On the wave-making resistance and lift of bodies submerged in water

Technical aiitl Research Bulletin 1',o. I-8

On the Wave-Making Resistance and Lift

of Bodies Submerged in Water

-by-N. E. Kotchin

Published in August 1951 by

The Society of Naval Architects and Marine Engineers

OF

THE SOCIETY OF NAVAL ARCHITECTS MARINE ENGINEERS

Steering Committee

Edward L. Cochrane, Chairman

Frederick E. Haeberle Kenneth S. M. Davidson, Deputy Chairman Paul F. Lee

Frank R. Benson

Wilson D. Leggett, Jr.

David P. Brown JQhn R. Newell

John E. Burkbardt

Hollinshead de Luce, Chairman, Hydromechanics Subcommittee Douglas C. MacMillan, Chairman, Ships' Machinery Subcommittee Edward A. Wright, Chairman, Ships' Structures Subcommittee

Hyd.romechanics Subcommittee

Hollinshead de Luce, Chairman

Walter C. Bachman

Louis A. Baler

E. Scott Dillon

C. H. Hancock

Allan B. Murray John B, Parkinson Lewis A. Rupp Harold E. Saunders Karl E. Schoenherr Frederick H. Todd

Through the courtesy of the British Joint Services Mission the translation from Russian (R.T.P. No.

666)

of N. E. Kotchin's paper has been made available for publication to augment the

literature on wave-making resistance. The manuscript has been furnished by the Analytical Ship-Wave Relations Panel and thiB

bulletin is published under the sponsorship of the Hydromechanics Subcommittee of the Society's Technical and Research Committee.

SEAPLANE SUBC OMMI TTEE

AERONAUTICAL RESEARCH COMMIT'i'i

On the Wave-making Resistance and Lift of Bodies Submerged in Water *

by

-N. E. Kotchin

Translated by A. I. (T) Air Ministry 17 March

₁₉₃₈

1. Introduction

The present paper deals with the wave motion caused by

a solid body of arbitrary

shape, in 8teady, rectilinear, horizontal

motion at a certain depth beneath

the surface of a liquid, The

forces acting on the solid body are also calculated, namely the

general formulae for both the wave resistance and for the lift of

the body. The usual assumptions in problems of this kind are taken

as a basis: namely, the fluid is ideal, the only force acting on

it is gravity, and that the fluid motion is established.

Further, the waves caused by the motion

of the body are assumed to be small.

In this case the subsequent assumption

is equivalent to the hypothesis that the body disturbing the flow

is at a sufficient depth below the free surface.

* Transactions of the Conference on

the Theory of Wave Resistance

Hitherto the solution of this problem is known only for

disturbing bodies of special form, e. g., for a circle in the

two-dimensional flow, and for a sphere or ellipsoid in the

three-dimensional problem. Besides this, a general solution for the

two-dimensional problem for the case of a thin profile has been given

by M. A. Lavrentiev and M. V. Keldish.

1

Lamb calculated the disturbance of an established flow

produced by a submerged cylinder, and the wave making resistance

of the cylinder.

]. N. Sretensky, in an unpublished paper, has dealt with

the more general case of a cylinder moving in a fluid with

circulation. He derived functional equations from which an exact

solution may be obtained of the mathematical problem arising from

the assumptions. L. N. Sretensky solved these functional equations

as a first approximation and calculated the wave resistance and

lift of the cylinder.

Using functions of a complex variable, M. V. Keldish

has dealt with a special case of wave resistance produc-ed in an

established flow by an eddy or some other phenomenon.2

The three-dimensional problem has been investigated in

a number of papers by Havelock.3

In/

H. Lamb. Hydrddynamics. 5th ed. .247-249, pp. 387-393.

Keldish. Notes on some motions of heavy fluid. Technical Notes. ZAGI No. 52. 1935. pp. 5-9.

T. H. Havelock. Some cases of wave motion due to a submerged obstacle. Proc.Roy.Soc., London. Ser.A. Vo. 93, 1917,

pp.520-532; Some cases of three-dimensional fluid motion, ibid. Vol. 95, 1919, pp.3514-365; Wave resistance, ibid. Vo. 118,

1928, pp. 24-33; The wave resistance of a spheroid, ibid.

Vol. 131, 1931, pp. 275-285; The wave resistance of an ellip3oid Ibid. Vol. 132, 1931, pp.480-486.

In these investigations the solid is considered as

replaced by an equivalent system of doublets for which the general

expression of wave resi8tance may be given. Fy this method Mavelock

was able to calculate the wave resistance of a spheroid and an

ellipsoid.

In another paper4 Mavelock calculates the wave resistance from the wave energy. Here, however, the wave resistance is

obtained not as a function of the form of the body, but of the

waves produced by the body. The dependence of the wave resistance on the form of the body remains unknown.

In the present paper, general expressions for the wave

resistance and lift of a body of arbitrary form are found and by

using the metriod of integral equations, the problem of wave resistance produced by this body is solved.

Chapter 2 contains a general statement of the proDlem

and the fundamental equations are given. Chapter 3 contains the

calculation )f the wave resistance and lift of an aerofoil of

infinite span. The essential feature of this method consiste in

the introduction of a special function by which both the wave

resis-tance and the lift are expressed directly.

The fact that an aerofoil mc'ving 'beneath a free surface

has a different circulation from that of an aerofoll moving in

an infinite flow, makes the calculation of the said quantities

highly intricate. Chapter 4 therefore is concerned with the calculation of the circulation of an aerofoil moving beneath a

free/

-. T. i. Havclock. The CticuThtion of Wvç csietancc, Proc. oy. Soc. London. V. 144. 1934. pp.514-521.

determined by the special function referred to above and with the determination of the functional equation for this function.

Chapter 5 deals with the special case of a plate moving at any arbitrary angle of incidence.

Chapter contains the calculation of the wave resistance and lift in the three-dimensiona. problem.

In Chapter 7 exam;.les are given of the three-dimensional

problem.

Finally, the last two Chapters deal with the problem of wave resistance caused by a body moving beneath a free surface, solved by the method of integral equations; Chapter 8 with the two-dimensional and Chapter 9 with the three-dimensiona:. problem.

. The Fundamental Equations of the Problem.

Let us consider ari ideal, Incompressible homogeneous

liquid of density p on which the acceleration force due to gravity of constant quantity g is acting. When this liquid is at rest its free surface assumes the form of a horizontal plane. Recti-linear, rectangular systems of co-ordinates are selected in such a way that the axes x and Qy lie in the horizontal plane

and axis Cz is directed vertically upwards.

A solid body is assumed to be moving beneath the surface of the liquid along a straight horizontal path at a constant

speed C. The solid Is considered a travelling In the direction of the positive axis Ox. This otcn produces waves on the free sin'face, which move

alr

th axis Cx as a whole, vith the same velocity e (it is assumed that the solid has been moving

at a

constant velocity c for an infinitely lon time. ) A certain amount

of enerr Is absorbed for the production of these waves,

-5-and is derived from the solid. Hence the solid Is subjected to

a certain amount o.f resistance called "wave resistance" and in addition derives a certain amount of lift.

The present Irohlem deals with the calculation of the

wave resistance and lift of the body nd the determination of the

forn of the waves producud by the motion of a solid of definite

shape.

For convenience the axes of co-orthnates, x,y,z are assumed as moving together with the solid. The axes x1, y1 and

z1, fixed in space and co1ncidin with x,y,z at the moment

t = C will also be considered. The following relations arc

obviously

obtained betweun co-ordinates x,y,z and x1,y1 and

Z1:

xl

X + ct, y1 = y, z1 = z. (2.1)

Besides these

two

additional

assumptions

are adopted.

The first, absence of vortices within the

liquid so

that the motion

ha3

a velocity potential. It is obvious that in a system of

-ordInates x,y,z moving with the

body, this

potential is a function of the co-ordinates x,y,z only and not of the time t, sinc. ts stntd above the whole wave asruct rLrnains Invariable

vthcn 'IiLWCd from th moving body. The velocity potential is hurufore a function (x,y,z). :t Is thus clear from formula

(2.1), that in thu fixed co-'rdlnatus

X1,

y1, z1, t, the pot'ntial of absolute velocity is of the form:

1(x1, y1, z1, t)

=

(x1 - ct, y1, z1).

(2.2)

-c

The second additional assumption is that the waves

produced on the surface of the fluid are small. _{Since these}

waves become smaller with increasing depth of immersion of the

body, this second assumption implies that the solid is assumed

to be at a sufficient "depth.

We no turn to the mathematical interpretation of the

hydrodynamical problem.

Owing to the absence of eddies, the motion has a

velocity potential (x,y,z) and hence for the velocity vector

we have the equations:

-* _cO

y = grad , v = -- , v =

--s-, v =

--Since the fluid is incompressible, the equation

div y = O

leads to Laplac&s equation for the velocity potential

=0.

Öx2

y2 It/ 2 (2.3) '1 2

=c

-t

g=O

-7-It is known further, that in the case of established

motion of an ideal incompressible fluid subjected to a force F

and having a potential V, so that F = - grad V, the pressure

is determined by Cauchy's integral which may be expressed by

variables

X1,

y1,

X1.

t

L

-

gz + f(t).

òt

i

It is assumed that at the free surface the pressurer

p0

is constant. It then follows that f(t) must be a constant value

equal to Po because according to hypothesis at a distance ahead

of the solid, that is: at large values of x, the components of

velocity, the ordinate of disturbed surface and, from equation (2.3)

/t all approach zero.

Thus

p

--y2

This equation holds for the free surface and since the

waves are assumed as small, y2 at the surface of the fluid is neglected

as a small quantity of the second order. Denoting the ordinate of

the disturbed surface by _, in co-ordinates x1, y1, z1, t and by r in co-ordinates x,y,z, we have

This equation is differentiated fully with respect to the

time t. Since

d1

_=v.

'z

òz1

and since in the derivative d

dt ?t

dx1 dy1

ô'1 dz1

+ + +

Ötòx1 dt

OtÖy1 dt

àtòz1 dt

i + V + V + V

-t

_Òt

_x1 lx

_Otòy1

1 'z

thu last three terms being small quantities of the second order

may be neglected, we obtain

+ g ---- = z1

as the result of the differentiation (2.4).

Owing to the relations (2.3), equation (2.4) acquires

the form: c g

Ox

arid (2.5) (2.5) 2 cT C

+g---

O.

Òz

Strictly speaking, this condition must be fulfIlled at

the froc surface, ut for the approximation accepted it may he regarded as fulfilled at th free surface when the latter is

in a state of equilibrium, that is, whLn z = O. Also,

the

rise

of tIF; fr.c surface may, within the reiuired approx±rntion, be (eterrnined by formula c ¿p(x,y,o) g (2.6) (::.7) Denoting/

9

Denoting the surface of the moving body by S, we have

in ad'ition to the boundary

condition

(2.6) the following boundary condition at S. The normal component of velocity of a point on this surface must be equal to the velocity of a particle of the

fluid on this point. Each point of the surface S is moving with velocity e parallul to the axis Ox, hence its normal component = c cos (n,x); and since the projection of the velocity of the

fluid particle on the normal is

cc/ôn

the boundary condition

is:

c cos (n, x) at s. k

"

Hence, with the given assumptions, the problem -f the determination of waves produced by a body moving in a fluid,

parallel to the axis Ox at a velocity c is reduced to the (1utrrnination of the simple function (x,y,z), which satisfies Laplace's equation

= 2 + + = 3,

(2.9)

z'

vhich is harmonic in thu region contained between the surface S an

tie

horizontal plane Oxy, and which satisfies the two

'oundar'. conditions: (2.8) at the surface S and

64Ç

c2

---r + g ---

= 0 (2.10)

òz

at the horizontal plane Oxy.

In addition the conditions at infinity, to ensure

absncu of waves in front of the tod' must bL satisfied. Hence thc/

the component velocities

Òc/òx,

5/'y, Ò/ôz

must remain finite when X2 + + c'c and. these components must approach

zero when

X -

+cyj.

2

(.\2

2i 11m

òyJ

= 0. (2.11)

In the two-d1meniona1 problem the horizontal axis Is denoted by Ox and the vertical axis directed upwards by Oy,

the moving contour by O.

The velocity potential cp(x,y) must again satisfy

Laplace's eq.uation:

- (2.12)

Öx2

ày2

and the two hoindary conditions:

c'---+g--- =

O when

_y

= 3, (2.13)

Òx'

òn

c cos (n, x) at

o.

(2.14v)

The condition relating to the absence of disturbance at a distance ahead of the body is satisfied by condition that

ÖÇ/Ò

x and

òÇ/ì

y should be limited when X2 + y2-. and that these quantities should approach zero when

x-4

+ 00

Iio\2

(o\2

11m

H---). + (---1

= o.

x_ì+c#L\Òx/

\Òy/

(. 15)

=

It should be noted further that in the two-dimensional

problem, the potential cp may be a complex function of x,y.

In this connection the circulation about the contour C must have

a definite value

r that is

cp must increase by

r when a point

moves round the contour in the positive direction. In the case

when the contour has a projecting angular point, the latter

requirement may be replaced as is often done in the aerofoil theory,

by the requirement that the velocity must have a finite value at this point; a definite value is then btained for T'

The ave form produced at the free surface is determined

in the two-dimensional problem by the equation:

w(z) = ((x, y) + 1(/,(x, Y)

(2. 16)

g

which is analogous to (2.7).

3. Lift and Wave Resistance of an Aerofoil of Infinite Span.

The solution of the problem set forth in the foregoing Chapter wIll be given in the last two Chapters of the present

paper. In the first part of the report it is assumed that this

solution exists and considerations are restricted to the calculation

of the hydrodynamical forces acting on the body.

We deal first with the two-dimensional problem. Consider

the aerofoil C as moving below the free surface with a veloáity

c, and parallel to the axis Ox (fig.1). In addition to the

velocity potential cp(x,y), the complex variable z =

X +

ly and the complex velocity potential

are introduced. Since,

2 2

dw

.--,

---i

ôy

dz

Ox2

yÒx

and introducing g

V,

(3.1) C', the condition

C'-+ÇT--- =

O when _y = O

may be written in the form:

( d"w

dw)

Inag i -- - y -- = O when y = O

) dz'

dzl

The complex velocity

dv'

W(z) = -- = y - iv

dz y

for which we have the condition

d1

Imag ¿ i -- - y W ₌ O when y = o (3.2)

dz

j

is now introduced.

We now take any point z in the lower half-plane and

draw two contours C and both of which circurnacribe

contour/

'-'1

13

-contour C, in the positive direction and lie in the lower

half-plane so that point z lies outside contour C1 and within contour

Co:,

Applying Cauchy's integral, we obtain the equation:

w(z) =

1

w () dr

27(i

J

z

-Cl ₀₀

since w(z) is a holomorphic function within the region D

situated between the contour C and axis. Ox. The integrals

along the contours will always be taken in the positive direction.

For the sake of simplicity we introduce the designations

W (z) = w2(z) =

i

_2ST1

_{J zÇ'}

n

2Tr1

I

( w(r)d

00

(3LI)

It is seen directly that W1(z) is a holomorphic function

in the whole plane of the complex variable outside contour C1.

which at infinity is of the order of 1/z and is analytically

continued at that portion of the plane of the complex variable

situated within the contour C, and that W2(z) is a holomorphic

function within the contour C , which is continued analytically

in the whole of the lower half-plane. The latter function will

now be transformed.

Condition (3.2) shows that the simple function

dW

-J

f(z) = i

(3.5

7

acquires considerable values along axis Dx and may be continued

into the region D', which is symmetrical with D with respect

to the axis

Ox

In accordance with Schwartz' theorem.

Thus

1(z)

will be a holomorphic function from

z

in the region between

cor.toirs

C

and

C' Í1syrrnetrical with respect to

Ox.

::.

V. Keldish applied this circumstance in determining

th.

v'ave motion produced by an eddy moving uniformly belovï a free

surfac:, in the following manner.

If a vortex of intensity

'r

i

suh'lLrgc'd at a depth

h,

at a point with a co-ordinate

- hi,

the only singularity corsponding to this case of the

functicn

_fr(z)

in the lower half-plane Is the polarity

r

and therefore

the only singularity of function

2711

z - r

is.

'r

i

i

-r

(z-

_)<'

Citi

z-Lfld SinCe

rr (z)

has complQx conjugate values at points

synnetrical in relation to

Ox,

the only singularity of

f'r(Z)

in the upper half-plane will bc:

T

i

'vT'

i

T

(z_)221ti

z-Putting

r

i

vr

i

'r

i

i

1T'

(z)

_2TT

_{(z - ç)C}

₂₇₁₁

_z

- Ç

27

(z

+

21t1 z

-and integrating equrition

dW

__L'_

_-

_VWr

₌

_{1r (z)}

dz

15

-with the boundary conditions

w

- O when

X -

+ O O

M. V. Keldish obtains the following expression for the complex

velocity of the wave motion, produced by an eddy moving below

the free surface:

z

i

+r

1

MF..

(eÍZ

e "' J dz (3.6)

- 2Jr1

z(

27T1

z-

J

z

+ cc

We now return to the general case. Together with the

function (3.5) we consider:

f(z)dz = l[W(z)

-J W(z)dz.

This function, however, may be a complex quantity when

the circulation over the contour C O. Let

W(z)dx ₌

r.

Then, denoting by

f

the complex co-ordinate of the point lying inside or on contour C, we obtain the function

dW dW

F(z) = i -- - W - i ----

=f(z)

-dz dx

(z

F(z)dz will be a simple function in the region between

Jo

the contours C and C' . Since in confoi'iiity with the conditions

the function W(z) in the lower half-plane is limited by a modulus

when

z -Dand since this applies 'also to function

Wp (z), it is clear that for

zjthe functiofl

r

F(z)dz

ifW(z)

-

Wr (z - i[w(o) - W (o)]

-(W - ')dz.

cannot increase by a modulus exceeding _Kizi, in which K is

constant. This means that when z =r F(z)dz can only possess

Jo

a pole of the first order; hence both F(z) and f(z) at the point z = will be regular and moreover of zero value, other-wise the result obtained from

dVJ

i -- - vW

= f(z) dz

would be that 'hen y O,

xb

+ the derivative 71/dz will approach a definite limit not equal to zero, and hence

s(x)

will increase indefinitely by the modulus, a result at variance with

the fact that V is limited in the lower half-plane.

By integrating equation (3.5) and taking into account the condition that '?(z) becomes zero when x + 'cre readily find that:

z

= -

ieV

f(Z)CiVz

dz.

(3.7)

+ X.)

In this case, it is Clear, however, that W(z) is an

analytical function of z in the region between C and C', but

only the point z can exist as a singular point and '.V(z)

may be a complex value in this r ion. That is to say, rhen

proressin: along C', '(z)

incrcìses by AeZ, in which

A is a certain constant, and "'hen progressing along C, '(z) remains invariable oiing to the conition of being a simple value. The

part of TT(z) of prL sent ìiterest is cetcrmincd

by the

cor.ition that the intrgration contour in formula (3.7) must

lit

cntirely

on th 1o,Tcr haf-planc.

The function W1(z) determined by integrai (3.4)

consista of terms of the kind B/z - in which B

IB

a cc'mplex constant and a point on the lower half-space. By using the

method of

ri.

V. Keldish It is not difficult to find the analytical

function (z), the only singularity of which In the lower

half-space is the polarity B/z condition:

when y O.

In the construction of the function

it is found that the only singularity in the lower half-plane is

the polarity

but since f(z) at complex conjugated points with respect to the real axis possesses complex conjugate values, the only singularity

of f(z) In the upper half-plane will be the polarity

Assunìng therefore, that:

f(z) =

(dci

Imìg ¿1 --- - PvW = o dz J IB

(z)2

17

--,

f(z) = dz

(z-)

vz

(z_?)2

and which satisfies the boundary

z-e.

B z - + (z

V

B

z-f,

and/

and integrating (3.7) we have

+ 2iv

_i Z Ç e1 dz. 'J

z-r

z-

z -

-+ o i

Puttig In this formula B ---- W( p) d and

2TU

integrating with respect to the variable over C1 we finally

find that function

C(z)

w( )d

f

W(ç)

z-Ç

2îti

z-'

cl

cl

-z

e1t

I

-

oive_ivz

dt

t-.

j +

also satisfies limiting condition

aû

Imag I -

vfl

_{= o}

dz

.vhen _y =

o.

Further, it is readily seen that if the contour of

inte-ration with respect to t Is assumed to be entirely within the lower half-plane, the function Ç'(z) will be holomorphic In the

egion situated between the contour C and axis Ox be limited

in the lower half-space when

_{tz I}

and approach zero when

X -

+ . But then the function Ç (z) - !(z) will obvtously

be holomorphic In the lower half-plane, will satisfy the same

limiting!

(3.8)

*

This formula is a special case of the general fonla obtained by I. V. Kcldish In his paper "Notes on some motions of heavy

fluid", loc.cit.

B

limiting condition, will be limited

in the lower half-plane

:ihon

I z

and approach zero when

x

Using formula

(3.7) in which f(z)

having no singularities In the plane

of the

complex variable becomes zero when

z

= ',

may be assuried as

= O,

've obtain the identity

Ç1(z) - 7(z)

O.

Thus we obtain the following formula for the complex

velocity

V!(z)

=

i

;: j

w()

cl

19

-V'(z)

=

v:1(z) + 'V2(z),

= -

2rc1J

z-cl

(3. lo)

z

ivt

1

i

_Oive-i'

I

dt

,

(3.11)

-In which case contour

C1

may approach contour

C

arbItrarily,

but the point

z

must he outside

C1.

The formula obtaincd

for

hc compx velocity is taken

as a basis for the calculation of the

hydrodamic

force

ating

on the moving contoir

O.

formula:

ThIs forcc wi. he

Calculated by the Chalyir.-Ths1us

(J

p - =

- : y:; (z) dz,

CÇ,

But

where P is the lift of the contour, . Its wave resistance,

C., _{an arbitrary contour situated In the lower half-plane and}

enclosing contour C; finally

_W0(z)

_{Is the complex velocity in}

the relativo motion, obtained by superposing on an absolute flow,

ari established fluid flow in the direction _{of negative axis OX}

and with a velocity c. _Hence

W0(z)

= W1(z) + W2(z) - c

and wo take as C1 the contour situated between C and C2.

We now calculate the integral

i

=

J

w02

(Z)Z

=

f

W

(z)dz +

f

(w2

-

c)2 dz + 2

f

w1(W2 - c)dz. C2

₀₂

Cr C2 r

f

W1 dz = C2

since W1(z) iz _{olomorpiic on and in t:e contour C,}

and this is extremely smal.

Hence also

Ç(w2

-

e)2 dz = 2

since

_W2(z)

_{is holomorphic on and within contour C2.}

Therefore

i

= o

[(w1

+

w2 - c)(w0 -

_{C)dz = 2}

fw(w2

-

_c)dz

= 2 ÇvÏz)w2(z)dz - 2c

Çwz)dz

De i gnat in g/

21

-Designating the circulation of velocity over any contour

encloaing C,

by r

80 that:

r

=

J

W(z)dz we h.ave: P - IQ

-e

JW(z)

W2 (z)dz + (3.14)

tJ8ing

(3.11) we have: W(z) W2 (z)dz z i .

r e#t

i i

2i

_{f Ç}

W(z) w (ç)

[z_e

2ieVZ

J dt I

ddz

+00

J (3. 15)

It is seen that 1f both points z and

r belorgto the

lower half-plane we have the ecjuatlon

00

r

-= i. dX .

z_Ç

Therefore

ff

W(z) W i. o

z-c2c1 C2C1 o (3. 16) dz =

j f J J

W(z)

()

e1e

d;dd:

(3. 17)

integration may be changed. _{Then, introducing function}

H(X) =

fe

W(z)dz (3.18)

cl

in which for the integration we may take an arbitrary contour situated in the lower half-plane and encircling contour C in a

positive direction we may write:

00 1 j

f

W(z) W ()

ddz

=

i f H(\)

E (X)dX = i JIH(Xfl2 dX.

z-CrC O O (3. 19)

We now transform the xmairing terms of integral (3.15).

For this we consider the ñmction

z

I

e_ÌVZI

-

at,

J

t-+ )

in which z and are assumed to belong to the lower half-plane. First, tp1acing the variable t = + u, we have

z

_jt

z-r

-

I

e1'

re

-ivz

t -1V (z- ') du. at = e e

Jt

+00

The integration contour is transformed as shown in

Fig..

Namely, the contour K in the lower haTf-plane proceeding from an infinitely distant point on the rositive axis Ox to

23

-point z - may readily be transformed into the resultant two

contours K1 and K2, the first of which is an arc of a circle

of infinitely large radius, situated in the upper half-plane and

the ends with one extremity lying on the positive real axis and

the other on the radius passing through the point

Ç -

z.

Contour K2 consists of the latter radius, succeeded in the negative direction by an infinitely small semi-circle with

Its centre at the origin of the co-ordinates and then by the radius

drawn from the origin to point z

-It is readily seen that:

reIU

j

du,

u

along contour K1, approaches zero when the radius of arc K1

Increases to Infinity. Assuming further that

u = (z

-)x

we obtain the formula

Z

ivt

('-1'(z-

)(i-x) J

e1'

I dt = dX.

t-

J X L

L is the contour shown In Fig.3 and consists of the negative

axis X followed In the positive direction by an infinitely small semi-circle, with Its centre at the origin of the

co-ordinates and drawn in the negative direction and a section of the

positive axis to point X = +1, It 18 seen further that

Ai'Vt

-e1'1Z(

e _dt =

Jt-r

(3.20) where/ +

v.p.

J

-

¡

e'(

)(1X)

'

where v.p. (principal value) denotes that the principal part of

the integral la taken

i

v.p.

(z-L

i

dX

_f

-

_dxl

urn

1Çe1V(

)(1-x) +

e1V(

)(1-x)

--x

_J x

£

We now readily find

z r 0lvt

f f

W(z)W()e

i -- dtd dz -

if

f

eW[je

VZ

W(z)dzd J t-c2c1 +cx, C2C1

i

+

v.P.f

JW(z)e_1v(1X)Z dzfW()eu1dÇ

X

-ooC2

cl i =

-

_{1IH(V)12 +}

v.p.

ÇIHY_xv2

dX

--p

X

and finally, with (3.14), (3.15), (3.19) and (3.21), we have:

-

_e

c

r

-

-

fIH(x)12

dX - 1VH( v)

i

E"v

C 2(1) + --- v.p.

IH(v- xv)l

--j X

-00

Separating the real and imaginary parts and adding

to P the Archirnedean lift

gES,

where S la the area enclosed by/

(3.21)

P - jfl

First, with very large values of the modulus we develop

the series

i

1w()a

i

_I

r

W (z)

_i

=

----

_I

I w(Ç ) d

+ ...

= +

21iJ z_c'

CflizJ

21iz

Ci

Ci

Re I

4w(z)_c12 dz.

C'-'

Thus for the moment of the forces acting on the aerofoll

':e obtain

L =

- g

Sx

+ Pe

W1(z) + '/2(z) - c]2 dz. .... (.25)

C'-,

This expression is calculated in exactly the same way as

25

-by contour

C

not counted in Chaplygin's formula, we find for

the wave resistance

Q

(3.23)

I

arid for the lift

i

dX

I p e

dX + ---

IH(Y- x)I2

-- +

gES.

X

o

-(3.24)

To find the point of application of the forces

Q

arid

p

o

ti.e aerofoil,

"e calculate the moment due to these forces

with respect to the origin of the co-ordinates.

If

X

are the co-ordinates of the c.g. of the aerofoil

S,

the moment

of

Archirnedear, :ift (considered as positive when lt causes the

wing to rotate in the clockwise sense), is gSx. The remainder

of the moment is caculated by the Chaplygin- Blasiu.s formula.

and thence

I-r

z W1'(z)dz , Re

f

z W12(z)dz = O. C2 02

Further, corresponding to (3.13)

we

have

fz

(W1., - 0)2 dz = O, C2 and thence L =

- gQSx

+ Re@

_I

ZW(W2 - c)dz

=

- gSx - p e Re ZW dz

02

02

zWW, dz. (3.26) Cr)

In place of

formula (3.19) we have

fzW(z)WÇ

;-er

C

___

-C CÇ C4

(3.27)

Thus,

8ince dii -- = - i zW(z)dz dX Cl

we readily

find: i

-f -f

zW(z) W ()

ddz

._JH(X)

H (X)dX CDC1 o In'

wr have z

etVt

dtd

rz

I H' ( V) H( y

S Ç

zV(z) W (

ev Z

f

-1

r

+ i.

V.P.

I H'(v - xv) H(v - Xv ) --J X 00 and since: z.N.., dz L

- gSx

27

-In the same way and cor'espond1ng to forrrnña (3.21),

vi

1 z

_,Tti

j

z

-cç C1

-co

Thus, in order

to find the values of the quantities P, Q and L it is enough to know

the

function

H(X) =

f

W(z)dz

(3.29)

Cl

which incudes

the

complex velocity of the flow concerned.

I '(z)

()cidz

Ç dt

eZ zVI(z) W

dz

J

t-Cl + wo obtuin

from (3.26)

00

e

Re[iH'(0)j

- eRe

i----

f

H'(X)

H (J dX

L2

J

o

dX

H'(V - Xv ) H (y - Xv ) -- .

(3.e)

and IH00(X) j e

W(z)dz

J

f

cb2 [(z + hi)2 C C

r

i

+ I dz = (1 + 21! cXb2) . (3.32) 2fl1(z + hi) j

Using (3.23) we find for the wave resistance:

and using (3d4) for the lift:

It is found that a very good first approximation _is obtained if H(X) _{in the formulae obtained is replaced by}

H(X)

=

f

e1

W (z)dz, (3.30)

where W

(z) _{is the complex velocity corresponding to the motion}

of the aerofoil in an infinite flow.

This consideration is now applied to the _{case of motion} of a circular cylinder of radius b at a depth h and with given circulation

r

about the contour. In this case, it is known that

Cb2

r

woo

z+hi

2rti log (z + hi), therefore cb2

r

w

_-

_(3.31)

(z + hi) 2Tti(z + hi)

2 2gh

--C

)

Q = --- e c-P/ (3.33)

and for its real part 29 -co

e

_Xh

(F+

7tcXb2)2

P 2h o i I + --- V.P.

f

(1-

_[r

₊

_2cvb2(1

_{- x)]2 --}

gEB. .

-oc

a siple calculation and using the symbols for the

intcvrLìtion of the exponential function

X

r

eU du Ei(x) _I J u -cx X

i

C e' du

Ce'

dA

Eijx)

ReEi(x) v.p. v.p. ¡

when x >0,

u

ix

- -cx (3. 36)

formula (3.34) may he reduced to the fi'ial form:

eT'2

CPCb2

2QFgb2

eltc°b4 Qgrtb4 2 gh + 2 e

-El1 ()

+ g p .. ( 7)

In the special case with

r =

o,

Lamb's formulae are

obtained from (3.33) and (3.37). In the other specia:L case

where the radius of the cylinder b = C that is, when deaing

with the motion of a cylinder under a free surface, (3.33)

and (3.37) lead to the expression civen for the first time by

L. N. 3rtensky.

Cacuating/

(3. 35)

P

Calculating the morflent _{ue to the hydroynamic forces}

acting on the cylinder from formula 3.28, we find that

L =

-VH(V)H'(v).

But from (3.32) it is seen that

H'(X) = - hl-I(X) + 2rrc'b2 e

the refore

L =

hQ - 2Tcb2ve'H(V).

The point of intersection of the axis OY and the resultant of hydrodynamic forces acting on the body is determined by the formula Yo L 27'cb =

--

-h+

Q -rgb2 C

It Is readily seen that this resultant never passes through the centre of the cylinder, when _{> O.}

4. Influence of the Free Curface on the Circulation of the Aerofoil. In the foregoing example the circulation velocity was

assumed as given. In the aerofojl theory, however, the aerofoils usually have a sharp trailing edge and the circulation is deter-mined from the condition of finite velocity at this edge.

The moving contour C is accordingly assumed having an angular point A. The tangents to the two arcs of the contour C

meet at the point A forming an angle smaller than 1Ç. It is

further assumed that any other angular points of contour C

sat i s f y/

30a

-satisfy thi8 condition and that contour C consista of a finite number of arcs of continuously variable curvature. The circulation

is detemined from the condition that the velocity is finite at point A. It must be noted at the outset, however, that the free

surface of the fluid which causes the velocity of the fluid

particles to vary, causes a variation of the value of the velocity circulation over the contour of the aerofoil. Thus when the

aerofoil moves below the free surface the latter value will differ from that obtained when the aerofoil moves in an infinite flow.

The value of the circulation entera into the formulae (3.23) and (3.24) where lt appears directly as the term

ec

r

in the expression for P and also in the function Ji(A) which Is also a function of r. The problem thus consists in the calculation of the circulation in the case of an aerofoil moving below a free

surface. We must first establish the formula for the complex

velocity W(z) derived from function Ii(A). !lthough E(A) may be derived from W(z) by a very simple formula

Ii(A)

f

W(z)dz (4.1)

where C1 IB an arbitrary contour enclosing contour C, the

expression for W(z) derived from E(A) is Intricate.

Assuming as before, that

i

W1(z) ---I

(4.2)

2fl1j

z-C i - i

fivt

W2(z)

---w(tJ ----e -

2ieZ

--- dt

2i4

z-

J

t-C '- -(4.3)

and referring to (3.16) and (3.20) we shall have

i_______

W2(z) = ----

J

W( r

f

21t j C o shall be satisfied. contour i - v.p.

f

)(i-x)

which after permutation of the order of integration in conformity

with (4.1) may be written

i______

W2(z) =

[e_i

E(X)dX + ive

H(v)

21t

o

i

-

v..Je_

z(1)

_{H(,'.- Xv) -- (4.4)}

Thus W,(z) is already expressed in function H(X). There remains

w1 (z).

We now transform conformally the contour C into a circle

of a given radius r in the plane of a complex variable u, so that an infinitely distant point of plane z shall be transformed into

an infinitely distant point of plane u and condition

dX + 21v

1e1V (z-

)

(du\

dz)

= z=co

Since W1(z) Is a holomorphic function of z outside the

3'

-r

a3

+ -,. + - +

21iz

z' z

Considered as a function of u, W1(z) will be

holomorphic outside the circle of radius r and in the whole of this zore will he given by the series

W1(z)

'Ie now take the boundary condition at the contour C.

The relative velocity of the flow at C must be in the direction

of the contour element dz, that is W1 + W- - c can differ from

dz only by the rea factor. -ience (W1 + W - c)dz has a reai

value at contour C. By transformation to the piane of the cornpex

variable u, in which the contour K _{wit.h lui} = r corresponds to contour C we have at K

u =

rei®,

du

ireid®

=

iud.

Hence at C the relation du/u has an 1maginary value so that the boundary condition may be written in the form:

dz

+ - c) u -- = o on K.

'lu

Since we have the series

Y1 i,-,

Z

= u+Y0+_:+

u u

which holds everywhere in the region

I ut )

1, 've have for

dz/du the series:

dz

-- / du

. -

-du

u2

u3

Function W2(z) is a simple holomorphic quantity and will hold within a definite region r

u. r1.

The same applies

to function

dz f(u) [W2(z) - cl u,

du

which thus adnits of development into a Loran series

f(u)

=

-n=- u

whence it is readily seen that the coefficient becomes zero.

With respect to the function

dz g(u) W1(z) -- u,

du

it le clear from the foregoing con8iderations that lt is a function

of u which is holornorphic outside the region _I u = r and when

Iu

>r may be developed into a Taylor serles i/u:

r

oob

g(u)

--- +--.

(4.6)

2Ti

n=1 u

The problem has thus been reduced to the determination of the function g(u) in the form of (4.6) in accordance with the condition

Re g(u) + f(u)] = O when lu = r. (4.7)

i

- -

f(u0)

y.

If, on the other hand, uu0 from the inside of

contour K, the value of (4.9) approaches

i

f(u0) +

-

34

-A function g(u) of this kind muet obviously be

determined as an approximation with an arbitrary, completely imaginary constant so that the circulation

r

In formula (4.6) will remain completely arbitrary.

The solution (4.7) le found by the formula

1

ff(Ç)dÇ

i

I

f(r)d

r

g(u) =

----

i

+ ----

i

+ ----

...

(4.)

2fli _J

' -

u 2lCi J

-

r'- 21t1

K

u

First, t is seen that in the region uf

>

r both

the integrals ori the right hand side represent holomorphic functions.

Further, when point u approaches u0 of contour K from the

outside.

i

2Trij

-u

(4.9)

K

approaches the value

i

27ij

, u o K __i__

:TlJ Ç-u0

K

but when u approaches u0 from the outside r'/i

ap;raches

r/0 from the inside and after repaclng

u in (4.9) by

/

and changing to complex conjugate quantities, the foTowIng

relation:

11m

i

_(f(r)d'

i

fr2\

i

urn

_i

_-

_-

_{flz- i}

_{+ v.p. ----}

r

u,u 211J -

_o 2

\u)

21tjJ -

r

KÇ---

°

u

u0

18 obtained.

Thus when

u

approaches

u0

from the outside, we have:

i

i

fr2

11m

g(u)

- f(u) - - f (

u - u0

2 2

i

r(Ç )dr

_rf(r)dr

r

+ V.P.

21[f

-u

-

r

K .

frrc

J

Using (4.8) we may write

i

_{([w2(z) -}

c]"dz

u

{W2(z)

-

c]dz

J

-

= J

r

-

_J

r

-C C

-

c]dz

21t1

I

-

r

21t1 u

I

-

r2

J K

u

C

u

in which case point

of contour

K,

correeponds to point

z

of contour

C. ,

We/

and, eizice

11m

uu

I.e.,

r2

u0

=

u0

g(u)

-Re [g(u)

Re f(u0)

+

f(u)]

+

---

i

Ti

O

u0

Re v.p.

l J K

when

Ircr)dr

'-u0

=

r

+

r

27x1

function: C dz L G(X,u)

JÇ(z)-u

note that an 36

-We now examine side by side with H(X) the following

f

r

2

Gx,

-r u

Ç(z) -

--C u

Using these relations eM tak.lng formula (4.4) as a basis we may write:

ÇW2(z)dz

_Çxu

H (x)dx +

ivG(v,u) H (y)

I_____._- =

J(z)-u

2T C o i G(V - xv,u) H (

- xv)

dA -- ..., (4.12) IT: X -(4.10) (4.11) d i

f

2\

dA - - v.p.

av- xv, -- JH(%# -

Xv) - .

(4.13)

uJ

X

-We now take the following formulae which may be obtained through the theorem of deductions:

C u o ( W2(z)d i r ¡ r

-

r2 X, -- I H(X)dX -

iiìc

L

-Ç(z) - --co I Ç

G(

2 i \%

;ij

dz r. i ¡ dz i

----I

=---

=1

21Li J r-

21f.i J

Ç'-

u

C K dz r dz i

21U

- r2 = r2

= -i when

u>r.

(4.

ç---

Je---C u K found that

In this way, since through the formulae obtained, lt le

i du i du I W2(z)-c W(z) = W1(z)+W2(z) = - -- g(u)+W2(z) = ---- -; J

-

u dz u dz C i r2 du W2(z) - c

r

i du I

dz + W2(z)

---27tlu'dzJ

- r2

2ltludz

C u

and putting in the values of (4.12), (4.1$), (4.14), (4.15), and

(4.4) we find the final expression for the complex velocity W(z) from W ( z) dz(u) du

T

idu

du Cr2 du

C--

+ C +

----27tludz

dz u2 dz 00

il

+ ---

I

+

---1 du

G(xu)1

()

2 _J

[e1

2i dz

o 2 / 2 i r du , r

+ ---G (X,

H(X) dX

21Uudz

u when

Iu$>r

(4 +

r

i

i du V

1(1

-ivz(i-x)

-G(v - xv su)]

H(Y - xv)

- - v.P.

H

_2ltidz

-38--

G(v,u)

2i dz

-+ 1 du i

r2du

f

+ ---- -- -- G

-2ltiu2dz

_\

It IB flOW easy to determine the value of T' . Let the

point u =

rel6O

correspond to

the

angular point

A

of contour

C. Then dz/du will become zero at this point. If the velocity

dz

is required to be finite at point A, expression 2lti W(z)u --du

tflUßt become zero when u = u0. When forming this equation lt should be noted that

u0

X U0

thus after simple transformations the following equation for the determination of r' Is obtained:

i_______

r

+

4ltcr sin0

+ Re u0

G(X,u0) H (x)dx

o

- 2

' Ima

{u0 Gfr ,u0) Ñ( '

i

dX

2V

.

f u0 G(v -

X

,u)

- Xv) --

= O.

... (4.17)

----Rev.p

_lt

X

In/

i r2 du f r --- G Lv, --j H( y) dz

_\

í/ r2\ dx

uJ

x X V, --J H(V - xv) -- . (4.16)

In conclusion a functional equation satisfied by function H(X) is derived. In accordance with the definition, we have

H(s)

f

e1 W(z)dz

Cl

We multiply both sides of equation (4.16) by d-j/Az

and integrate

along

C1

to which contour

correspcnds

in

the

plane u. It is seen that in the double integrals obtained the

order of integration may be changed, and that owing to the equation

fe/AZ

dz O

cl

after integration those terms derived from e

and

of formula (4.16) and the Integral whose sub-integral

fimction

contains e.Z and

eZ(1)

as factors will disappear. The following relations are introduced:

K1

le

i e/z(u)

_{G(X,u)du F(X,p.),} ;iti _J Kl

I

1,z(u)

/ r2' du e

r' G,

-if)

-K1

so that the

unknown

functional equation

will

acquire the form

(4.18)

40 -00

r

i. i

-

F1(,)-c0()+crF2(p.)

--- f

F(X,M.)E(X)dX

2T1

21t o r

li

+ _--- F1(X,p.) H (X)dX + i/F(V,/.4.)

(y) - IVF1(v,p.) H (y)

27J

o i

I

dX - - v.p. F(v- XV,,IA.) H (y - Xv) --X -i

i

I

dX (4.20) TL J X

-c

in which all the functions (4.19) are assumed to be known and, in place of

T'

its expression from formula (4.17) must be put.

As a first approximation we may take:

r

HÇi.i.) = ---- F1(p.) - cF0(p.) +

cr2 F2().

21

This is equivalent to the replacement of W(z) in

formula (4.18) by the complex velocity of the motion of the contour

in an unlimited fluid assuming a given value r for the circuation

over the contour.

By putting (4.21) into (4.17) we are enabled to deterriine as a first approximation the value of r corresponding to the rotion of a contour in the case of a free surface.

Having determined function H(X) we find the corpex

velocity 'V(z) from formula (4.16). It is important to find the

form of the frei surface. According to formula (2.16) we have for

the profile of the waves produced

C

-

Re i;(x) (4. 22)

g

'e consider the profile of the wave at a long distance ahead of the body, that le, with large positive x and a long distance behind the body, that is, according to the modulus of large negative x. From (4.2) it is seen that

11m W1(z) = O

IzIx,

Further, lt Is seen from (4.3) that when x-+oo

WC

have:

11m. W2(xì = O.

X- +00

Conversely, "hen x- -oo we have:

-vr

_lvx

5lvt

11m 2(x) + -

W() e

----a dtdÇ j = O,

1 +00

and since according to the theorem of deductions

C

W()e1VX e1r

d= O,

27t1 e1',

we find 11m -or finally

fe1

'ut t

--00

1W2(x) +

2lf

dt = Ci

42

-w ( x) -, O when x -p +

-ivx

W(x) + 21'H()e

._ when

X-5 - oo.

U8ing (4.22) we find that far ahewi. of the body no

waves are formed ani far behind the body sinusoidal waves are formed

2cJ

2CV

Imag (H(v

-[H1(s.')einvx +

H2( v)cos'.'xJ

g g

(4.23)

where

H1(y)

and

H2(Y)

are the real and imaginary function

H( s'):

H(v

=

'') + ui( V).

(4.24)

The amplitude of the waves produced is:

2cVIH(V)J

2

a =

-1H(V)I;

g C

that is, it differs only by the factor 2/c from the mo1ulus of g

the principal function for the value of ) - where c is the

C

velocity of propagation of the waves.

5. Example. The Case of the Rectinilear Plate.

As an example of the application of the formulae obtained we take the case of a submerged rectilinear plate AB of length

21, with Ita centre at a depth h and inclined towards Its

direction of travel at an angle a (Fig.4). The travelling speed

Is denoted as usual by c.

(4.25)

The whole of surface z = x + iy, with a section through AB -te transformed to a profile of radius

i

r = - _(5.1)

2

lying In the plane u, and with its centre at the origin of the co-ordinates, by the well-known formula

r2e2 z

= u-hi+

u Using formulae (4.1) we calculate

=

21'U

(Irefl )k1

=

fe

Juj=r

and putting u =

1re't,

we find that functions Fk(M) differ

from Bessel functions only by the factors:

eM'

_f

le

Ikl

j

(Ire)

I u =r i e k-1 (2p.re

)

Since J_1(z) = -J1(z), we finally have

= 21!re1

_{e/1' J1(,te)}

,

2ie/' J0(te)

e1 ebh

JCL

tel)

r -We/

J

(5.2) r2e2 I4A. u-hi + u

Jdu

, u

u =r G(X,u0)

r

j 1.4 L = F1(X) - re F2(x), and. finally ici. 1

= 21e'

[i.0xte

- J1 (xte )j. .... (5.4) 44

We now determine the function H(J.4.) by formula (4.21):

HÇ, ) e {r'

(tei0)

_{- 2lUtc sin}

J1

(.te1a)].

(5.3)

We have still to find the value of r using equation (4.17). First we calculate the value of G(X,u0) from formula (4.10). Here u0 is the point of the contour K corresponding

to end A

of the plate and it is assumed that at A the velocity of the flow remains finite.

It is then seen that

u0 - re = T + o.. '-'o Hence

f5-iXz(u)

-- du u -

u0

du [u_hl + (1 r2e2

fe

- u2 u + re

Iu=r

/

¡

re

lX (u-hi +

_i

4

u

fi

re

e du u 2 2lcL

re

u

Fori du

which gives

For e1mplcity we put

r

= _(5.5)

2lUc sin a

and obtain the function

u0 G(X,u) E (x) =

- 1le

e2

27tc sin

[YJ0(Xle_1a)

+ iJ1(Xle) j [1Jo(.le) -

J(X2.ia)]

_(5.6)

00

Ç

i fL

e2' j (Xe)

J

(Xleia)dX

= _{- A (}

-m n

h o

i

V.P. f

e_2Vh(1

m

[v(1 -

X)le]

Jn [v(i -

X)lea1

dX

X

-cx

= a

and from (4.17) we obtain the determination of '(: ( X is assumed as real). We introduce 00

fe2

(ateic) _J o i v.p.

f

e2b_

m

[ab(1

-00

=

B(a,

(ate)dt

- X)ela] j b, a),

[a(i

-dX 5.7)

( t

t

+ sin A0 + -

Re(e A10)+2v7tte2h

h h

- 2V1U&2Vh Imag

46

-jj

(site) J0

(Vte)]

- 2%'t ein a

i

_la

-

2 vi 5(5ta B10)

h h

- i + - sin a A11 - - Re(e A01)

-2 Vh

+

2,ite

ces aJ1(V

le)I2

+

2vrTte2h

_Imag

[5ia

J(vte) J1 (Vte)]

- 2Vt sin a B

+ 2V1 Re(e

B01)

The functions A00 and A11 are obtained as Legendre functions of the second order, from the general formula5

+ c2)

o

whence lt follows that

1 /2 A

(a,)

= _--- Q

1-

+ eos

00 _7T.a 4

f

A11(a,a)

_Q(;'

+ cos 2cL). Tra

It should be noted that with Re(n + 1) > O

i

fa+b

(ct)dt _Q L Ç

5at

_{J(bt) J}

=

\

Q(z)

= i i r (1 t2) (z - t)' dt -1 and,1

5. Watson, G.IT. A treatise on the theory of' Bessel functions. Cambridge 1922, p.389.

and with arbitrary n other than completely negative we have in th region _{I z}

>

i, jarg z

k

r the series:6

fT

r(n+i)

i (1. 1 1 3 1

Q(z)

= 2n-i ( 3 F - n + - - n + 1, ri + - , -2 4J ' z

r

(F - hypergeornetrical function).

Functions A and may be developed as a series in increasiìg a, for example:

A00(a,a) e2t J0(ateta)

J0

(atea)dt

o

c \2k c

e2

(_1)k (atela _{(-i) (ate_1}

dt J k=0 klk: "s 2

J

1=0

::

: _j o 00 '

(_1)k+l a22l

k=0 1=0 k!kll

2k+2l e dt o 00 = (_1)k+l _{a2k+2'Z e21} (2k + 2l)

k=0 1=0 k!k11

r'

(1)Tfl

_(2m) 24k+41+1

e2(21m)

s'

e.. Thittaker, E.T. and Viatson, G.N. A course of modern analysis. Cambridge 1920, p.317.

5.10)

1T O i

ib

kk(m - k)'.(m _{- k)}

Similarly, we find

'cZ

(_1)m(2+i)a21»l

e1a(4k+1m)

A10(a,)

24m+3 _k=O

_{k(k+1)!(m-k)!(m-k)}

rri=O c 7

(_1)m(m+l)t aCm

A01(a,a)

A11(a,a)

=

B01(a,b,a)

=

_1)m (+2)!52m+2

2

-

48 -co _{/ m}

''

_'L

a2m

_(ob

B00(9,1D,a)

= 2m

E

It is readily seen that all these series are similar when J a

In exactly the same way the following series are

obtained:

e(41_0

k!k (m-k) (m-k+1)

e2h12m)

k(k+1)!(m-k)(m-k+1)

.J rn

e1a(41m)

(-1) _2rn+1 a f

(2b)'

4m+2 2m+1

e1a(412m)

2m+ i 2m+1

()

t'

_{kk(m-k)(m-k+1)}

24m2

(5.ii)

Ciri(2k-m)

(_i)m

aOm+O

r22

_(ob)

k!(k+i)!(m-k)!(rn-k+i)!

?11(a,b,a) _{= 4m+4} m=O 2

(5. 12)

in which x = e

v.p.

j

eU(x du

u

is introduced.

It is seen, however, that

e21a(2m)

X

x

U

xf1 (x) - e

e (x -

u)1 du

-

xt

(x)

m-1 o0 00 -

[e_t

_{tm_i dt = xf (x) (m} -J m-1 o and that f (x) e El (x), (5.13) f1(x) =

xf0(x)- 1,

and in general frri(x) in -x E11(x) ,m-1 -

i-2

-- 3,rn_4 - _-

(in -

_i) . (5.14)

Both serles (5.12) and (5.11) are In agreement when

Having found

I

the wave resistance la determined by

fbrnula (3.23):

Q=

but H(X) =

27c sin ae'1

[YJo(X1e1a) -

iJi(Xle)]

and hence

Q =

42 t2Qg

sin2 ae2'11 IIJo(v

- _{IJ1(vte1°)J2. (5.15)}

For the lift P using formula (3.24) makIng a few simple calculations we obtain

2T.Qt2c2sin2a r

P =

27c2Ysin

a.

- h

V2Aoo+21 Imag A10A11

+ 41tQgl2 sin2 a. B + 21 Imag B10 + B11]. (5.16)

we have:

7ÇL2C281fl2cL

₂

(

1.

L0

= 2

+ 2QIU

C2 sin2 a

- cos

a(A11+A)

- YvL 21e1n a

e2"1

te)I2

₊

J(v

e1.)I2)

- 2Vrcos a(0 +

B1) +

_{2e_2vhIJl(Vte1a)J21}

-

e2c2a1n2 a. Re

{.r AJ

2ThÇ1a

+

-- 2V%ie(B10--B01)

+ 2le2''

J1(Vtela)J0(Ve0.)

-

_2Vle2

e1a[J1(v1a.)J

(

vle1)0(

(5. 19)

In'

-00

(_1)m aT»l

_b

=

iD

2m

(2

Finally, using tormila

(3.28) we can deteinine the

moment due to the fórces acting on

the plate.

For convenience the

moment

L0

18 determIned In relation to the

centre of the plate.

Without going into the

detalle of the calculations we give

only the

final result.

Introducing the relations:

4 (a, a)

Çe_2t

J1(ate1) J0 (ate1)

(_1)fl(2m)a»1

m

e141)

nO

2»

ô k(k + 1)(m -

-i

_dX

B(a,b,a.)

v.p.

_Ç

e_2b(1_Jl[ab(1_X)e1a1Jo[ab(1_X)e_1a](

e1(412m)

-

(5.i)

k(k + 1)(rn -

-

k)

(5 1.7)

r when b h = oo , a = - = 0, we have: h

i

I A -, A = A A 0, At1 = Y) Ç) 1( 01 10 'J Q ß(1) Boo 0, B10 = B01 = B1,1 = ' 10 = 0,

and the equation assumes the form

'Ç

i, r= 2Ttic sin a.,

0,

= Tç.ic'sin a

i

(_) ()

= -i'J

c'J sin a. cosa = - - i cos a, that is: the known formuae for a rectilinear aerofoil are

re-established: in particular, it is seen from the equation for the moment, that the centre of pressure is situated at 1/4 of the

cilord from the leading edge.

For the aerofoil moving below a free surface we now

develop 'r, p, Q and L as series with progressions of the

Q p

parameter a = - , the expressions for '(, ---, --- are given

h P, _P

with an approximation of the second order with respect to terms

including a, and with an approximation of the first order with

respect to LQ/P

For this it suffices to have the expressions for the

functions A and as an ap-roximation of the first order with respect to terms including a. Elementary calculations with this approximation give:

52 -i a a -,

A10(a,)

= -

ei,A01(a,a)

-2 8 8

e',A11(a,a)

= o, -2b BD a,b,a) = f (2b) = e E11(2b), o a a {e_2'O _{2b E11(2b) - i]}

B10(a,b,a)

= - e f1(2b) = - e 4 4 a

Í-2b

4 Le 2b E11(2b) - i]

B01(a,b,a)

= - e

E11(a,b,a.)

= O,

A(a,a)

a =

-e

4 ab ae

B(aba) =

-- e f0(2b) = ____

be2b

Ei1(2b). 2

Noting also that

t

= h and that e ₀₁

a'e

that abe J1(abei2) = r)

after simple calculations we find from formula (5.9) when

i gh a -, b = .'h --h

c'-J0(abe)

= i 4

+ t..

i

i - a _{- sin a + cos}

-2'b sin a

f (2b)

2 0

i

+ a' - - cos

2a +

Ttb(b+1)e2b _{sin 2a}

t

e'

cos c 4 C + bf1(2b)cos2

a -

4T1j' e2b f (2b)sln 2a - Cb f0(2b)sln a + 4b2 Sin2 a f0' (Cb) 1 3

+aj__+_sin2a+b(2+b)e2bsin2a

4 4

+

4T2 b2

e4b cos2

a -

6b sin'

af

o

- 8TCb2 eCb f0(2b)sin 2a + bf1(b)

+ 12b2 f 2(2b)sjfl2 + _{(5. 24)'}

nd

for the moment of forces relative to the entre of the p'ate

r

L0 ₌

i

sin a fcos a - a [ - sin a cos a - bf

o

(Dl

- 2 e_2b _{sin a + ...} The/ (5.22) (5.25)

'e can now easiy calculate the wave resistance by formua (5.15)

Q =

4T

i2g sin2 ae2b

i + aI(b - 1)sin a -

4be_2b

cos a

+ 4bf0(2b)sin

] +

(5. 23)

For the lift, formula (5.16) gives the value

P =

_2lec2sin

a

The distance of the point of application of the resultant

of the forces p and from the centre of the plate is determined by the formula:

54

-with an accuracy of the first order -with respect to a we obtain:

r' cl

-

i + al-

1n

Ic

-2b _{- 2be2b E11(2b)sin} ai +

..4

..

(5.27) + 27tbe cos

_a

When angle a is very small the formulae become much

s imple r.

'Je now develop the formulae with an accuracy of the second order with respect to terms including

a

/1

r

2lca

ji -

a[2be2b

_{+ a} ( - -

2bfo(2bj

+ a2 + 47t2

b2

e-4b + bf1(2b) + a

(n(i

+

b)e2b

- _b2

e2t

f (2b+

o

Jj

=

4gt'a2

e4b

i + a [- 4be2b +

(b i + 4bf0(2b)

i

P =

21t.tEc4

a -

a[2be2b

+ a

(i

-

4bf (2b)'

)ja

L C 2 -4b + 4 b e + bf1(Cb) +

a(2b(2+b)e_2b

-i '1 d =

_{1+a2be_2b+a(__2bf(2b))+....

Ç,

°

5.28) Cne/ L d o (5.

46)

P cos a

+ Q sin a

i

4

L-One more 8pectal case must be considered, namely that of motior

at very high velocities, that is, at very large values of

C2 i

gh b

For this we assume b O in the previous formulae. Whereby they become greatly siriplified and we have:

i i =

i--aaincL--a2c082a+...

2 4 Q 47

gQt2

ein' a i. - a sin a + i 2 2

p.

2e2 sin a

- a sin a - - (1 - 3 sin a)a +

4

a

= -1 + - sin a + ...}.

2

6. Calculation of the Vlave Resistance and Lift in Three-d1mensiona

Problem.

We now proceed to the calculation of the wave resistance by a surface S and moving below the free Burface in the direction

of the positive axis ox at a speed c. It should be recalled that In the three-dimensional problem, the oz is directed vertically upwards and that the co-ordinate eystem'R x, y, z become displaced

with the body. The potential of absolute motion p(x, y, z) must satisfy the Laplace equation:

----+--- +---

= 0, (6.1)

x y z

with/

56

-i n/

with the boundary condition

Oq,

e cos(n,x) ut s

(5.)

the boundary condition at the free surface

o2 + V---- = O when z = 0, (6.3) òz where g = - , (6.4) C

and finally the condition that the derivativa

p/òx, ò/òy,

are all bounded at infinitely distant parts of the lower half space and approach zero when x .- +oD . As in the two--

-dimensional problem it is assumed that such a solution exists and

that it is unique.

We take an arbitrary point P(x,y,z) of the ower half space, outside surface S, and two surfaces _S and S

encosing

surface S in such a way that the point P(x,y,z) lies outside

S1 and inside S

Applying Green's formula for functions (x,y,z) we have: i i ò-p(x,y,z) = -

1'

ò r

---dS+

41t i r n

;;T

-

dS si s1 i

r

r

i

i'

J r

----dS---1 ---d3;

47t

rn

4flJ

òn

S S

in which z' i tue distance between points x, y, z and the

variable points of surface or S and n the direction

of the outer normal to one or the other surface.

It la seen that p1(x,y,z) i

o-i

iO

i

1' r I

- ---

dS + --- t --- dS

4T Jròn

411. Jòn

Si si

is a harmonic function In the whole of the space outside

on the other hand

i 1 a-i. t

i

i.

I

r

=

---

J

-

---dS -dS 4r

r On

41T

ôn

So0 00

Is harmonic function within

Thus (x,y,z) is developed as the combination of two

functions

p(x,y,z) = 1(x,y,z) + 2(x,y,z) , ... (6.7)

one of which is harmonic outside S1 and the other within So0

It is seen, however, that when S is Increased the value of

2(x,y,z) remains unchanged at point (x,y,z); similarly with

decreasing S1 the value 1(x,y,z) at point (x,y,z) remains unchanged. We thus obtain the function

cp'(ì,y,z)

in the combined

form of (6.7) and C(x,y,z)

_{Is a harmonic function In the} whole lower half space.

We now obtain function 2(x,y,z) in a different manner.

The function Ç1(x,y,z) _{determined by (6.5) is represented by}

the/

.... (6.5)

r =

J(x

- + (y- + (z (6.8)

We form a function g1(x,y,z;

Ç)

of the independent variables x,y,z which is harmonic In the lower half space since

it Is know-n that

i g(x,y,z; ,-r1, )

=

-+ g1(x,y,z;

Ç)

satisfies the boundary condition

2g

Og

+ V

O when z = O,

òx

Oz

and the derivatives

òg1/òx,

g1/ôy,

g1/òz,

are bounded at the infinitely distant part of the lower half space and approach

zero when

X-4+ 00.

For the doublet

I i i i

a-

o-

ò-

o-. r r r r = -- cos(n, + cos(n,

r1) + ---

cos(n, );

an analogous function g2 is obtained from g1 with the formula

g(-)

58

-the

combination

of the potentials of a 8irrtple layer and of a double

layer, in the form of a continuous distribution of doublets and

source

8.

Let the source be at point Q(,-r,

) in the lower

half space and let

Òg1

òg1

òg1

- - cos(n, ) + -s--- cos(n, r) + .--- cos(n, q').

We/