the abscissa x/s of the points are

-- -

'

2. I

r,

)7

og_.

-At the free surface i .e for

B

we have

1 (,\

B)"\B)

syCD

7r

s

_(X)

2

_SYCD

and log 2

ti-+1

cos =.. + cos

To each value of G corresponds one value of , corresponding to

1-F

_F1

S 1-n cos n 9 D n=1 7. (1.9.)

S

=

T

io3J

8.

The upper surface of the cavitation pCket 1 is determined by

a similar expression. Finally for the lower boundary of the pocket

)(i.

= (..og

(A-XB)(?\_1/AB)

+ Ti

sL

A

(ZcD

n=1 (i .io.)

the values x/s are obtained in the same way aS above.

1.4. Ca].cu].ationof the drag

The drag of the hydrofoil can be determined by the formula of Blasius

f\2

r

_r

_(&dz

= 2 =

- 2 OB

BA.+r+ CF

from there we deduce for the drag coefficient

1 2 CD = -. v2 OB B A +

r

) dz

+C

F 0

The houndary a half circle of an infinitely radius connecting A

and C . On OB!, , B

A, and C.

F

; dz = dx is real as well as (-) - v2, the integral taken along these boundaries is zero. On B dz = idy and z = iv = - i , the integral is therefre

2

dy=

At infinity z = i

fl+P/7z

and

I

ffl+Lp\2

=0(_i

)r

_.\Z

for R x S

and finally

=tg2

D

2

0

0

in the same way for the function

9.

and

tgC)

r-

_S

y'

_X

(1.13.)

=c

This shows that the drag of the foil is due to the jet originated at the leading edge and does not depend on the pressure distribution over the lower surface of the profile.

1.5. Remarks concerning the analytical expressions of 1.3.

All the relations which determineS the ordinates of the free surface the cavity and the lower surface of the profile., consist of two expressions of which the first does not depend on the chosen load distribution, but is oiy a

function of the valueF, i.e.

of

i/s,

while the second depends on the impo-sed loads multiplied by the coefficients An.

CD

p2cç

pP0+r

I

6

'Pycle. j'>=cie

r ;

B,

(z)

OaA

=r

g()

9=0

;

C.,

;

I'nO

B..

B.. B'..

OA

(A)

c= rg(e)

Fig.1 _1

ÀY

Pc=Po

6

P=Po

Vo

A0

S

7

2.. analogue simulation

2.1. Boundary conditions of the function and

+' 2

The boundary conditions for the in the (z) plane have been in-vestigated in 1.1. Now we shall consider, according to

1.5. the functionaS

being the combination of the two functions _'f-'1 and and we shall define

their boundary conditions. On the free surface and on the jets limitingS the cavity the functions and ' satisfy

jJ I

Ty

On B B we have according to 1.1.

= cte

10,

and on AF, the lower surface of the profile,

4)

iy

= a =

r'/S g'()I1/S i()

2,2. Analogical representation of the functions

f)

₁₂

The electric set up requiredfor the analogue simulation is simple

The boundary CFAB

is materialized by a plate of plexiglass carrying

The side walls with exeption of the one repreenting the free surface are co-vered by a cOn±jnUOuS large electrode, while the latter consists of an impor-tant number of small electrodes of 1 cm length which remain disconnected but may serve to measure the potential of the field at the corresponding points.

2.2.1. Representation of the function (figure 2.1.a)

We dot not think it neccessary to translate in detail the mathe-matical boundary conditions into conditions for the electrical field. However

it is easy to see that in the case of the function we impose on the valls of the tank a zero potential with exception of the section . On the lower surface o± the profile as well as on the cavity and the free surface the electrodes remain disconnected, they take an electric potential Lfrn1, which is proportionnal to the ordinates of the streamlines and which can directly be measured. According to the slip condition we have

()

9'i

(1

--0

The formula connecting the CD and (1.12) becomes

and I

V0

S

\JS

vs

11. (2.1.)

The value of is related to the electric current flowing through the electrode B B' , but as it has been notices before in similar mountings (3) the electric current measured does not exactly correspond to the current flowing through the electrode, error which may be caused by the capacities present. Therefore, we prefered to measure directly on the electrodes.

We so obtain with a

antcLnt

precision

where V stands for the difference of potential between two points of dis-tance

AS.

2.2.2. Representation of

fi2

(Figure 2.1.b)

The electric conditions determining the electric potantial are the same as in 2.2.1. with exception of the electrode B, B which is now represented. by an insulating surface which automatically satisfies the condi-. tion ¼.f,

2x = 0, while the electrodes between A and F are connected to large resistances allowing to satisfy the Neumann àondition

y=

;-f(----)

As before, we have 2 but, V = Os CL from where

2fl

12.

(2.2.)

VY\

CL

CL

)vc

13.

if

ER

= /f()

is sufficiently large the measured voltages on the electrodes

should not excead

_{0<'f'2 < -5}

measured in the scale 0 - 100. Hence

lOOs

IThK

With these values of , we. calculate the values of

2 2' (

YC/c)

2.3. Composition of

4)

The shapes deduced from are those of a non lifting profile unique for each value of the submersion cS/s, while those obtained from the function +'2 depend on the load distribution as well as on the depth of imzner-sion. It is to note that the latter has no physical meaning in itself as the upper and lower boundaries of the cavity are interchanged. Only in combination with the values of _{makes 'f'2} physicaly sense.

ç

The shapes of the supercavitating foil section at a submersion

d/s

and a given load distribution

(2.3.)

where CL/.CD is the square root of the sharpness chosen for the profile. The values of CL!' CD will have a boundary determined by the condition that the ordonates of the streamlines representing the upper surf ace of the poc-ket let be larger than those of the profile.

For each chosen pressure distribution it is possible to define a maxiin.un lift-drag ratio, which we may take as a first cx'iterion for our choice.

2.4. Section modulus of the foil

A second criterion for selecting an advantageous pressure distri-bution is the foil strength. We consider the wing as a beam and write for the section modulus

= Z/s3 _{= li/c h/s) max}

where _{signifies the moment of inertia of the section and (h/s)max, the} ma-ximum distance between the neutral axis and the farthest fiber (7).

To calculate the values of Z we used the method of Boudigues (8) which gives the shape of the profile, at the points

(x1/5

; Yk/S) for two pa-rametric equations of the form

fx

= -- (i - cos

Sifi

(t - k

2ntg j(tk/t)

s _{represents thechord in thecase of profiles with a sharp trailing edges.}

In our case, where two "trailing edges" exist, s _{stands for the horizontal} prodection of the distance between the leading edge and the traling _{edge of}

the lower surface. '

We must remark that this methode, if it is _{not applied to foils} with sharp trailing edges might lead to _{some errors which will be very small}

(n = 40 in our case).

The calculation of the caracteristica]. shape and _{the moment of} inertia for each profile

yhds

_{the change in the magnitude Z in function of}

and CL.

k = summation index

2.5.1. Hydrodynarnic condition

The value of

(cL/V)

must be according to 2.3. maximum for the considered series.

2.5.1. Mechanical condition

The value of the section modulu Z must be comprised between the two limites (2) : 2 x 1O

K

Z <12 x 1O. The section modulus does not

present an exclusive condition but rather an upper and lower limit for Z.

2.5.3. In order to make a rapid first choice of foil section we believe that also the following points should be taken into consideration.

The profile thickness, i.e. the distance between the upper 11-mit of the cavity and the lower surface should continuously increase.

The incidence angle of the foil, i.e. the angle between the line connecting the leading edge to the traling edge and the stream velocity must be equal or larger than zero.

3. Results and conclusion

3,1. Numerical verificatic,n of the analogue calculation

In order to simplify the numerical calculation we have chosen a distribution of the form (figure 3.1.)

'I

-1

Ansin nO/

where the only coefficient which remains in the Fourier development (1.4.) is A2 = 1/2.

It is now easy to callate by using the above expression the va-lues of in the (z) plane needed for the analogue represention,

TI

I L

___j t n=1

A..

OF

I

OA

0!,

H

PM

0!A

I-I

x

2n

1PM

vn0

F

v,=o

a) Funchon 1i

ioIaI1ng

B..

B.

C..

B'

v=-100

vn=o

v=0

-100

-100

t

F

_4'2n=0

cc.

vn=o

_isolaling

A..

'p2n=0

B..

4);f,

_r

dg(9)

y

dg(9') dG d (x/s) d 9 d(x/s) and dg (9)/d() is given by d g(Q) / 1 1

+ - cos. 29

d(x/s)

)

(cos29 -1

\

2s

sin 9 ₎

L

>F

2 1

-1

senQ

L

16.

The function dg(9)/d(x/s) becomes infinite at the leading edge of the foil in the (z) plane. In our particular case this is not very important as the intensity of the singularity is small. The figure (3.2.) gives the trend. of the function f(x/s) for the two depths chosen in the numerical calculation namely = 0,2.569 and cs/s _{= 0,5414.}

The figures 3.3. and 3.4. g.ive the numerical results for diffe-rent values of the parimeter CL

ItT

_CD The difference to those obtained by

the analogue method is m.ich smaller than 1%.

3.2. esults for different load distributions

The figure 3.5. represents the shape of a non lifting foil at an adirnensional depth of a/s = 0,25. Shapes of non lifting sections for other submersions /s may be obtained from 3.5. by a simple change of scale.

The figures 3.7. to 3.10 show the function '/J for the submer-sionO'/s equal to 0,25, 0,50, 0,75, 1,0. These results correspond

to the following four load distributions : uniform load, Tulin Buckhart, ('a', (z)), three term law (B) and finally the five term law (c) (2), figure 3.6. It is a simple matter to combine these results by using the method of the paragraph 2.3. As an example we show in the figures 3.11, 3.12 some

corn-17.

binations for /s = 1.0 and

/s.= 025.

According to the criterions of selection for the optimal load set forward in the paragraph 2.5. it results that the uniform load and the Tulin-BUckhart law are the best compriinises. The load distribution (B) and (C), although they lead to a somewhat higher lift-drag ratio.., show as indicates. figure 3.11 variations in thickness and incidence which necessary present a lower section modulus.

We did not insist longer' upon the analyseof these four distri-butions as in general we agree with the conclusions of Auslaender presented in (2).

In order to show the possibilities of the proposed method we have calculated three sections having a linear load distribution. The foil shapes obtained that presents the same maximum value of

_C11/'lJ

= 3.0 are platted in the figure 3.13 together with the three load distributions.

The figure 3.14 reproduces the values of Z for the three pro-files. Taking into account the limites for Z stated in 2.52 we notice that the three profiles present approximativevly the same limites for

Q.

We

con-sider therefore the load distribution E to be optimal, as it yields a profile with a sectioninferio± of 6% to 12% to those for the distributions D and F.

Therefore we are led back to the results obtained by the much simpler method of the paragraph 2.5.3.

3.2.1. Finally as a practical application we present in figure 3.1.5. a wing section for a tulin-Buckhart distribution at a adimensional submersion /s =

1,0 for the same CL as Auslaender used in (2). The obtained results compose

very satisfactorily to those of Auslaender.

3.3. Conclusion

18.

The analogue method makes it possible.. to analyse and compare in a minimum df time any load distribution, The foil shapes are directly

ob-tained by the experiment, and the calculation of the optimum although it takes most of the time of the analysis leads relatively Last to the results. For

example, the experiments and calculations corresponding to the three linear load distributions (DE and F) have been executed by one person in two days. This time could still be reduced if elaborate means for solving the equations for the shape and modulus of the profiles were available. But, even more im-portant seem to us the versability of the analogue method which, once proved their accuracy by numerical comparisons can be used for solving more complex problems, almostcornplicatedby numerical methodes. We illustrate this aspect by calculating a supercavitating hydrofoil at zero cavitation number which does not develop drag.

4. Not resisting supercavitating hydrofoil at zero cavitation number

We have just seen that the supercavitating foil at zero cavita-tions number with a cavity starting at the leading edge develops a drag due to the created jet, This jet-drag may be avoided if we f ix the beginning of the ca-vity at the trailing edge of the upper surface. This can be achieved by imposing a negative pressure on the upper surface of the foil. The analogue methode is very appropriate to solve this problem. This type of profiles is known as the base-vented-foils, which to our knowledge have not yet been studied theoreti-cally If they are near a free surface.

4.1. Boundary conditions

The figure 4.1 .a gives an idea of the physical flow pattern. The houndary conditions are identical to those given in figure 1.1. with the excep-tion however that at the upper surface we assure now

p(p0'

The boundary condi-tions which are to satisfy are indicated in figure 4.16.

4.1.1. At the free surface and on the boundaries of the cavity we have accor-ding to the Bernoulli Theorem

dg(e)

'

_d(X\

51

0.1

02

0.3

04

05

Fig. 3_2

=0.2569

Fig. 3_4

O=A

free

surface

upper surface

lower surface

_O.5414

S

CL

I

S '0

2A

1'IL

20

10

AECp

CL

te pressu re

01

02

A (Tulin-

Burkart)

/

03

I

04

05

B (three terms

Fig. 3_6

/

C(Five termS

/

_I

I

/

/

06

07

08

09

XIS

60

V MD

90

..P ,V\O1

aLL 1(1 SOa11

-.

L0

0

VU

b0

I0-0

VU-0

VU I

-

--8 M07

-

I

-

I Lb

90

(10\gAs

\122A)

Li

V.

sp=v

.

4

60

90

a.p MOl

.

tL

Os

I_'3\!A

A)

-0

L0

I0-0

v0

0-0

I'0-0

MO

:.__

_

i I

3

-- --r

9 M0i

0

- -.-.-iiii__

YMO1

- a

V0

ED

90

10

- - - -____

6'O.

9O

a3 MD

S S a

SLbOT

.

f,z

tILT

0

I'0

tO

vo-oo

Ii

vo-0

'0-0

I'0

MO

--_I

9

VM07

----

-

-Lb

"0

b0

I'0

90

60

13

6V

--90

90

80

14

118 0,6 0.4 112

sV

Fig 3_12

I

0.6

CL5

017

CL

2

CL4

ió3

A

/

'I

_/

/

/

/

/

4

1/

/

/

/

,

,,

z

01

02

0.3

04

c3

0.6

Fig. &.14

0.9

'1D

/

/

/

/

1,2

_CL

0

-'The circulation over x is given

by\f

_(u+

- u)

x, hence 19.

u=0

or

=0

4.1.2. On the upper surface of the section

+ _ul 100

C

- = h

£

₍₎

and on the lower

C

=

cik

From these we deduce the well known expression 100 s

=

4.1.3. On B, B. ,

according to the expression for the jet-drag of superca-vitating resisting foils

vc

CD =

-O

V

0 S

The value of CD will therefore be zero when the perturbation velocity v in the jet (flow between upper cavity boundary and free surface) is zero, i.e. we have v ₌ 41 = 0 on B B

From this we draw the conclusion that the stream lines down stream of' the foil become horizontal.

where

/c_

c+

\CL

CL 11 _C..

_C+

z r=

-

_L. S _{CL CL}

?C

C+

= -\CL CL ,1 C =

d(--)

0 20.

As the load distribution C /CL remains constant at aiiy point of the profile we may vary the ratio

c+/%-

by increasing and contracting the cp+/CL and

cP_/cL laws. This process leads to a family of sections presenting at any point of the section the same total load.

The distribution C+/CL has a minimum value which may be used to define a (CLt)max, where (TI P0 - is the cavitation number of

1/2

r

the physical flow. This "efficiencyt' ratio given by

1

'JmaX

(2J

\ CL JT

lends itself to determine the values of CL which assure a small risk of ca-vitation.

Hence, the ratio c +/c - is a limit for the condition that the

p p

upper and lower surface must not intersect and it may therefore help to make a first choice of an optimum load distribution.

4.2.2. Mechanical criterion

Different load distribution may satisfy the above mentioned condition. To make a further selection we plot the variation 02 the section

modulus (c/Ti)

over the lift coefficient CL. The best load

distribu-tion is then determined by the maximum Z situated between the limits

2 x io< Z(12 x

4.3. Arialoque set-up

The analogue set-up is the same as the one employed for the simulation of the £unction'412 in 2.2.2. (figure 2..lb) with the only diffe-rence that the electrodes representing the upper surface are now connected to large resistances.

5. Results arid conclusion

5.1. Results of the experiments

We have calculated the shape of non resisting Polls for three different load distributions (figure 5.1.) and for two different adimensional submersions 6'/s = 0,5 and 1,0.

For the two chosen submersions, but no doubt also for any other, the constant load reveals to fulfil test the conditions 4.21 with the values

(CL/.)max:3.33 and 2.85 for

S/s

= 0.5 and 1.0. respectively. It follows the distribution H with and 2.5. and finally the distribution I with (C1/j--.)max = 2.85 and 2,22 for /s = 0.5 and 1.0. In the following

table we give an away of the corresponding results :

*

V0

Po

F'

A P>Po

F

I5

(a)

n

₎

B

At

_'F

C

41' jpiXr

_fl=

_'-)

(b)

P = Po a

P P0

_x

B

/ CL Values of (. max 0,5 3.33 : 3.00 : 2.85 22.

In the figures 5.2. to 5.7, we reproduce the shape of hydrofoils having different

C+/C

ratios and two different submersions.

In the figure 5.8. we have plotted the van ations of Z and

= s , where s is the cross section of the foil, against the lift

coef-ficient CL fr different load distributions, but the sane values for (CL/Ti)ma. = 2.22. We find that for the above mentioned limits for Z and

for the usual values of CL (o.124<CL<0.290) the constant load distribution is superior to all others.

For the values of 0.29O CLKO.384, the distribution H Satis-fies best the condition 4.22 while the distribution I does not present much

advantages except in the case of very large CL.

The figure 5.9. shows three foils without d submersions and constant load distributions over the upper The profile which corresponds to

6/s =

c has been taken

Fabula, mentioned above.

.5.2. Conclusion

We have found that for the usual values of lift the constant load distribution presents the most advantages in so far as the hydrodynantic

rag, for different surface (C+/CL = 1). from the paper of

:1

2.85 2.50 2.22

23.

or mechanical view point are concerned. The only inconvenience of this distri-bution is that the Joukowsky condition at the trailing edge is theoretically not satisfied. But, as Fabula (9) has already mentioned, this will in reallity not be very important for in a real flow and with the technical possible ma-chining of the trailing edge there will only be slight differences to the theoretical law.

LS 6U

0

H

I

V

13

so-0

5'O

I,

d3

0.4

0.2

0

0

0,

0.

fl2

SCI.

A

0

0.1

LciwG

08

S

(CL

_=2.85

\OIJmax

2,5

nax

2L)m=2,22

ax

.6

4

(CL'

2

0,6

0,4

0.4

03

06

(CL

₌₃₃₃

max

0,4

05

06

0.7

08

0,9

I

X/5

0,4

0,2

0,2

03

01

02

03

-n

th

0

0

0.

A

x's

112

SCL

'S

LawH

_;

4.0.5

285

\Pi)rnax

(CL

_=2,5

\0' /max

=222

max

max

6-4

(CL'

"0-i'

.2

0,4

0.2

0

₀₁

0.2

0,3

04

05

06

0,7

0,8

09

1

0,4

0.2

0

0,1

0,2

0,3

.04

05

06

07

08

09

I

0.4

0,2

0.4 0,1

0,2

03

04

0:5

0,6

07

0.8

09

1

0

0.2

S

LawI

U,'

UI

(CL'\

_2.85

\aI

X'S

:2.5

ax

2.22

max

max

(;

0,4

T

(cC

0,6

0,4

0,2

O091

A

1]2

_SQ

U,

0,4

0.2

0.4

0,2

0,2

0

0,4 0,1

0

02

0,3

0,1

0,2

03

Lawc

04

04

05

0.6

0.5

06

I

07.

0,8

0.9

(CL'

_2.85

\)max

(CL\

_2,5.

)max

.2,22

max

I /rnc,x

04

)2

(CL

(C1

07

08

0.9

1

1)2

3CL

A

0.4

0.2

0.4

(CL\

₌₂

¼j

I

(CL

₂₂₂

S

5

01

0,2

03

0,4

05

0.6

₀₇

08

0.9

0.4

0.2

0.4

0,2

0.6

0.4

0.2

X/

0

01

02

03

04

01

0.2

0,3

04

05

0,6

07

08

0,9

(CL

"a.'

(CL'S

_=2,22

knax

(CL'\

₂

\a. )mcix

=1,67

max

112

5L

A

_Law

_I

09

I

07

0.8

05

06

07

0,8

09

0.5

06

0.2

0.3 0,1

S

0,2

0,1

A

I

I

/

I

I

/

/

H,

/

/

/

-4

L 2,22

0l

I

I

REFERENCES

Chapter I

(i) Frurnan D. "Analog studies of hydrofoils near' the free surface" Final report US Navy Contract n° 62558.3645

(2) Luu Thoai Sum "Trace de profils supercavitantS répondant a une charge locale imposée"

J0urnal de Mécanique, Vol 1.1 n° 1, March 1963

Ghapter II

(i) Thlin N.P., and Burkhart LP. "Linearized theory for 2 1ow about liftings foils at zero cavitation number"

Rep. C 638 DTMB Navy Dept, Feb. 1959

Aus].aender J. "The linearized theory for supercavitating hydrofoils operating at high speeds near a free surface"

J S Oct. 1962, vol. 6 no 2

Luu T.S. "Contribution a la théorie linéarisée des effets des jets minces et de cavitation .."

Public. Sciences et Techniques du Ministère de l'Air n° 389, '1962

Luu TS. "Trace de profiles supercavitants répondant a une charge locale imposée"

Frtzman D. "Analog studies of hydrofoils near the free surface". Final report US Navy contract n°

62558-3645

TimosherLko, "Strength of materials" Vol n° 1

Boudigues "Dimensiozinement et construction des turbo-machines"

ENSA 1958

Fabula i..G. "Linearized theory of vented hydrofoils"

Navweps report,

1637

NOTS TP 2650 , March

7,

1961.