Super cavitating flow past bodies with finite leading edge thichkness

NAVY DEPARTMENT

THE DAVID W. TAYLOR MODEL BASIN

WASHINGTON 7, D.C.

SUPER-CAVITATING FLOW PAST BODIES WITH

FINITE LEADING EDGE THICKNESS

by

J.. N. Newman

RESEARCH AND 'EVELOPMENT REPORT

SEPTEMBER 1956 Report 1681

Lb

y. Scheepsbouwkúnde

Thhsh Hccdo

Deift

SUPER-CAVITATING FLOW PAST BODIES WITh

FINITE LEADING EDGE THICKIJESS

bT

J0 N0 Newman

TABLE OF CONTENTS..

Page

Introduction 1

T,heory . 1.

Source Distribution Solution .. 2

Deterrninatiòn of' Lift anOEDrag 6

Lift and Drag at Zero Cavitation Number :io

Appilcation'to Foils with Finite-Leading

Edge Thickness 12

Comparison of.Reults With Known Data ₁₃

Discussion of the Resúlts 16

Recommendations . 17

References . 18

SUMMARY

STATEMENT OF THE PROBLEM

The prob.em tO be considered is the effect of increasing the leading edge thickness _{of a} super-caviating foil bèyond the cavity thickness, such that a portion of the back _{near the leading edge.} becomes wetted andcav1tatjon does not commence until some point downstream from the leading edge.

FINDIÑGS .

The investigation has resulted _{in an approximate.} solution for the lift and drag on a foil with a finite

leading edge0 _{The solution shows that there is a very}

large decrease in

the

_{lift-drag ratio due to the wetted} portion of the back, and as such it seems advisable whenever possible to avoid using a foil with a thick

leading edge. _{It was not determined however, at what}

angle of attack this type _{of flow, would occur for a} given foil with some arbitrary _{leading edge. thickness..} RECOMMENDATIONS ...

It is recommended that the _{above-mentioned type of}

flow be avoided by.keeping _{the thickness of the leading} edge less than the cavity _{thickness0 It. is also}

uggested that further research be devoted to determining _the

critical angle. ofattack at which the back will be

par-tially wetted, and to determine an exact solution by a

CO1O;màpping.tchn1quee

NOTATION

b., A constant, undefined in tems of physical parámeters CD Drag coefficient =

CL

A

Lift Coefficient

Drag

D1 Drag component, due t& the back

Drag component due to the

face

L Lift

L " Lift coiponent due to thé báck':

L2

Lift

component due to the

face

CaVity length measured from the leading

edge

m ' .StÑngth of a source distribution

p LOCàI static pressure

CavIty pressure

PO Pres3ure' on the body

Static pressure of the stream at infinity

R. ., ''Radiu5 öf

the

arc of the' 'Circular Arc SectiOn,

Re ' Denotes "the real part of"

r ' s Ratio'.. otupper to lower chord lengths:; r

'.2

Chord length

Chord length of' the back (upper

suiface) measured

to, the point of cavitation' inception

- Chord length of the face (lower surfáce)

t ' Dummy variable

The x.component of the velocity on the cavity wall

The x-component of the perturbation velocity Tie x-component of the peiturbatiOn velocity

wM.ch is induced by the cavity source dietrbutioñ. u The i-component of' the perturbation velocity which

.18 induced by the body soirce distribution V . The velocity of the fluid at any point in

the flow field

The local perturbation velocÍty... . - u..

y The y-coinponent of the perturbiOñ e1ocity The y-component of the perturbation velocity on, the cavity .

The y-compoùent of.the perturbation velocity.,

.onthebod7

.Aspacé cóordinate parallel toU A dummy variablò' . :.

T. Aspace. coordinate, orthogonal to the x-directIon

:

Y-or.dináte of the.f oil surface !-Ordlnate of the back

t. Y-ordinate of the face.

Z ..

duy variable

The slope of the back When it is constant (iip to s

The slope of the face when it is cøn5tant

:F1uld density Cavitation number, R0 -P0

=ptJ

A dummy variable VelocitY potential L. dummy variable. .

ABSTRACT

An approximate method is develóped for determining the cöefficients of lift and drag.ror a foil whO5e face

1 completely vetted and whose back Is wetted up o 'a

certaIn point 'and 9avltating. downstream of that point,'

at zero àvItat1On flnber. 'The "so].ution obtaInedi$

compared Withexc Qryfor flow past á flat plate,

with exaej theQy

for 1lo

st a non-symmetrica]. wedge,

I NTRODUCT1 QN

The effects of cavitating flow past flat plates and thin,foilshve been the subject of recent research by Thlin _{and Wu. Both developed linearized theories} for the flow and solved the problems by _{conformal mapping.} In this way they determined the lift and drag of _the cavi-tating hydrofoil as a function of the drag and moment of

the equivalent airfòil,

The limitation that the thickness of such _{a foil}

be less. than the cavity thickness _{presents practical}

difficulties near the leading edge where the cavity approaches zero thickness and leads to vibration _and buckling of the foil0 It was, therefore, considered desirable to investigate the effects of increasing _the leading edge thickness to some value greater than the

cavity thickness. _{For such a section the theory based} on thin foils will' not hold since the back _{Is no longer} fully cavitating. _{Consequently, an attempt has been}

made to solve the problem using potential flow theory in _{manner similar to that used by Tulin to}

Investi-'

gate the flow past a symmetrical wedge30 THEORY

Two methods have been used _{to develop approximate}

solutions for cavitating flow. _{The most popular is the}

determination o _{a mathematical model for an equivalent} flow. _Blrkhoff

and Gilbarg and Rock5 give accoünts of early work along these lines to determine the flow past

fla't plates perpendicular to the stream. _{More recently}

Tulin and Wu developed linearized solutions for flow pst

thin lifting surfaces. _{The other method is the application}

of linearized potential flow theory _{to determine a source} distribution and from this to determine the velocityatd pressure distribution on th _{surface of the bodya This} is the method used by Thilni to determine the cavity shape and drag of _{a symmetrical wedge.}

Theequivalent airfoil was not _{used In the preset} problem bcause of the àdditional singularity at the

leading edge which is not present in the thin foil problem. Çonsequently, linearized potential flow theory was used in

a maìiner similar to that of Tu11n30

1References are listed on Page 18

-1-The presence of a finite leading _{edge thickness prevents}

the inception of back cavitation _{at the leading edge, Back} cavitation does not commence until _{some point}

_Sj

and cÒntinues

from' there to the _{xtremity of the cavity at some point L.} For sufficiently low cavitation _{numbers the cavity length will.} be large relativè to the idwer _{chord length, a condition}

necessary for the development of the theory0

It is assumed that no cavitat'ion occurs on the pressure side until the'terrnination of the chord length,. _{2, and the} cavityextend.s out to (Figure 1) _{It Is a10 assumed that}

the slopes Of. the upper and lower faces,dY _{ard .dy2}

dx

respectively9 and the chord lengths _{s. and 2 are such that} no cavitation occurs.on either the _{face or the back In the}

intervals O x s _{and O x s,, For this to be true}

the Stagnation

po1n must

_{be at th leading edge0}

The problem then resolves _{to that of finding}

_{a soure}

distribution which will produce _{this flow, and to déternzine} from it the pressure distribution

_along

_{the wetted pOt1On} of the face aridj back. _{:,The pressure on the back}

beyöfld the

point S1

(I.e0

_{the portiOn of the back}

lying Insidé the

cavity) is assumed to.be at the cavItypressure,

_CônSeuet1

the pressure at any poInt on the surface of the _{foI1wi1i be} IQiown an.d...Dy suitaile _{lhtegratiorj the lift nd drag caribè}

determined. i.:

SOURCE DISTRIBUTION SOLUTION BOUNDARY CONDITIONS

On. the wetted portions _{of the face and back}

the sreamÏIne slope is equal to _{the slope of the body:}

-()

_t)-.«Mfcf

(1) L/.

_..

_{oc.. J} v(Y, y)f

_{j -}

_{í !2LVc}

o()-___cjy)r ex.. U..'u.(cjy)

_U

L Ve.. Li

.öhthecavity'wfi the ròsiì.ipand using

BérnouflPs equation.this determines the avitation number as a Í'tthetion of the cavity wall velocity and the stream velocity:

2u.(x3y)

_Ö(vYc

(Jo0 Uøo

(&') o(')

where Equation (3) holds over both surfaces. When linearized1thése three equations give:

v(xOt)

_o<x<s,

lJ.

y(O)

2u(XO+) _¿ 1)00

2at(iO-)

u00

The remaining bounda7 condition is that at the two

points where cavitation commences the flow must-be.. smoöth.

Consequently, the slope of the cavity must be coñtinuou with the slope of the body0 This dondition is satisfied by

eliminating solutions for the source distribution which g.o to infinity at these poinis0

SOLUTION

The following solution is based on the simplified assìimp1on that.,1th the s'agna-t1on point at the leading

edge, the

1o'

ô the face i& indépndent of the geoetry of the baák. Conse.que.rtly, theflow on each surface is. assumedtobetbe sainé-tas that oria symmètrical body ofthè

same slope and chord length as the surface in question. The lift is then assumed to bé the difference between the lifts of the two surfaces considéred separately, and the drag is

assumed to be the sum of the two drags. This is a crue assumption since it assumés that the stagnation streamline will be horizontal as it approaches the leading edge, and

this will hot be the case unless the foil Is symmetrical, In which case of course therewIll be ño lift. _However, this procedure should give an approximate solütlon and

was followed0

-.3.'-(3)

(la)

Since the

two

surfaces can bé treated sepárately,'. the

d.evelopmnt is based on a single surface of slope dy0/dx

and

chord

iength s, and which may be applied to either the

face or the back0

4 distribution of sources of s°trength m(x) along the

x-axis

'in the interval O<x gives the following:

and thus: if:

g(x,y) J

d(

In w ¿Ji.' 2vr o L. r8 tn6rUx-w) dx'

2irI

vfr) y)

.L

. ..L

f

1

nfr)y dx'

y

2cr]

_o

At points o the

xaxis

the perturbation velôcitiel u(x,O)and v(x,Q will be:

L

u(xO) 2ct11 _,-x'

o

o)

tm()

The' boundary

conditions on the x-ax.1 will 'be satste4

t (x'x"

j)(-)('

_s where :4v(x_xI)2

?

ci._o )i.-X'

-

-'I2Ù.dx'

iCOr4)

= -21h y's)

'

: '(12)

dx

nd thüs

I[2t/'fr)d' 'Sfai(xIM'

¿/ç.

2irJ'

x-X' 2ctJ '

2

"o)».

Using the thin airfoil thory inversion formula6 gives m(x) from Equation (il):

I

1Tqxç)(ex)1f

(xix)

=

-2s)(y

fSç/

jd1+]

To satisfy the juncture conditin the term in brackets must vanish at xs: therefore: £

Q=b1rUof

s

(J_i))'+

k

[woo2

s.

-!fI'

x)fr Lico

EvaluatIng the integrals as shown

2V

J14-K J0 49é I' ---

(x-i)

For.the case of zero cavitation number,

0=

_{Q,.0 =}

and,,00

.

(13a)

rn(4

_Ç13b)

2UCfsfo

dt

DETERMINATION OF LIFT MID DRAG

DRAG

Assuming the foil to be deeply submerged in an idea].

flow (i.e., neglecting frictional and wavéresitance) the drag on one surface will be given by the integral:

p.c

J

(po r)

X.? dt p[U+ v]Ji - St.

(11i-)

u:

dt

o

dt

o

which may also be written:

- i

t=

r[ut]Í

í()

¼t)

_pj

o

D

and after linearization this becomes:

O t [u.(-t Y) - u ( c)l '!Z cl 4.

Jo,

-Idt

but

u(t,y0

=

u0_0(,t,y0) +

Where u_0(t,y0) Is the x, component of. that part of the: dis

turbánce velocity òn the body

which Is

Induced by the c8vity source distribution and

1_0(t,y0)

is the x COmponent of that part of the disturbance velocity on the body which

is .induce4

by the cavity source distribution. . . ..

sInce

..

J0 (yò)°

c.i.t

o

the equation for drag becomes:

p =

pVc. cr LJ0 o (s)

- () yo)

dye

(1+c)

The Integral term ir (11t-c) is evaluated as follows:

using equations (7) and (l3a)

()

r,

_2-'-dt

J 2n'

I o

[Ç

i.

+i/if4ic:t

-7-( s)

UsIñg Appendix A the first term of Equatioñ (15) is

evaluated: s

_!1tffd

(J 15d

-iJ ;;

Oy

()

2.wj d

o S

And using Appendix B the second term of Equation (15) is:

2 d 2 L

i If

¿.

\/(-sX

- e) 'J Lo

therefore:

Ie

*>)

)it d*.

dt

The lift is calculated in the same manner as wa

drag. Assuming small axtgles o attack, the lift of

face wL11 be r

In Equation (17) and hereaftér, the. following _convention lsused* ina.flo from left. to right(F1gureJljft

_{up ia}

6sitivé;slopeu

ndio

_d.slópëi

dö'ntand.:toj ti è.r1ght.is .Lpositie

ôrponding4oùthLuuai

angle of attack0 . .

dt

d.

_1a)

And Equation O.c) becomes:

12= p(W

-the drag coefficient, _CD

=

+

Linearizing as in Equation (l+) gives:

L =

PU

I

rand. u(t,y0) = 0_0(t,yo) +

The integration that follows is consideratll simplified by making the second assumption; namely, that:

) J ¿-w' 6-c'

t

=

opvcos 9t1oe

d

Equation (17a) thus becomes:

L

Oc.

ffrUoQs

fo.o (iyJdJ

(ib

and proceeding as for drag:

r'

f

S.

f

f

(od'

1T

-

l.d

-

Yt-

,e-t.J

J7

S8 +

The fi'st

term

of Equation

(18) is evaluated from

And the second terth becomes:

2u

f 5!th,1'.tí1éY

-JI

t)

/

(

o

The e quátion for Lift is then:

L =

_/]

--.10'

3d.

+

__

o

-

2p

_{L1C f}

/4'

(i)

_d-r

ir

J

V'(#-)(s-t)

AtcL:

LIFT AND DRAG AT ZERO CAVITATION _NT314BER

The expressions for lift and drag can be considerably simplifiedby assuming zero cavitationnumber.,

_{Thus the.}

cavity length is infinite and the-ve1ocity on the cavity wall equals the stream velocity _(U

_{U00 ).}

_{Using Equation}

(13b) and proceeding as before, th _{expressions for lift and}

drag become: -

L:ff,0coV'f5droíz

-- _cit

_!'t

dz'e/é

(19)

dt

_{i J,) ir(t-'.)j dr frt)y.}

°s

o (1.74) (18)

°°L(

4 ..j)(

-

yj'J

t-s

where in the last equation the variable has beén changed

from t to t, to conform with _{the other equations0}

Equation (19), likéwise, becomes: 90.

_roL.

!IJ_)/vI '4_i)t.0 1t

D=

rj0

(

And. using the

procedure of Appendllx

L0

dt-fi

Dividing Equations (18c) _{and (19c) by l/2 gives the lift}

ánd drag coefficients: S 9(/oo I I

= - ;rJ

dJ0

(-t-À

t(

1 11

_1Jod..

L

f

Sf øo °! I yo I ,Wt ' ch

L

j_ (°7ff.51

x'-s.

-i)

_)ôI('d.td:

/-s I 1!. JO

G1V

cii]

g+j')jC

J0

dtY;.

d4

.2. (

18b)

(.18c)

Ç19c)

and changirg the order of 'integration,

CL

Cb

(18d)

Equation (18d) can be simplified by .furthêr linearizatiön 'using the seri?s:

Neglecting al]. but the first-power term In

the

series gives:

I- irs) dtL'/S_t

b

Final llnearized l±ft and drag coefficients at zero

'cavIttìóñ number are then:

«z' U'J. 2(s-j+fr(:..tj

Fçr the case of a

the assumption is made sums of the, components sepàxatelr on the face

.

;_ff(id11

i±i'

.2.,i jJ°

d*j

:

(4ki)

*

Ut]

CD,

[S'j2

APPLICATÍON TO FOIÏJ WITH INITE LEADIÑG. DGE THICKNESS AT ZERQ CAVITATION NUM5R

foil .sUòh as.is shown inFIgrG 1,

that the tötal lift and drag are the of thé lift and. drag detèrminé4

and the b8.ck0 ls:

L.L1 + L

0{J?Y

( (-t)S)

+j5

i'

o F. 18e (18f) Ç22) (23)

where the chord length in the denominator of the' coefficients is s _{By considering s2 to be unity the expressions may be}

further simplified without loss of generality0 _Denoting'the ratia between the upper and lower chord lengths

= r we have: = ''

U0

(r

I(

IL0

a:;:

+J''

o

4"'

+ If

ay _d4.

U0 d-tPf'

CONPARISON OF RESULTS WITH KNOWN DATA FLAT PLATE

Since several assumptions have been made, it was

con-sidered advisable 'to _{compare the results obtained from}

Equations (2h) and (25) with known _{data. The 'simplest:} comparison is that of a flat plate _{where r = O and}

= 2o Equations (2+) and

₍₂₅₎

_{thus beôome:} dx p, 1i f' d

.

= ir

The coefficients

C = 21TÇhc eòsO..

2

, i,4o 4 2.

ir0

()

62

fróm exact theory8" _are:

(2+)

=

o')

' I.57°

Thus fo th _{case of a flat pláte, th}

approximate

solution for the _{ift coefficient is}

seen to be low by about

12% and the solu _{on for the drag coefficient Is low by about}

3%o Actually the smaller _{error in the ase of lift Is due}

only to the series _{approximation, Equation '(18e), which}

raises the lift coefficient

ratio. 1n3 to 1, or about 10%. Applying Equation (18d) t9

the. flat plate gives:

.-(27a)

Another exact theory which is

known and

is more

pertinent to the foils being considered is that fora

zion-synfinetrical wedge (Figure 2). For this case the exact theory7 give.s:*. C 2a3

(_,)2

D _LfrSa ¡oc. C

-

(-oi1)in-rr(_i---L

_(Ls2

the constant à is determined from the integral equatIon:

U $2

.For a thin wedge (-,<7r ).Equation

becóe

(Appendix. C)'

QÇ...r(

_X)-

_{Cos CA)) 6i}

_X

_dZ

where:

and:

I

r

j.

(39)1 and

(31).

.3_

fr

(I_acsti

(30a)

(31a)

*

Note tha.t the sign of _{has been changed from that in} Mì.1ne-ThÖmson to conform with the convention of this paper. Thus

as

shown in Figure 2 has a negative value.

This solution also is based on the condition that the

stagnation point is at the leading edge, and the condition for this is that the ratio of chord lengths be:

_____\2.

Z - cos cj)S'i.i

dZ

' (

J4co&t),

fWec(cos_cO)JX

- HCDSt

Using EquatIon (32), the approximate solution

(EquatiOns 2+ and

₂₅₎

gives the following results:

(33)

'w' dt

-cD=

(3+)

For the purpose of comparing the. exact and approximate

solutions, values of CL, ,

and-'

have been plotted

vs. r. (Figure )+). _{As was the case with the flat plate, the}

approximate solutions for 'both lift and drag are somewhat

lower than the exact solutions0 Both coeffic1nts' approach

the exact solutions as r approaches unity, or ,a synunetrtcal

wedge.

A third comparison has been made with experimental data

for a circular arc section,9 .'igure

_.3.

The experimental data

(?igure 5)

is shown in a plot of 11f t,and drag-coefficients.

vs. cavitation number for 8 degrees aigle of attack0 At this angle separation did ñot occur at the':.iead'ing edge, but

in-stead at the discontinuity of the uppe' surface0 Thus the

flow corresponds exactly to that of Figure 1. The experimental points are marked as circles on the graph, and the solid lines represent the predicted coefficients assuming ,ull cavitation

from the leading edge according to Wu's theory.

'Applying thé approximate solutions to the circular arc at 8. degrees angle of attack and zero cavitation number gives

the following results (Appendix D): CL

(32)

Separation at Leading Edge O.29

_OO+57

Separation at the Bend

_Ol65

OO69l

-15-These points have been plotted on theQ O ordinate

'OEf Figüre

5.

Again the'lift coefficient is low,by about

13% when separation occur's at the leading edge'0 Unfor

tunately9experimenta]. datais not available belowr= 0.2,

but the general tì'end of the points seems to be close tó that given by CL +

C0

The approximate solution gives a drag coeff1cet whih is about 10% high for the case of cavitation from the leading edge and the general trend of

the experimental points is in the direction of C,.

+ _CD

Lu"

_'2

DISCUSSION OF,- TIE. RESULTS

In the developent of the, solution it has been assumed

that the stagnation streamline was horizontal9 thus' allowing

the face and back to be treated separately, and that the

lift and drag forces.4ue to the velocities induced by the body

source d1s,tr1butipn'were zero0 As a result this. solition is,

only an approximate one, the accuracy of which can only..be determined from comparison with iown data, such as is dOnò In the preceding section0 The greatest divergence from

kilown' data is in the case o' the flat plate, where, the

approximate solution 'for drag' is 23% 'less than the _exact

solution. _{Strictly speaking. this comparison should not' be}

made since.the stagnation point is not at the leading

_ege9

"unless _{2Q, in which case the comparison becomes trivial.}

The difference between the two solutions should not be takèn lightly since it is not _{own to what extent the' position of}

the Stagnation point will 'influence the f maI accuracy. If. nO other conclusion can, be da from this, it can at least

be seen. that it is possible' for .a solution based _{n completely}

different conditions to give an answer which is approximately

correct.

Comparisons with data for foils whe±e the stagnation 'point Is ..at, the leading edge. are more' encouraging. _{For the}

case of 'the non-symmetrical wedge, or equivalently the foil with, flat surfaces', the lift and drag coéfficients aré _both

1ow, "but for. . r 0, the maximum difference is only abont

7% for 'both, and as,r increases the difference becomes 'smaller.

There is ;o 'assurance,of coursé, that the _{same accuracyri11.}

exist for foil with curved surfaces _{arid this fact' is borne} 'out by the

comparis9n

_{with experimental data, although this}

1n any event, the accùracy of the results is Sufficient for the conclusion to be drawn that the performance of a foil with a partially wetted back is greatly Inferior to a fully

cavitating foil. This. Is shown for the wedge in Figure 6 where Lc is plotted vs0 r In this case when the back is

D

wetted over % of the chord length, there will be a loss In lift-drag ratio of 3O as compared to a fully cavitating

flat plate0 The obvious conclusion to be drawn therefore, is that a partially wetted back.should.be avoided if at all

possible.

BEC OMMENDATI ONS

There are two prificipal 'sho±tcomings in the solution as it now stands0 The first is that assumptions have been

made which diminish the accuracy. and thus the authority of the results0 The second is that the assumption is made that the stagnátion 'poInt is at the leading edgewithout

stating the conditions undrzMchhiiLbe.o0

The approximating assumptions can best be overcome by

solving the problem by conformal mapping. This has been attempted without success but further efforts woul.be

worthwhile.

Determining the conditioñs at which the stagnation point will be at the leading edge, or particúlarly, the angle of

attack f or any given foil, wOuld also contribute greatly to

.*e

value of the solution0 Mime-Thomson solvedthls problem for the..non-synixnetricaLwedge (EquatiQn

32),

but no solution has been obtálned fòr curredurfaoes0

Ì Ç

Further rsearcb could also be directed towai,'d extending

the soltion'toc6vèr finite cavitation númbersG It is quite

possible that the effect of the partially-wetted back is not a great under such condltions;and this infórmátion would be. most valuablè .f or désign urposes. . .

ACKNÓWLEDGEÌITS

The author is indebted to A. J. Tachmindji for his

constant help and encouragément throughout the investigation.

REFERENCES

1. Thun, N. P., "Supercavitating Flow Past Foils and :StrUts,1 Proceedings.o.f a .Sympostum on Cavitation

in Hydrodynamics, National Physical Laboratory,

September

_1955,

H. M. Stationery Office

2. _{Wu, T. Yao-tsu, "A Free Streamline Theory for} Two-Dimensional FullyCavitted Hydró*ils," Report No. 21.47, Hydrodynamics Laboratory, CIT, Pasadena, Califox'nia; July

_1955.

.

..

3.

ThUn, M. P., "Steady Tto-D1ninsional Cavity Flows About Slender Bodies," TMB Report

_{83I., May 1953}

1+0 Blrkhoff, G0, "Hydrodynamics," Princeton University Press for Universityof Cincinnati,

_1950.

5.

Gilbarg, D. and Rock, D. H., "On-Two Theories of _Plane Potential Flows with Finit Cavities," Naval _O±dnance Lab.. MemOrandum

_8718,

August 19+6

heng, H. L and Rott, N., "Generalization _{of the}

Inversion Formula of TI

iñMrfóil Théory

:

of Rational Mechanics and Analysis, Vol. _{3,No.3,19514.} MUfle-Thomson, L. M., '!Theoretical Hydrodynamics," _The

NeMillan Company, New York,

_{1938, 1950.}

: Iamb0 R., "Hydrodynamics," Sixth Edition, Dover

Pub1i-cations, New York,

195

Pardn, B. R., "Experiments on Circular _{Arc and Flat} Plate HydrOfoils in Noncavitating _{and in Ful]. Cavity} F1ow,'! eport No. +7-6, CIT, _{Hydrodynamics Laboratory,}

t

SOLUTION OF ,FREQUENTLY VSED INTEGRALS

Let I(m, n,

S) =

Substituting z

sin2O. gives:

a.

Ib Çb1

d6

I(m)v)

=

J

-cos'e+I- n,',,,

I.(m,n, (1

)= 2

for the solution of.

i---1

I

f±'o'X'

J

_s

V )Lg x'-x

APPENDIX A

Re

md for the solution of

₍

-Is

-l9

.-I

='_2.j/'!&

_ùc'(j

_.tt9)+2e

syt

>»s

f' ('-x)fr-)

4-c)

,

es) I(

£S

!(2!)

-'---çP) e- O) e-.

j

=

t)r J/*.

.

x'..s'f'o

EVALUATION OF THE SECOND INTEGRAL OF EQUATION (1g)

.it IViJ;r;

,s.

i. dyet

dv0 L

(5.Z!

/' {

_fr

₍e-s, c-s ¿-s, o) J

it)0

therefore: f

4f7_i_j

$4,

Were:

s

r..!.

¿'

.t-.)e')

dv y-r-s X1,/

¿ir'

-jrI.

APPENDIX B dc o

i(;

-tç.S

J. I

_{o -}

-

d-

\

-t-s

< 4tA

I s,

4)

n

but:

ÍkC

'.

=É e (

(t)o

_i :;i0

c1

(

aJt

dt

X

J is determined as follows:

/((\t.(s(

OJO

dt

-c-..

j

_{I d-t} .ç .

2J J

d-6 O

j'T

o -o Sd

_dt

T

f L

o o

(5

I

(-c-Lb-s)

-

(--)(c-s) i

j

2

_{d-t ck}

th! 4

c

_J-c

dd

-21

SOLUTION OF EQUATIONS (30), (31), and (32) since.:

=(

SI.yi

vc')j

\ fv

/a(X#4)

and:. Thus:. :°° .'

then:

f(cosx-o.c

)siIid

=

_2f 1s14)Va(%'t(A))J

1

Assuming a thin wedge, 1, and the integral may be

approximated by:

-

f

.-cq&u», z

dx

/-eos

f

and by the same method:

dZ

(j- Cotw)'

/,,I+ cos '(A)

I I

i-

coS (A). APPENDIX C 2 -22-I +co.s ¿A))

fra1

2()

r-x)1

APPENDIX D

CALCULATION OF LIFT AND DRAG COEFFICIENTS FOR THE

CIRÇULAR ARC SECTION

At.8°

ángle

_{of attack, the fáce is tangent to the x-axis} fat the leading edge:

Therefore,. sine the face isá circular arc of radius R:

-lx

= tanQ where O = sin.

-dx

.. .

=.

tan(sin)

.And:

= =

tan(8°tan1

O.1i00)

dx

_ll62

Also for equations (21i-) and (25) to hold,

2

must

be unity which determines R:

R

_=1:

=

and: r

= ____

= _____

= O)+88

Equations (2+) and (25.). become:

CD..= i frr

d'/12

1Uo

YrxJ

_L0

Ir'

j0 3fjz_x 2:-x)

vrr

The first integral in each equation is easily evaluatê4.to give CL1 and.. CD]: : CL1 = P

-O, 133

= .23 -=

Oli9'1.

(2+a) (2 5a)

But thé evaluation of the integrals. for and

considerably more difficu]t. Consequently, they were

siinp]4fied by assuming constant.. Since the

maximum value is R =

359

and the minimumvalue Is ij' =

the exror invOlved is small.

he aver&ge value was taken as 3 2.

huss 4

-(I

wdx__

T J0 2fr-r)4 ,ri-;:, '

j2

O298 -. o

= + OO23)+

OO69l

xs,

yco A

X

Figúre i -. Öiagrani of the Foil In Super-Cavitating Flow

400

8.550 R

FIgure 2 - The Foil as a Figure

3 -

The Circular

Non-Synunetrical Wedge Arc Section

.25...

2

2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r

Figure 4 - Lift and Drag Coefficients for Thin Non-Symmetrical Wedges

26

- - Approximate Mime-Thomson

Solution

-1.0

0.9

0.6

0.5

0. 4

0.3

0.0

KEY

o c)

FRI EXPEHIMENFAL DATA WITH

SEPARATION FIIOEI THE DISCONTINUITY

V CD) IN THE UPPER SURFA

- CL PREDIC11]J FR! YIU' S THEÇY FUI

CD SEPARATION FRCTvI TIE LEADING EDGE

ODEFFICIErSifS BASED ON

FINITE-LEADING EDGE THICKNESS SOLUTION

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

CAVITATION NUMBER, o

Figure

5 -

Comparison of Approximate Solution with Experimental

Data for Circular Arc Section at 8 Angle of Attack

27 /

t

/

_o

/

/

o

/

/

/

/ /

/

o

/

/ /

'I

L2 Ö

0.2

0.1

1.1

1.0

0.9

0.8

cl

4

0.7

0

0.6

0.4

0.3 0.2

0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 6 - Lift-Drag Ratios For Thin Non-Symmetrical Wedges

28 I "

\'

\'

-Approximate Milne-ThomsOn Solution

-\

N

INITiAL DISTRIBUTION

Copies

8 Chief Bureau of Ships, Technical Library (Code 312),

for distribution:

5 -

Technical Library

i - Civilian Consultant to the Chief (Code 106) i - Preliminary Design (Codé +2l)

i - Propeller and Shafting (Code ₅₅₊₎

2 Chief of Naval Research, Fluid Mechanics Branch,

Code +38, Attn: M. L. Tulin

Director,- Ordnance. Research Lab., Penn State

University, UniversityPark, Pa.

1 Commander, Naval Ordnance Test Station, Pasadena Annex, Pasadena, California

1 Experimental Towing Tank, Stevens Institute of Technology, 711 Hudson Street, Hoböken, N.J. 1 Head, Department of Naval Architecture aùd

Marine Engineering, MIT, Cambridge

39,

Nass.,

&ttn: Prof. L. Troost

Director, National Advisory Committee for Aeronautics, LangleyFleld, Va., Attn: Mr0 Parkinson

1 Director, National Advisory Committee for

Aeronautics, Washington, D. C.

i Gibbs & Cox, Inc., 21 West

St0,

New York 6, N. Y., Attn: Dr0 S. F. berner

1 Director, Iowa Institute of Hydraulic Research, State University of Iowa, Iowa City, Iowa

Director, St0 Anthony Falls Hydraulic Lab., University of Minnesota, Minneapolis 11f, Minn. 1 Director, Hydrodynamic Lab,', California Institute

of Technology, Pasadena +, Calife.

9 BrItish Joint Services Mission (Navy taff), P. 0. Box 165, Benjamin Franklin Station9 Washington, D0 C.

3 Canadian Joint Staff, 1700 Mass. Ave., N.W.

Wasblngton.6, D. C.