Analog calculation of the hydrodynamic characteristics of an infinite span hydrofoil submerged near the free surface

BUREAU D'ANALYSE

ETDE

RECHERCHE APPLIQUEES

B.A.R.A.

6. RUE LOUIS PASTEUR

BOULOGNE-BILLANCOURT

(SEINE)

Office of U. S. Naval Rozoarch

London

v.

Scheepsuouwkum

Technische Hogeschool

Mt

o6'

--- e,;_

_ v-SC.

/

.1

e.--ANALOG CALCULATION OF THE

HYDRODYNANIC CHARACTERISTICS OF AN INFINITE SPAN HYDROFOIL SUBNERGED

NEAR THE FREE SURFACE.

2 By Daniel FRUNAN

FINAL REPORT

Contract U.S. NAVY n6 62558-4483

ANALOG CALCULATION OF THE

HYDRODYNAHIC CHARACTERISTICS OF AN INFINITE SPAN HYDROFOIL SUBMERGED

NEAR THE FREE SURFACE.

By Daniel FRUMAN FINAL REPORT

Contract U.S. NAVY n° 62558-4483

HYDRODYNAMIC CHARACTERISTICS OF A HYDROFOIL _{SUBMERGED NEAR THE FREE} SURFACE

I. INTRODUCTION

Analytical methods have been developed by various authors with a view to determining the hydrodyriamic characteristics _{of an infinite} span hydrofoil submerged in a heavy fluid, _{near the free sur}

Isay (1) and Nishiyzna (2), in particular, have described two which make it possible, within the framework _{of conventional} to take into account the effects of _{the free surface on the} distribution of a submerged flat plate

; however, they present few

numerical results, so that obtaining further values on the basis of their formules still requi-res a considerable amount of calculation.

The method of analogical representation of. non-rotational flows permits a quick Solution of this problem, _{provided that the hydrofoil} induced disturbances fall within the scope of the linear theory. The problem can be solved by means of a series of operations which include

two phases. Firstly, for given intumescence and hydrofoil shapes, one determines, by means of rheoelectric analogy, _{the intensity} distribu-tion of the hydrofoil attached vortices which _{fulfils Joukowski 's} condition at the trailing edge and admits the free surface as a stream line, without, however, complying with the invariable pressure condition. Secondly, as already indicated in the (3), _{one determines the free}

surface ordinates induced by the previous vortices

; this defines

the intuinescence used in the analogical determination of a new vortex intensity law, etc. _{; until the intumescence ordinates and the vortex} distribution _{Simultaneously converge towards functions representing} the solution to the problem set. Instead of introducing an arbitrary free surface into the first analogical approximation, it is appropriate to calculate that which would be induced by the vortices attached to

face. methods assumptions, pressure

-2-Although the method-' may be applied to any form of mean line, in the first part of our work we have limited our study to that of

the flat plane with incidence. The results yielded in this simple case can be easily analyzed in order to obtain the lift of a mean

line of any shape by extending the characteristics of inverse flows in an indefinite fluid to the case of a flow limited by a free surface.

1.1. Boundary Conditions (figure i.i)

Let us consider a flow around a hydrofoil with an ordinate

y:i

)

. The flow is represented by the disturbance flow function 1 , harmonic in )( y and defined by the following

boundary conditions

On the hydrofoil section

Lii'

i)

(i.i)

Or, if one integrates between the leading edge and abscissa

(1.2)

taking

On the free surface, function 'i-' cn be written as follows,

in the analogical approximation of the p order

i'!4fl

:

i;

ci

('.3)

-'

-where represents the value of the circulation in the analogical approximation of the p order, and cs (

)-

_I

'() S

the vortex

t 41

distribution in the previous approximation. I

I-

4

At thetrailing edge, Joulcowski's condition has to be fulfilled

-I

-t

For '-? - O and )/_

-l

1.1.1. Assuming the Froude number is 'n$inite, conditions (a), (c) and (d) are identical, but the condition on the free surface is expressed by

-3-1.2. Analogical Representation

The representation of the boundaries in the tank is identical to that described in paragraph 3.4 of the Final Report

(us

Navy contract N. 62558 - 2545). Function is identified to the electric potential of the flat tank, the boundary conditions being

imposed by means of electric Set- up which are different accor4ing as the Froude number is finite or infinite.

1.2.1. Case of an Infinite Froude Number

The electrical set-up is illustrated by figure 1.2. Electrodes E' E' of the plate are supplied by the electric potentials proportional

to y by means of a 1000 point dividing bridge. The tank wall

representing the free surface is insulating in order to meet the zero normal derivative condition. The overall electric intensity produced by the plate electrodes is collected at the infinitum point by means of a conducting half-cylinder connected with a potentiometer P in

shunt with the secondary circuit of a transformer

T

of ratio 1/3, having a common primary-secondary, terminal Joukowski 's condition is adjusted by varying the infinite singularity potential until the

electric intensities i produced by the plate electrodes and measured by a zero resistance amperemeter define a decreasing monotonous

function exactly reduced to zero at the trailing edge (figure 1.3). On the basis of expression (1.2), coefficient X. _{is obtained by}

( 'ii

---

"

(1.4)

where

'.'()

andó(',yresPectiVelY represent the electric

potential as measured between two electrodes of the hydrofoil-plate, and the corresponding ordinate difference.

The circulation around the hydrofoil C ₎

will

be yielded

where ₁ represents the intensity, as measured on the

sinularity. This intensity

I

is of course eual to the algebraic sum of the intensities produced by the hydrofoil electrodes. The lift

coefficient

< _{coefficient r1tlo is then obtained}

( L

-

.r JL(

)

vc(( cI't'

and function5is yielded by

'j

r

It will be used in the problem developed in the following paragraph

to calculate the first approximationto the free surface.

1.2.2. Case of a Finite Froude Number - Approximation of the p order

In the preceding case, we fixed the value of V0 _S

determined the circu1ation for Joukowskits condition to be met. For a finite Froude number, the cycle of analogical adjustments is simplified if, instead of imposing the value of we fix the value of The analogical set-up resulting therefrom is that of figure 1.4. Electrodes E, E .... which cover the taik wall, simulating the free surface, are brought to potentials _g , short of 2actor by means of a 1000 point dividing bridge p . The pótentia]. scale

is adjustable at will by varying a resistance connected in series with the divider. Ordinates proportional to those

'( _.)

of the proposed hyth'ofoil are produced at the centres of the plate electrodes by suitable electric potentials, obtained on a second divider with an independent current supply ; the potential sca].e

O(

is, there also, adjustable at will by means of a resistance ₁₉ The overall intensity produced by the plate electrodes is zero as long as the secondary circuit of the transformer

T

does not include an auxiliary feeding system : on the contrary, if one introduces an intensity I derived in a suitable way from the primary supply (for instance, by means of a current generator ( allowing an adjustment Of I independently of the potential scale defined by _), this current is entierely produced by the plate electrodes and

-5

The intensity I being iinFosed, the value of 1-', is determined by (1.5) and can be produced at the terminals of dividerP _{by means}

Of4 .

The potential scale of divider is then modified until Joukowski's condition at the trailing edge is fulfilled. By means of formula (1.7) it is possible to obtain the new intensity distri-. bution of attached vortices, to be introduced into the calculation

of the free surface in the next approximation.

1.2.3. Calculation of Hydrodynamic Characteristics

The measurements of the electric values - intensities

on the plate electrodes, overall intensity I., and potential difference on the electrodes representing the hydrofoil, enable us to determine the values of the following hydrodynaxnic characteristics

Lift coefficient

-L-L

2L1L)

_ciJ()

I4oment coefficient whith respect to the leading edge

Cr

or, by partial integrations :

Zi'i d

---0 V5(

with

as circulatin disibution

Vortex Distribution

which can be expressed in relation tov

.' ()

P.

f p

Circulation

Distribution6'

or

V0$(

-

ecry()

-r

1.3. Analogical Calculations and Results

Rheoe].ectric calculations have been carried out for three values of the reduced depth,

_h

= 0.25, 0.50 and 1.00 ; and Froude

numbers included between 0.00 and 6.00. The selected hydrofoil was

a flat p1ate:

, so that the coefficient Qç may represent the

inci-dence of the flat plate in relation

toV0

directio

Figure I.5 gives the variation of the lift

coefficient of the flat plate in an indefinite fluid, for various values of the Froude number and of the reduced depth.Divergences between our own results and Isay's results plotted in the same figure, can be ascribe to the defectiveness of rheoelectric simulation, by discreet values of the boundary conditions.

Figure. 1.6 shows the values of the moment coefficient

C-

, related

to

134

C , the moment coefficient of the flat plate in

an indefinite fluid..

Figure 11.5 requires comments, in view of the unusual aspect of the lift variation with the Froude number. For high Froude numbers, even F = __

the lift of a submerged hydrofoil always remains lower than that

obtained in. an indefinite fluid. This difference becomes more pronounced when the reduced depth decreases to reach the borderline case when the

depth equals zero, and when the lift is theoretically half of the lift existing in an indefinite fluid.

When, for a given depth, the Froude number decreases, the lift continues to decrease down to a critical value of the Froude number,

and then starts increasing. This value of the Froude number seems to indicate, simultaneously, a limit to the validity of the linear theory. As a matter of fact, it has already been observed (4) (5) that for low Froude numbers - of the unity order -a hydraulic jump appears on the upper surface of the hydrofoil and thereby the small disturbance assumptions previously admitted seem to lose their validity.

7

This relative lift increase for low Froude numbers is justified by the considerable variation of the vortex distribution curves -figure 1.7 to 1.9

Figures I.TO to 1.12 give the values of %f

(

₎

/%/

which

will be used later on in inverse flow applications.

1.4 Law of Inverse Flow in a Flow Limited by a Free Surface

In a report on hydrodynamic boats, Tulin (6) mentions the work carried out by Ch. F. Chen who resorts to the properties of inverse flows to determine the uniform translation lift of an infinite span flat plate equipped with a flap, submerged near the free surface of

an incompressible fluid. Nevertheless, this application is limited to the case of an infinite Froude number, that is to say to high forward speeds, and, as far as we know, it has never been generalized to finite Froude number flows, It is however possible to prove that the basic relation between direct and inverse flows in an indefinite fluid (7) remains valid in the case when the fluid is limited by a free surface, even if, in the latter, waves are produced downstream of the hydrofoil, as is the case for finite Froude numbers.

In the most general case of a three-dimension flow, let us consider

a submerged hydroplane, fully affected by speed \/ of the direct flow and _- of the inverse flow. Let us use starshaped index to

characterize the function related to the inverse flow, and, for the sake

of convenience, let us consider a trihedronX)Z, y connected with the

hydrofoil, Funôtions

U,

the disturbance acceleration potential of the direct flow _; and

f*the

disturbance speed potential of the inverse flow, are defined in the usual linearization assumptions around the uniform

moton of

\J.

speed according to Ox by the following boundary

conditions. On projection

₅

of wingy : on plane

and \/

P,'

_; on projection )of

the free surface on plane 1 b

V 2f

tL'y

and

on both sides )'

t C

of projection £ of the direct flow wake

i..

.Land F''.

fk

; and on projection of the inverse flow

wake

-r" (zj

; for :

and ' ; for x . and

7 -

u

o

and grad

Green's form.ala, connecting functions U and , , harmonic in the

whole space, is written as follows :

/

'

The contribution of is zero, as well as that of s established by an integration by parts ; then, there remains

tu)

;L

Ic

V5.L

\

If we take into account the slipping conditions along the hydro-foil, and if we integrate by parts, we obtain

)j)dc-fjp (:) d

which is identical to the relation existing between a direct

_{and an}

inverse flow in indefinite fluid, In the two-dimension case, this relation becomes

which

can

also be written as follows, if we consider that

fr

()::._j

or else

(1.8)

C

A2

'-'4

*1

0

_V0.

if we integrate by parts, we obtain

For a flat plate with a flap

/j

(r)

and

(!!

_{() -}

for(

_{and f'"(}

),_

; the lift

coeffici'ent will be :

V0S

(i.io)

Figure 1.13 gives,tfor h = 0.50, the values of

C /T

for various length of flap C, and various Froude numbers, Figure 1.14 gives the varition of CL as a function of the Froude number and h 0.50 for a flat plate with a flap in the case of C = O2 and 0.4.

It can be noted that the lift variations are approximately of the same order as those obtained for the flat plate ( = 1.00). 02 course, on the basis of figures 1.10 and 1.12, one can obtain the lift coefficient for any shape of the mean line.

1.5. Application to a NACA.65rnean line

The NACA 65 mean line is given by

where

3'max is the camber-chord-ratio. By putting the first and second

derivatives of (i.ii) in ( 1.9) we obtain after some operations

4CL

+4tç)

_(1.12)

where CLIS the lift coefficient of the flat plate for. = I at the same immersion-chord ratio and Froude number.

10

-into (1,12) the values of the analog calculation. for a flat plate. In the sa1ne figure we plot also the values of the lift coefficient for

a flat platewith a 20% chord flap. We can see that the effect of the

free surface is less important in the case of the NACAS5 than in the case of the flat plate whith flap.

1.6. Analog calculation of the velocities on the upper and lower surface

of the hydrofoil

-The method given in this paper can be used to determine the ve-locity distributions on the upper and lower surfaces of the hydrofoil they are proportional to the pressure distribution. The only differenc between the analog Set-up given in the figures 1.2., 1.3. and this analog representation is that the electrodes of the lower and upper surf ace of the foil are independant.

The figures 1.16 and 1.17 give the values of the velocity distri-bution obtained for a flat plate

(ö

= 0.20, h = 0.25) and the

REFERENCES

(i) A. ISAY, Ingénieur, archiv.

27, 1960,

p.295-343

T. NISHIYANA, A.S. N.E., Journal,

1958, p. 559-567

U.S. NAVY, contract n°

_62558-2545

- (4) LAITONE, E.V. "Limiting pressure on hydrofoils at small submergence depth"

Journal of Applied Physics, vol.

25 no 5

₁₉₅₄

PARXIN

B.R., PERRY B., W U. T.V.

"Pressure distribution on a hydrofoil running near the water surface" Journal of Applied physics, vol. 27 no

3, 1956

TULIN M.P.

f'The hydrodynanic of high speed hydrofoil craft" Third symposium on naval hydrodynamic

La Haye 1960

VINCENT J.C.

"L'écoulement inverse en analogie rhéoélectrique" 0.N.E.P.A. (note intérieure) Avril

1954.

'l'n=O (Fco)

Ap

order approximation

4(,F)

A

4Vosa fj)

grad 4'o

y .-J

Fig. 1.1

no correc1 adjusement

Approximahon

For a correcI representahon of )()

near the ftailing edge.

Fig. 1.4

correct ddjustemen)

last electrode

1.0-ACL

;--

_VO

h

Ii S

_v

Fig. 1.5

o analog calculahon

Isays values

5

I

E

s'o

sc

1..

9 I

&

D)L_

OA

s.o

O'L

0

A

Voa

h0,25

F_V0

gh

0.25

flg. 1.7

0.75

V I

Voa

h=O.5

gh

F

₅₆₄

=3,99

282

= 1.785

= 1,130

0,25

₀₅

Fg.I.8

Q75

Voa

I

i=ioo

F_V0

gh

F =5.64

= 3,99

2,82

=1,785

= 1.130

-,%. __%

-0.75

05

Fig. 1.9

0.25

L)

VO sa

-2.0

0,25

h=O.25

05

Fig. 1.10

F=co

= 5.64

=3.99

= 2.82

=1,785

0.75

i

0.5

2

Vos a

/

,1

,,

-.

/t

/

//

_y.

0.25

05

Fig. I .11

-=1130

0.75

.3

F

=

=

=

5.64

3.99

2.82

1.785

Vo a

fci.o

Fig. 1.12

F =5,64

= 3.99

= 2.82

= 1.785

= 1,130

0125

0,75

I

h

01

0.2

03

Fig. 1.13

Fco

=5,64

=3,99

2.82

= 1.78

= 1.130 0.5

ACL

CL

1,0

0,5

S

-'

=0.2

Q3,4896t

C

\Igh

=0,4

CLOD =4,5986r

£CL

CLa,

-1.0

-0.5

C

=2ita

for

flar

pIaIe

Lco

CLa,

41tS7max

For NACA 65

F

plal whfth flap

O,20

F]g.I .15

NACA 65

6

F

-20

F1a1

plate whilh Flap

i=O,25

E=O.20

S

Fig.I.16

U

NACA mean line

h025

F_V0

0D

2.82

Fig. 1.17