The forces and moments acting on a body moving in an arbitrary potental stream

NAVY: DEPARTMENT

THE DAVID W. TAYLOR MODEL BASIN

WASHINGTON 7, D.C.

I THE FORCES AND MOMENTS ACTING ON A BODY

MOVING IN AN ARBITRARY

POTENTIAL STREAM

by William E. Cummins June 1953 Report 780 Th

THE FORCES AND MOMENTS ACTING ON A BODY

MOVING IN AN ARBITRARY POTENTIAL STREAM

by

William E. Cummins

REFERENCES TABLE OF CONTENTS Page ABSTRACT . 1 INTRODUCTION ... - . ' 1 ASStJMPTJONS . . .; _{* 2}

- EFFECT- OF TRANSLATION OF AXES .. - 3

HYDRODYNAMIC 'FORCE. ., . 6

THE "LAGALLY FORCE",F1 11_

THE FORCE F2

....

. . . 18

TIIE.FORCEF3 .' , 20

HYDRODYNAMIC MOMENT

-

22

THE "LAGALLY MOMENT'; N1 . '

..

23

THE MO ENT DUE TO ROTATTON,M3 . '.

..

.

..

. 26

MOVING SINGULARITIES . . . .27

P.OTENTIA1 OF THE JNDISTURBED STREAM3 . . .. . . :29

SOURCESAND.DOUBLETS ' .

34.

CONCLUSION . .: '

36

ACKNOWLEDGMENTS

-;.

_:3

APPENIX.1 - EVALUATION OF INTEGRALS IN F1AND W1 . 37

INTEGRAL IN F1

..

-. 39

INTEGRALSl(s,s±1)

. I :. 41

-INTEGRALS J(a, 8±1)

..

: 42

APPENDIX 2 - SUMMARY OF FORMULAS . 43

ns Xs

_,,

NOTATION

A Vector doublet strength o' a singuiarity

A' EuivaIent vector doublet strength of a moving singularity.

'ii

'S

Coefficients in the expansion of a potential in spherical harmothcs .

Coefficients lii expansion far singularity equivalentto

moving singularity

Net hydrodynamic force acting on surface S

- Force on-S in a moving system .

"LagaUy forCe" on S ,

Additional force on S. due to' change of' the flow with time

F3. AddItional force on S due to a rotation of the body

$

F1(i), F2i), F3'i)

' Force integrals evaluated over

Components of F1(i) parallel to the

; y, i-axes

i,j, k

' ' Unit vectors parallel to the z, y, a-axes

i(p,.p±1)

Integral appearing in F1(i)

J(p, p± 1)

Integral appearing in M1(i) ' ,"

Hydrodynamic moment acting onS

Mm Moment on iii a moving system

M1 ' "Lagally mOment" on S '

-M2, _{Additional moment on, S due to change of ,the flow with time}

M3 ' Additional moment on S duO tO a rotation of the body.

M'1i), M2i) . Mdment integrals evaluated over

'(i), M17(i), M(i)

Componentsof M1(i) parallel to the; y, a-axes Momentum of a mass of fluid

"ft

_{Moment ofn,omenturn of a mass of fluid}

n. Unit normal vector to a surface (directed inward to 5, OutwardtoS')

q8 R R

s', s'

Si SO, S) .yo T/' iv Fluid pressure

PrOssure in.a moving system

Associated Legendre functIon .

Fluid v.elocity . .

Fluid velocity relative to the origin of a moving System Fluid velocity in the undisturbed stream

Position vector

Position vctot of the origin of a moving system

Fm Position vector: of a point referred to a moving system. rg Position vector of the ôentroid of the body

r1 Position Vector ofhe singularity (i)

Distance from Ithe point cr1) ..

Ra4ius of tie sphere S

Position vector of a point referred to (re)

Position vector of moving singularity referred to (re)

Surface o the'b'ody

Control surfaces surrounding the system of singularities

Sphere with center'at (r1) .

Unit ectors in polar coordinate system

Time .

.

Velocity of the origin, of a moving coordinate system

Velocity of point fixed in the body. '

Velocity of moving singularity relative to tb.bódy

Space occupied by the body :.

Space between S .and S 'and 8" respectively . -.

V2

vi

z, :v, 2 xo, ?/' Zo 'm' 2rn 1(s) 0, A p

Velocity potential due to the presence of the body

4) . Net velocity potential Of the flow

Potential of the velocity qm

Angular velocity of the body'

V

Space within Sd and exterior to S

Space occupied by a given mass of fluid Space Occupie,.d by

Volume of S

Rectangular. coOrdinates

Coordinates of the origin of a moving system

Coordinates relative to a moving system

Coefficients in the expansion of the potential of a con-tinuous distribution of singularities

-,(0) =2;s)= 1,8 >0.

Space polar coordinates

cos0

Mass density of the fluid

'References are listed on page 48.

ABSTRACT

The force an4 moment on a body placed.in an:arbitrary steady potential

flow were. found by Lagally when the bod.y can be represented by a system of

singularities interior to the surface of the body. They were found to be.si.mple functions of the strengtls of th singularities and the character of the undis. turbed stream in theneighborhood'of the singularities. In the present paper, this result.is rederived and extended to the case. in which the, body is subject

to.. an arbitrary non-steady motjon (includjng. rotation) in a stream which 'is óhang.

ing with time. The force and moment are found to be the "Lagally force and. moment" plus additional components. These additional components are given for the force as simple functions of the singularities used in establishing.the

-boundary condition and of the motion' of the body, but an integration over the sur-face of the body is required for the' moment.

INTRODUCTION ..

The determination of the force and moment acting on a body placed in a non-uniform

po-tential flowof an ideal. fluid'has receive4, considerable attention, the problem being of

consid-erable. importance. in both aerodynamic and .hydrodynamic applications.

There have been two essentially different approaches 'to tlke question. In the first, the

flow is assumed to be only slightly non-unifotm, and the dynamic .action is, found. in 'terms of

virtual mass Thus the problem is reduced to the case of motion in a uniform stream Lord Kelvin1 solved the important special case of the sphere as early as 1873, but G.l. Taylor'3 was the first to make an extensive study of arbitrary bodies His analysis, which applied to a steady state system only, included some discussion of the moment. Tolirnien developed a' solution for the force and moment'in terms of the,"Kelvin impulses" and extended the discus-sion of force to include the case of uniform translation in a steady non-uniform stream.4 These results have been rederived by Pistolesi who found an error in' Tolimien's formula for the

moment. .

The second approach considets. the boundary condition at' the surface of the body to be

establiehed by means of a system 'of singularities within the body. The force and moment are then found in terms of the strengths of the singularities and the character of the basic stream in the neighborhood of the singularities. Hence, this method is not restricted to slightl,ynon-uniform streams,but is limited to those cases in which a suitable system of singularities can be found which simulates the presence of the body This is the approach used in the present

2

Munk wasthe first tofind the forceacting on a body generated by sources.6 Lagally apparently solved the problem independeiitly. about the same time.7 Since Lagally's treatment

was far more comprehensive and included a discussion of the moment,, the statement.of the force and moment-in terms Of the si'ngularities has come to be known as ":Lagally's Theorem."

Glauert applied the method to the study of bodies 'in a converging stream in order to find a correction for the force on a body when tested ma wind tunnel with a pressure gradient.8 Betz derived the torce and moment with a somewhat less mathematical approach than Lagally and presented the 'results' in a very convenient form.9 Mohr discussed distributions of singu-larities over the surface of the body.1° Brard has recently attempted to extend Lagally's meth-od to unsteady flows' but was unable to present formulas of the same simple type as those which hOld for the steady state case.11 .

It i's evident thatif a singularity distribution is known which establishes the boundary condition, the flow is completely determined, and, in principle, the force and moment can be immediately found by integrations over, the surface. However, in addition to possible diffi'-áulties in perfqrmingthe'integrations,' the fact that the pressure is a nonlinear funãtion of the potential is a' severe limitation. It is desirable to be able to superimpose known flows to ob-tain new flows and to obob-tain the resulting force and moment in some simple manner. For

-' steady flows, Lagally's theorem provides just such a formulation. '

In the present report, Lagally's theorem is rederived 'for general singularities, and the -analysis is extended to the case of non-steady streams and non-steady motions of the body

(ro-tation as well as translation). The force and moment are found to consist of the steady state "Lagally force and moment" plus additional components due to 'the changing flow. The addi-. tiona force is stated in simple form in terms of the strengths of the singularities and 'the

mo-tion of the body, but the moment is found to require an' integramo-tion, over the surfaOe of the body.''

However, in the latter case, the intégrand is a linear function-of the potential, permitting the

superposition of known flows. ' .

-.ASSUMPTIONS,

The velocity field is 'irrotational and has a velocity potential cbfr, l' , )

,If the body were not present, .the.stream would have a velocity potential ç, which we call the potential of the "undisturbed stream." :

- _{3. There are no singularities of the} undisturbed stream in the region occupied by the body. 4. The boundary condition at the surface of the body is satisfied by.superimposing a sys-tem of singularities upon the undisturbed stream, such singularities falling within the region

which thébody would occupy. The potential of the system of singularities is designated by

Then ,

' ' '

EFFECT OF TRANSLATION OF AXES'

A point fixed in the body js selected as the origin of a moving system of coordinates;

the axes remain parallel to a second system which is fixed in space (see Figure 1) The posi-tion vector of any point.of space with

respect to the stationary system is designated by r and with respect to

the moving system by Fm ,The

posi-tion vector of the moving origin is r0, and its velocity is v0 The following

relations hold:

r-r, =r,

= [2] = Ym = 2m

where the subscript rn. refers to the moving: System.

The velocity of a fluid particle with respeet to the moving origin is related to its abso-lute velocity by :

=;q- v0

where q is the rlative velocityand q is the absolute VelOcity. Since v0 is a function of time,

only, it satisfies the identity

=

orm)

where Zm aYm m Therefore

q=-V+

Vm(Vo rm) or, Since =.v Figure 1

'1.

ô()

1'

From Equation [3]

and

.: '.

q='_v(+v0r)

Hence q satisfies a velocity potential which we designate by ct, It is related to s1' by 71m' = (Zm + Zo, o' + t) V0. [31

The pressure at any point, not considering the gravity field and an additive function of 'time, is a". m' dv0 arm

at

VmIm '

--But .rrn dr, _{= - V} a t dt so ä dv

_'=q v +__.!

+v v.

at rn 0 dt rn

o

0

Substituting in Eüation [4]' and collecting terms,

p = _'rn

in which a term containing v . has been dropped. This is permissible, sincev0 'is a

func-tiononlyof. time, and the net force or moment due, to a constant pressure acting on a closed surface is zero.

If the. velocity field q' were to be considered absOlute rather than relative, and the pressure were calculated accordingly, we should have _:.

.1

am

-1p q

qm

_at

so we can write

'dv

PPm.P

rrn

By'means of this relation, the flow relative to the moving axes can be considered as' if it were

the actual, flow,,and the forces and moments so pbtained can be converted to the true values.

Thus, for the force exerted on a g'i'ven surface 8,

[:5]

where fl is the inwardly directed unit northal to S. By Gauss' theOrem

and since

where-jis the Volume of S. Hence

F

:Fm+p3z.;Q.

Similarly, for the moment about the origin of the moving system,

By Gauss' theorem again,

and.we have B

)

,dv0

_\

_dv

VI----.r I=.

_\dt

_mj dt / V

n.da-

I

(dv0 r dt d.v

f

I di--dt J V

)fldo

-. dv dt

rmx nda

xndo- p

j

rm) Fin x nda

1 fdv

/d0

\

)

r,,j =-r,,,

x V_-_2

r)

k- r,p7.x

Fm

=rrn

{81

where' rg'is the centroid of the body relative to the origIn of the moving srstem. Therefore

since V x 0. Therefore P /:dy _dv C Idv r.

)r

in .m

xnda=-2x

I

rdr=(-_--2x

r)

i

dt

dt.

j

dt

V.

-Figure 2a 6 = r M5 Mm+p( dv &

dtl

Thus the problem has been reduced to the case of a body at rest in space or in rotation about some point fixed in space. This is the case which will be considered in the remainder of this paper.

HYDRODYNAMIC FORCE

We suppose the singularities generating the surface of the body to be enclosed at time t0 by a control surface, 5', which everywhere lies within S (see Figure 2). At present we

Figure 2

specify only that S'possess a clearly defined nornl at each point. This control surface is

considered fixed with respect to S. The portion of the body between S and S'is designated by

V ' The body being in rotation, the space occupied by V 'changes with time, and we designate

this region by V'(t). In the following discussion we consider the particular set of fluid parti-cles which at time t0 occupy the region V'(t0). Since the fluid is also in motion, the region occupied by this set of particles is also a function of time, V1(t). By definition then

V'(t0) Vf(t0)

but at any other time, in general

V'(t0) V1(t)

Figure 2b

{9 I

The net force acting on this set of fluid particles at any time is dfdt wherem is the

F5 ₊

£

0

where F5 is the net force acting on S and F5 is the net force acting on S'. Since

F5=JS

atJ and SVf

pqdr

we have 7

Fs=J;P(.)fldcr_

_f

p --nda+

JS'

which is precisely the force in which we are interested. At a point fixed with respect to the body,

= V V +

dt

_at

where v is the velocity of the point. As the origin is supposed stationary, and is also fixed with respect to the body, the point being considered is in the most general case in rotation about the origin, so

V=jxr

[12]

where is the angular velocity of the body. We have

I

.35. 3t dt

We can write

f pflda=_J

dt dt

_s'

pndc- f pt-!!da=i f pnda-ø

.. dt dt

p q d,-f

[10]

since - _{= 0 X fl} dt Then and aiid 8

r

_{actflda=..$ pndueox $ pcbndu+}

ç(q.oxr)ndo

[13]

dt

_s'

The last term of Equation [10], the time derivative of a volume integral whose bounds are changing, must be converted into a more convenient form. At the time t0+&t, the initial

bounding surface S will still be a bounding surface, but it will have rotated by an amount t about the origin. The surface of V1 which coincided with S' at time t0 will have become some new surface, S"(see Figure 2a). The portion of the body between S and SI' is desig-nated by V '. At time t0 fl2(t0)

=f

_V pq(t0)dr At time t0 +

6t

pq(t0+8t)dr

V

om=f

pq(t0 +

_{8t)dr -$}

pq(t0)dr

The two surfaces 8' and S" are considered fixed with respect to the body. The portion of

the body interior to 5' and exterior s" is designated by V1 and the portion interior to 5" and exterior to S'.by V2 (see Figure 2b). Then

V"=v'.+vl V2

am=f.

V'(t0+ôt) pq(to-i-Bt)di-_JV

pq(t0)di--$

)Pt0atdr

V1(t0+t) V2(t+&t)

9

The velocity of a fluid particle on 8'-relative to the body is

q+rx

Therefore, the normal distance between S'-and 8", the amount the control surface is deformed relative to the body in time 6 t, is

l(q + r x o)

"I 6t

Accordingly, in the expression for 6Tfl, we can-write for

di-dr=-(q+rxea)n&tda

mV1

-

dr=(q.+rx)n6tda

mV2

where da is taken on S'. Then, since the portions of 8'-which bound _{and V2 complement} each other,

(f

_V1(t0+6t)

-

f

_V2(t+8t)) p.q(t0 + 8t)di- ₌

_-$

P{q(to

+ 8tq n)

S'(t0+ôg)

+(rx.e.

n)q(tO+6t)]otdc7

Substituting this in Equation [14] and allowing ôt to approach zero, we have

=--

f

pqdi-

-

J pq(q

n) do-

J

p(r x i' n) qda

[15]

dt dt

_'

_'

We can further reduce the volume integral which appeari in [15], since

S-"

$Vdr=P

by.Gauss' theorem. The unit normal can be written

By Green's reciprocal theorem

az ay

_{k_}

n = I - + I - +

an 4'an

an

$

nda

s+s'

do- I

zdo

$

an

an

s+s'

s+s'

since is regular througH V'. TherefOre, since,

r =.ix -i-jy+ kz-- we have

But on S we have the boundary condition

so Using this, since and Therefore, Then. qndO-= s+s'

pqdr=-Jv.

10 1. r.a_1?.dc,.=_p J_s+s' an n x r

n

,. r.(q

n)da

s+s'.

Jr(, xr

n)da._J

r(q -S S

Tlie first termon the right can be easily reduced. One formresultingfrom Gauss'theorernis, (Reference 12,p. 52)

-jci(n.

b)dor.=_j

[a(.b).+

(b

.V

fr(,ixr.n)da-i(rxea)dr=(rxoYV

J.

Jv g S

v(xi.(Vxr)-r(Vxgo)=P

(rxob.V)r=rxo:

=(rg

x

')'f.wrg

.ø)r(ci.

a] p!(q

n)da

[171

sinc

= - r )

(o-dt

Summarizing, when We combine Equations [10], [131, [15], and [17], we have

J

P[(

q)n (q n)q] dci .fl) + cbri] d

r

_f

p.[(r x . n)q (r x q)n + (n x _[18]

Fr x.!fto(r

có)+r

_(riw)1p$

L _d g

g.

The first term in the above expression would give the force if the body were not rotat-' ing and the undisturbed stream were-steady, i.e., the "Lagally force." The second term is due to the change of the flow with time, and the last two terms arise when the body is in rota-tion. Since -these various components will be discussed separately, we -call them F1,F2, F3,

respeótively. ' - - -F

q)n(q .

n)q]7

_[19'al

-f.[r(q.

ii) +

n]d0.

[19bj

F3=J[(rxei. n)q._(rx.q)n(nx-)Jda

[19c]

_[rgx_

(rg +rg( . _.)I1P*

If the origin of the system of axes is taken to coincide with the centroid of the body, the last term of the expression-for F3 vanishes.

The above forces are defined in terms of integrations over the control surface 8'. Since S'has not, been specified, it is evident'-thát the forces'.are. independent of the particular choice of S', as long as it satisfies the conditions necessary for the integrations 'to be carried Out.

ThE "LAGALLY FORcE", F1

Initially, we suppose the singularities generating the body 'to be discrete, isolated, and fixed with respect to the body. Their locations are designated by the set

&

position vectors r.. For the control surface, we select a set of spheres S' with their respe9tive centers at the

singularities and their radii R chosen sufficiently small o that no.two spheres overlap. -We designate by F1 (i) the integral in evaluated over the sphere S. Then

- F1 =F1(ij

.

.. [201

Since F1 is independent of the particular choice of S, F1(z) is independent of

We refer the region around the singularity at r. to a system of space polar coordinates with the origin at r1 (see Figure 3).

Figure 3 cm + ct.0s0

+ sin0

12 [231 [241

dci = R2 sin OdOdA : _[251

q q =

11/

n + 1 sin2 0 L\

z_x=l?

ôos.0 [21a1 - y. = J sin 0 cos A [21b1

0-

: =Rsin0 sin A

[21c]

The quantities appearing in Equation[19a] may be written

where

,

n -,

....acb

L!

ae

and and are the unit vectors in the (R = const., A =const,) and (R = const., 0 = const.)

directions respectively:

S0=_i _{cos Ocos x+k cos OsinA [22b]}

=

jsinA

+kcosA

_[22c1

The components of the force parallel to the i,

_j,

k, directions become:

2ii ir

F1(i)=f

_f

13 E-(Rcb)2 + cos e+ cos C 0 sin20

+ 2R; I,,I sin O]sin OdOdA

2u ir

F'1(i) =.f0 J

sin2 0 cos + 4 sin2 0 cos A + ( cos ,

2R cb b sin 8 cos 0 cos A

'no

4' 4 sinAl dOd A

j

, 2n.T

Fiz(z)_..J

J

OsinA + sin2 8 sin A+ 2 sin A

A

- 2R _e sinO cos OsinX

-

b cos

A]dOdA

In the region o <1? <

I - r. , where r3 is the position vector of the singularity nearest

r., the potential is analytic, so it can be expressed as an expansion in spherical harmonics

which converges throughout this region,

n

c1 '1?

Pz)(a cos sA +

sin sit)

n=o s=o _[27]

where j = cos 0, and the'P(jL) re the associated Legendre functionà. In this expansion, the

first double summation, in which 1? appears toa positive power, represents the potential of the undisturbed stream combined with all the other singularities within S and outside Si, and the second summation, involving R to negative powers,represents the potential of the siñgulaiity at r The first summation is convergent for 0

_{R <r - r}

and the second is convergent for

allR>O.-Thö functions of .which appear in,Equation [26] are:

n (a,cos. S + sin s A)

I

n=o s-=o 1Z=O .S1

gin±i )n

d

(n

i)R"'

cos sA + sin SA)

cos sA+b8 sin sA)siñ 0

II"

,(c7,in sX b,cos sA)

sin 8A bcos sA)

When these are subtituted in Equation [26], the resüling expressions become quite cumber-some. However, since F1 (i).i:. known to beindependet.of R, it is evident that the net co efficient of R must Vanish unless t 0, so. only the latter terms need be considered. A fur-ther reduction can be made by taking account of the integrations ith respect to ). since all

terms contain products of the type cos or _{sAl I} or _tA sin

_{J L'}

-[28a] n=o s=o

-'

'i._f" d sA

bsinsX)sin0

no. so

For 1(j) a term must vanish unless this product is of the form sin2 sX or cos2 sA. For the

other two components, there is an additional factor, cosA or sinA. Those terms with cosA

vanish unless the above product is of the form sin s A sin (8 ±1)A or cos s A cos (s ± 1)A. Those

terms containing sinA vanish unless the product is sin sAcos(s±1)A or cossAsin(8±1)A.

The components can now be reduced to:

where dPS dP8 + n+1

(1

n + 2 p5 1

+(n+1)(

dPs _dP PS

_{____}

n+1 pS ___!!.1(1 12 n+1 d 00

fli

F17(i) = -! +1 + 12=0 _S=0 12 +

(s)(a'

n+1 na5 + n-i-i )

!(8

+ 1, s)

50

00 r n=0

_L='

-X

(s)(a_{n+1 n n+1} S =0 15

a,) J(s + 1, s)

(0) = 2; 77(8) = 1, 8> 0 b_n+1

6S+1)7 (s.s

_fl

_n'

_- 1) - b a1)J12(s, S + 1) n [29a] [29b] [29c)

F(z)

'rp 00 fl + 68 6S

)f

n+1 n

j

jr

1(n + 1)2 pS₁₂₊₁ pS 12 -iL 12=0 S=0 1T 2

F1z()

r

and I(p, p +1)

-=n.

p p±l Pn+lPn

_{[(1_2)(n+1)2+P(p±i)(fl+i)(2P±i)]}

( , dPP±

p±idF1\

Pn + (n + i)(] -

_1n+1

d /

+dP+l!±1

,L2)2} d ____

(1-d d

The convention is adopted in Equation [29] that

1,0

-

1,0 _{- o}

n n

These integrals are evaluated in Appendix 1. The components of F1 become

n E1(2) = 2np _fl+1 S +_{n 'i+}

1=0 50

a

'77(3)

iS+1 I n+i

50

+ n 16 + S+1 _{8+ 2)!} n+i

a)

_(ns)!

8 + 1)! (n

-2

(s)(as+1

_{s_bs+1 s(n+s+2)!}

n+1 a n+1

a,

,

(ns)!

These components must be evaluated at each singularity. is then given by Equation [201.

The Equations [301 are bilinear forms in the coefficients of the expansion of the total potential, excluding the singularity, and coefficients of the expansion due to the singularity. The bilinear character of these expressions has a number.of important consequences:

1,S

J(ns--i)!

n+1 n 63+1 ₆ s+1) (fl + 8 + 1)! n+1

(nsi)!

[30a] [30b] [30c] p

i-'i!

i(s)(a+3+1

n=o 15=0

A inguiaritycanbeoonsideted tO be composed of a number of superimposedsingulári-ties, 'i1), ('i2), (i3) and the forces F1('i1), F1('i2), F1(i3) ... determined independently.

Then

F1(i) = F1(i1). F1(i2)+F1(i3)

Similarly, the Fiotential xcludingthe singularity can be considered to be composed of a number of superimposed potentials, and the force due to the interference of each of: these with the singularity can be determined separately and F1 (i) found by addition.

Consider the net fOrce on the body due to the mutual interference of two of the singu. larities withinS. By 2, these forces can be determined without consideration of the effects

due toall other components of the flow. Instead of evaluating these forces separately over

the spheres S and S, let the integrhls e taken over a larger sphere S. witi its center at

r and 1? > r1 F1 _. The combined potential may be expanded i a form such as Equation [27]

which will be convergent for > - r1j. However, since the combined potential must vanish

at infinity, all of the unbarred coefficients must be zero. Since'the integrals will have pre-cisely the same form as Equation [291, the components must be zero due to the biliüear nature

of Equation [30].

- 4. In evaluating 'Equation [301, the unbarred coeffiCients may be determined for çl, the potential of the undisturbed stream only, rather than the total potential exclué]iig the singu-larity at r, sinceby 3the net force due to the mutual interference of all the body generating singularities is zero.

5. In the case of ontinuous distributions, we may suppose.thC region over which the

sinularitiesre distributed to be subdividédinto small elements. The net potential

of the portion of distribution within the element 4 r, containing the point r can be written

b

'4'g_(n+1)ps(,)(&s

cos sXi-

sin sA)

4i--

fl0 30

-which converges for all greater than the -maximum distance from the point r1-to the bounds

of i- _{This has the form of an isolated singularity at r1. Hence Eqjiation [20] canbe written}

b, c. 3)

If the number of elements is increased indefinitely, the dimensions of each approaching zero, then the coefficients , f3 will in general approach limits, and the sum becomes an

inte-gral.

17

18

The corresponding formulas for line distributions and surface distributions are immediately

evident.

THE FORCE E2

The same general procedure is followed for F2 as for F1. We again consider 8' to be composed of a set of spheres surrounding the singularities, and define

F2(i)= r(q . n) + n] dci [31]

so

If we make the substitutioii

[31] becomes

F2 = F2(i) [32]

r = + R [331

F2(i)=

_-

p1J

R(q. n)dci +

rJ

(q _{n)dci +J}

nda

dtl

s s s

L

Remembering that these integrals are independent of R, it can be seen from Equations

[23], [25], and [281 that

JR(q

Si

can involve only the coefficients a', a, b. These are

the strengths of doublets with their axes respectively parallel to the x, y, axes. The potentials of these douLilets are

a7coso asin9cosA

bsinOsin

If we regard these coefficients as the components of a vector, this vector will have the direc-tion of a single doublet equivalent to the three doublets, and its magnitudewill be the strength of this "resultant" doublet. We designate this vector by A, nd call it the vector doublet

strength of the singularity. The potential and velocity field of a doublet in terms of its vector

strength may be written

and

)n.A]=3(A .R.)R._.A

R3 where. n has, the same meaning as in Equation [22a]. Then

and by Gauss' theorem,, reniembering thatn is directed. outward

fR(qafl)do *:f, V(A.R)dr

=J_ I

A.dr

R31

-

i.,vi

3 A [36]

SimilarlYJ' nd&depends only upon A, so

J ¶fldcr,=_Lf(A. R)nda:=L!TA

(37]

j3

. 3

-by Equatiøn [34]. he reai5ing integral q.. fl do 4epends only upon

and is sip1 the.

total. flow from a source of stzength a, so

jq. ndri=4'?r

dr

Then, since

_{-= - r. x i}

dt

F2(i)= 4irp. [

-

(r1 x +

The extension to continuous distributions is evident:

=

_2TPj[_

_{o' I}

+

'dt

dt

[391 -[3(A 19 fR(q _{n) do =f}

2(An)

R

A R)nda

-[35] [38] [39].

Then

20

THE FORCE F3

We define

F3(i)=_J

p[rxw n)q(rxo q)n+(nxo)]dc

[401

Si

F33(i){r

_g x

-

(rg o) + rg(tai [41] dt We can write

Jp[r x

. n)q (r x

. q)n]da

=J p(r x

x (q x Si _Si [421 and by substitution [331

=J

x o) x(q x n) +(R x w) x(q x n)]dci = p(r w) x

x n d _JP(R x

. q)nda

since R x n = 0 on S. We evaluate these integrals separately, again taking advantage of the

fact that they are independent of 1?. It is easily seen from Equations [23], [25], and [281

that only the term with the coefficient can contribute to q x n da

But this term represents a source, and for a source, q x n must vanish on S. So

x nda = 0 [431

The integral

q)ndo

can similarly be seen to involve only terms with coefficients , and b1', namely the vec-tor doublet strength A of the singularity. From Equation [351

21

R3s.

Using Gauss' theorem (remembering that n is directed outward from S)

J(R.

x A)flda

=JV(R.

x A)dr

=1 (wxA.V)Fdr

J_vi

(ioxA)dr

vi =-I /i13(i1 >< A) ,. q)ndg= --lT((o x A) [441

[r

x h)q - (r x )n]d a =- (to x A)

By Equation [31],the remaining term of Equation [40] can be written

f(n x 01)d7-= -

x J b ndc,= -

x A) [46]

Si s 3

Combining Equations [441 and [46] we have

and F3(i) 0 F3

-

_[rg

x _to(rg ')

rg(ø

dt [47] [481 and so

- or

HYDRODYNAMIC MOMENT:

Up to a certain point,-thé development for the hydrodynamic moment is exactly parallel to the develøpment forthe force. -The net moment .actingon--a given.thassof.:fhiid j5.:

- dt

wherelfl is the total moment of momentum of the fluid about the ôenter.of thoments-(in our

case, the Origin). Then

22

fp(r.xn)dci=-f

tv

M

=_f .p(r x

n)do+ff

p(r x

which can be written

qrx

p

2±r

x

nda+-_f

& xq)dr

. [49]

s : -

s'

aè (It v

The Second term of Equatidn [49] can be rówriUen as before

xn)dof,P(

xr)(r xn)da

p(r x

n)do

xf

pF(-r x n)da p(q xr)(rx'n)dcy

dt

_s'

_s'

_s'

The last term of Equation [491 can be transformed to

2_m=i_f

_{p(rxq)dr_f}

p(rxq)(q. n)da

+f

p( xrn) (r xq)da.

- [51]

dt -dt _v

.

s'

.

then

23

which is analogous to Equation [15]. The volume integral can again be transformed into a surface integral,

fp(r x q)dr=

Vx

(rb)dr=f

b(r x

v'

v's.

With this step the correspondence between the two developments stops, for the surface inte-gral in Equation 12J cannot be transformed by means of Green's reciprocal theorem, as was the surface integral in Equation [161.

Collecting the results in Equations [491, [50], [511, and [521,. we have

Ms._j[(

. q)(r x n) - (q n)(r x

q)]dcl +/j.f

pcb(r x n)dor

[53]

x r . n)(r x q) + (r x o' q)(r x n) + x (r x n)]}da

This is again divided into three components; the first would be the moment if the flow were steady (Lagally moment), the second arises when the flow is changing with time, and the last is an additional effect due to rotation of the body:

While the component M2 cannot be reduced, it is a linear function of 'I, allowing the superposition of solutions. It should be noted that the components M1, _2' M3do not

corres-pond exactly to the forces F1, F2, F3 since the integral for the force correscorres-ponding to was broken up into two parts, one becoming part of F2 and the other part of F3.

THE "LAGALLY MOMENT",M1

We again suppose the singularities to be discrete and isolated. The moment M1 (1) is [521 M1 q)(r x n) - (q . n)(r x q)dc7 [54a] N2 = p (r x n) do [54b] M3

=j p{(x

r h)(r x q) +(r x

q)(r x n) + x(r x n)]}d _[54c]

Mi(i)=f p[_.(q

. q)(r x n) - (q . n)(r x _q)] dor Si [55]

and

Making the substitution [831, we have

24

Mj(i)=rxFi(i)+f

p[-.(q

. q)(R xn)(q.

n)(R

x)]dor

Si

since R x n 0. Again using polar coordinates,

2ff ff M1(i) =(r1 x F1(i).

)._J

J

p sin OdOdA 2ff ff

M1(i).=(r1xFi(i)J)+J

_f

pR['gsinOsinA

+ cos 0 cosA]dOdA 2ff IT

= (r x

F1(i)

k) .45

5 pR2[cD04' sin Ocos X

x'-'

osinAldOdX

The sane procedure used in obtaining l'1(i) is followed. The integrals in the above

expres-sions then become

=r1 x

Fi(i)_f

p(q. n)(R xq)da

Si

SoS:

p

4Xsin0d0dX=ffp

(.1

xl

(2n+1)s(P)2d

1

l,s ns

fin

) [57] [59a] M1 = M1(i) _[56]

p

g2 (0sin 0 sin

₊ cos 0 cos Ad0dA'

2*.ir

p9bsIri 0 cos A +tt cos 0 sin A)dOdA.

n=1

Lso

25

ii(s)(a1b-b'')J(s+ 1,8)

1

0

where ) has the same meaning as before, and

.1 r dPs±1. _dPs

J(8,

8±1)=f

-

(7+1)P,±1L__fl](i_

[ns±1) + 8(71 + 1)1

pSpS1

d.

r

Evaluatingthese integrals, (see Appendix 1), we find that

00

(8) (a' b

+i) _{J(8, a + 1)}

'

rj('s)(a + ₎ a + 1)

(i) = [r x F1(i). i]:_'27T

i$117(i).=[r x F1(i) _] +iTp

n=1 s=1 n1

s0

(ab b ?)s

(n.+ a)!.

(ns1!

+ 1)! '

(nsi)!

2) - [59b [60a] (s)(a b

+ a 4'

_[60b] [59c] /

and

26

ill1

(i)=[r1xFi(i). k]+irp

(8)(a1

+ b5 b3

a

-

bs

s+1\ _{(n + 8 + 1)!}

n n n ri

The total moment is then given by Equation [561.

Since the expression for M1(i) is a bilinear form of the same type as that for F 1(i), the discussion of the latter applies equally well to the moment. Hence, for continuous

distri-bution

M1 =fMi(i)(a

b, a,

[56'!

THE MOMENT DUE TO ROTATION,M3

We define the moment M3(i) to be

M3(i) =f((w x

r n)(r x q) + (r x w .q)(r x n) + [to x (r x [61]

Equation [351

[60c]

M3 = M3(i) [621

The first two terms in the integrand can be reduced as a triple vectorproduct,

(r xd n)(r xq)(rxw q)(r xn)rx[(rx)x(qxfl)1(rx.o)(r

q xfl) [631

since r r x a = 0. Making use of substitution [33],

(rxeo)(r q x n) = (r x

w)(r. q xn) + (R x.)(r1

q xfl)

But by Equation [43],

f(rxco)(r.qxfl)da=(Fjxw)rj.fqXfldc70

Si Si

Also, since it is evident that

_I

(R . q x n)d,. involves only A, we have, using

Si

[64]

27

f(Rxcd)(rj.qxn)da=_f

(Rxc)(r1. Axn)

Si S

=-io xf (r1

x A. R)ndo Si =

xf

y (r x A

R)dr

vi

1 1'

=--wxJ

(r1xA)di-V.

_4n[ti.x(r1xA).]

[66] 3

Using Equation 3], the last term of the integral in Equation [61] becomes

=

tIfldc)

since R x n = 0. Using Equation [37], we have

fx(r xn)]da=[, x (r1 x A)]

[67]

3 Substituting these results in Equation [61], we have

M3(i)=0 [68]

MOVING SINGULARITIES

The cases which have been discussed so far are (1) discrete singularities which are fixed with respect to the body, and (2) continuous distributions of singularities. While these cases include the most important applications, flows exist which can be discussed in terms of discrete singularities moving within the body. In the present section, the analysis will be ex-tended to include this case.

The control surface S enclosing the moving singularity is taken to be a sphere with center fixed at r(t0l, the instantaneous position of the singularity at time ç. At the time

+ 8t, the singularity will have moved to rL(to + t) or referred to the center of the control sphere R0(8t).

or,

(2R.Rog\h/2(A.R_A.)(2R.R_R23

1

/

Expanding by the binomial theorem and collecting terms, we have

0

a

-+ (A + a R0) R + terms of higher order

I? R3

Therefore

'(t) [701

and

28

Let the coefficients for the expansion of the potential due to the singularity about r(t)

be a, b. This potential may also be expressed as an expansion about r(t0) which will con-verge for all RI > IR0(t)I. Let the coefficients of this expansion be a ,

. The latter

expansion is precisely of the form due to a singularity fixed at r(t0). If we find a , b in terms of a, b, we may insert the values directly into the formulas for the force and moment.

It is evident that

,

_,

a (t0)a(t )

bs

(t )=bS(g)

0 fl 0 a

Therefore, the formulas for the Lagally force and moment, which depend only upon the

instan-taneous values of the coefficients, remain unchanged. Further, it is only necessary to deter-mine and A', the source and doublet strength of the equivalent singularity, since the time derivatives of these quantities appear in the expression for F2 (i) but no higher order terms

appear.

The potential about r(t) may be written

A[RR0(at)1

+ terms of higher order

[R - R0(&t)I _{jR - R0(ot)}

[691

where Differentiating, dA' dA dt -dt 29 dt ,

°dt

=

V )(r1)

(R. V:)2(r) +J_(

ax ay

'a.s

=R(dos 0 -_+ sinO cos A + sinO sin R(n V)

. .J ..

At time £0 this becomes

dA' JdA

dt ₀ Ldt. _[72.]

where _v is the velocity of. the singularity relative b the body. Equation [39] then becomes

-

da°

_dA

C 73]

POTENTIAL OF THE UNDISTURBED STREAM,

It has been seen that the coefficients a, b need be determined only for q, the poten-tial of the undisturbed streair In general, these coefficients can be found in terms of the po-tential and its derivatives at point r1. Since is analytic in the.neighborhood ofr1, it can be. expanded in a Tayldr's series about r. .

Hence, the expansion can be written

çS(r)

_'i!R

.V)7Z(r1) _[741

n=o

Equating coeffiáients of 1? .intthi expansion and the expansion of the potential in terms .of spherical harmonics, we obtain the system of identities . .

71+ s=o.

s=o

30

n+1

:+1çcos O)(a1 cOs sA+

O)[cos sA(fl .V)45 +sins)L (n .V)b]

When the operations are carried out in this second form, it can be reduced to a sum of tern's of the type A cos aX, B5 sin aX, which are linearly independent, so we ma equate coefficients of the twO forms. We have then the further system of identities,

= 2(n-i-1)

[2Pcos

O.--

+ (1 -

1)P1

sinO

abl)

ax [77a] aas abs +pS+lsinO( "- ₊

-'I

'

a,

Pfl+I b1=1

1;[2P:coso4+ii(s

i)P1

.O(8a1

₊₁₎

aa'

P:'sine-

_az The special case in which a ii + 1 is easily solved,

2(n.+1)

'i':

siii(

8a

-)

S

a b5'

_n

ay'

[76] [77b] 1

V )' (ri)

=4'P(cOs

O)(acos s A + 3siñ aX) [75]

s=o

which permit the determinatin of a, b.

The solutions are most conveniently found in the form of recurrence formulas Sinee

[75] are Identities, and the a, b have explicit values üi terms of the derivatives of d?, we canwrite

and since Similarly

b'1-

fl+1

_(2n+1)(2n2)

PS n+1 /

-The special case, a = 0,. js also easily solved by setting 0= _{0. Since P_1 (con- 0) = 1,}

o . aa,° =

n+1

_n+'1

_a.x

The identities of-[77a and [7Th] can be transformed by _{means of the recurrence formula (see}

Referencâ 13, page 360)

-ps_

_{+ s) sin U [80]} We then have PS

(a

1

71+1 dz

= _{2(n+ 1)}

{i

sin 0 abs n n+.1 3x-pn+1_ .(2n+ 2)! _{sin'0= (271 + 1)P sinO} n+1 _211(n -i 1)! 1 (2n+ l)(2n + 2) 31 in 0

/a'

ôi

+-_

aa ay

ça"

ay + a s ax

+ Psin

0(-faa341 ao-'

sIn0(---+

.'

ay na I:78a] {?8b] [81a] a b [81b]

{:

_dx

Then

We shall prove by induction that

a -3 r9 z 32 a .1

m+1m+t+1 3x

a bt 1

m+1m+t+1 ax

We have already shown this to hold for t =0. Assume it to hold for all t < s and for m < n.

when t =s. We first prove

1 13as_l

ab1\

3x (n-i-a)

\

3yj 3z 1 By Equation [78a1

77(8 1) (321 a2o\ ij(s

1) (Bar'

_abr'\

ax

(2s_1)2s\3x3y

az32

I 2s By Bz

I

Assume Equation [83a] to hold for

B a ₁ 77(8 - 1)

(3 a1

a

b1''\

az 1

-

(m + a)

rnt

- - -1 3am 77(s - 1) _/ 3x

(m+s+l)B2

(m+s)(m+s+1)3x3y

_BxBa

77(8_i) IBa

abs_I

-

m+1 m+1

(m+s-i-1)\

3y

Therefore, by induction, Equation [83a] holds. Similarly,

1

1)(Ba'

abs_i)

(ni-a)

By {83a] 83b] [82a] [82b

Substituting thes resu1t inEqttation[8i1; we.hav Then

jaa\

1 :+1

(a+i

;;1:;;J=;:;

ôa5+ sin0

(.

Ps n+i \. _au + P+1 33 1 abs - n + 1 = 2( + sin 0 ( + 8)

The associátéd Legendre function is of the form

C sine 0.f(óos 0)

iii which fcos 0) has the property that

f(cos0) # 0.

Hence, we can divide Equation [84] through; by sinsO and set 0 equal to zero to find

We .make use. of the recurrence relation

2s cos 0 P = sin 0

Pt'

+ (n + s)(n - s + 1) sin 0 P [851

s:-' sin O(n

ax d2

I

öa5.

at

sin 01 --+ __2_

au

/

p5

_\

1 (

p1

\

(ii + s) (n, - 8 +1)1

pS'i

cos0I--I=s1n 01

_Sifl 1+ 0/ 28

S1fl' 0/

2 8'

'sin'O

frOm which

sinO)

- _{- 28}

O)02s!(n.$)!

,

P

(n+ 8)(fl- s + 1)

.P'

_\

_-

(71+8)! [861 [$4a] [84'b]

sincO F(1) 1. Using this result in Equation [84],the relations [821 f011ow. Hence, by

induction, these relatioiis hold for all n.

By Equations [78] and [82], a and b may be easily evaluated. Since a ç!(r) and

=

0, a and

_b: can be found by repeated use of Equation [78] Then Equation [$2] can be

used t

find a and b.

-The values of the coefficients a, b are tabulated below for s 4.

or 2 aTh-1 (n+ 1)! = 2. - (n.+ 2)! ax (ç77 34 0 -

ni

(ii )!

x°1.

[87a] [8Th] 2 n-2

(2;)

[87c]

(n+2)!

ax-2

3)

b3 2

3i3

(3qi

*

[$7d]

(n.+3)! 3x-

(n+3)!

ax3

-n-4 2 c' _{- 8} ) b

-

'

U - (44. _-... 4 'rxxxx

'yyzz

fl - -

''yyyz

-+

4)! dz

+ 4)!

ax4

SOURCES AND:DOUBLETS

When the singularity at is a source or doublet, the expressions for the force and mo-ment take particularly simple forms. Using th values for the coefficients giveii by Equatiàn [871 we have the sOurce

-

F'i(i)=4a:[.±]

[88a] 4IT t7

[--]

[88b]

Pi(i)=41TPa[±]

[S Sc] [87e]

F1(i)=-4npq(r1)

J88'l

For the doublet

which can be conveniently written in vector form

F1(i)=_4p[(A. v)s]

[91'J dA

F2(i) = 4

()= (rxF1(i). 1)-i-

47(1±_P ±)

₁ az 1 i

-

a M1(i)= (r

x F1(i).j) + 4

-_ a1

az

Miz(i)=(rxFi(i).k)+4fTP(

.±)

ay

'

a2 35

where q5(r) is the velocity at r due to the undisturbed stream. Also

= r x F1(i) + 4 lTp(q5 x A)1

a2s

₁ [9 la] a2

dzäy

_

' azaaJ

a2

8y82j.

[91b]

axay

-i-a

'

a

ii

a21 a2

al

_{[9 id}

aza

1

ayaz

1 2] da°1 F2(i) xc1,) _{+ [89]}

dt J

Ml(i)=rLxFl(i)

[90] Also or [92] [93 ci

C

36

-CON CLUSI ON

We have shown how the force and màment acting on a body with an arbitrary motion

through a fluid subject to a time varying potential flow can be found if the body can be repre. sented by a system of singularitiesplaced within the 'body.

The force can be considered to consist of three components The first, which would be the total force if the instantaneous flow were steady, is simply the "Lagally force." This is

found in terms of general singu1rities (Equations [20] and [30]). The secànd component de-pends upon the change with time of the singularity system generating the surface of the body. This force (Equations [32] and [391) is found to be a function of the strength and orientation of the sources and doublets in the singularity system out not of the higher order singularities

The third component is the force which would be required to generate the given motion of the

body ina vacuum, if the body were to have the samedersity as the fluid (Equations [81 and

[481).

The moment similarly consists of the "Lagally.moiñent" (Equations [56] and [60]) and additional components, bu it has not been possible to resolve these additional moments in the

same manner as for the force. 'they consist.of two terms; the first, appearing in Equation [9],

is simple enough, but the second requires the evaluation of asürfaèe integral (Equation [54b]).

However, the integránd is linear, so it is permissible to superimpose potential flowswhich

satisfy the b,oundary conditions.

ACKNOWLEDGMENTS

The assistance of Mrs. Alice Thorpe has been of great value in the preparation of this report. She aided in the reduction of the many integrals and carefully checked the algebraic

operations, greatly increasing the author's confidence in the results. The author is also very.

grateful tO Mr. P. Eisenberg and Mr. M Tulin. for their careful review çf the report.

37

APPENDIX 1

EVALUATION OF INTEGRALS IN F1 AND M1

REFERENCE FORMULAS

The associated Legendre functions satisfy certain difference relations which are tabu-lated here for reference (Bateman, Reference 13, _{p. 360).}

(n_s+1)P:+i_(2n+1)P:+(n+8)p:10 v/i ,2

:'

28ups

- (n+ s)(n 8 +

1) _v/i

-

_fL2 P_iIP+(fl_8+i)/i_,z2 p:1

=pp:+(n8)v'1_2p_1

n+1 (1 - 2)dP,. (n + i)aps _{- (n - 8 +} 1)

'n+i

dP$

(1_2)_._!.=(n+3)p:_1 nfLP,

d1i

We shall also need the following integrals:

1

IP,P,d=o

J 1

$1 I

(P:)2d=_--_

.1

1

2n-s-1 (77. - 8)!

1_,20

I1

(P,)2

dt

_-

_{8 (n - 8)!} fl [101] [102] [103] [104]

(1)

38

2(n+s+l)!

pSpS

n n+1 _{(2n + 1)(2n + 3)(n s)!}

The above formulas are also from Bateman, pp. 363-364. These relations are supplemented by certain additional integrals which will now be proved.

ji

PS1241PS ₁ 2 _{8(21 -}

(n+s)!

1

This may be proved by induction. Call the above integral K,, and assume [1061 to hold for

K,,...1. Then making use of [94]

1 1

1

ps _p5 1 _{1(2n + 1)}

f

(P)2

(n + pSpS pdp

1

n#1 n 1 - S + 1 L _{_} 1

1

1 2]

The first term on the right is easily integrated:

1 2 d 1 _d

1'

(P)2d-

-

8) + 1 (n a)!

-f

j

(P:)

1

1 2

1

1 - L2

₁

a(2n + 1) (n - a)!

using Equations [102] and [104]. Then

K

[2(n.-s)+1](n+s)!

(ns)(n+8)!

(n+8)!

s(n - a + 1)!

s(n - 8 + 1)! - s(n -

a)!

so Equation [106] holds for K,, if it holds for It is easily shown that it holds for n equal

to 8:

p ---

(1_2)5/'2

(2s + 2)! ps ₍₁ 2)S/2 _{=(2s +} 5+1 - 2S+1(8 _{+ 1)!} and 1 1 K

pS pS

(2s + 1)

f

(pS)2_ d (2s)!

f-i

S

s+1 S1_2

S

1_z

s.O! [106] [105] [1071

We make the substitution, using Equation [06]

/1-2p5

V. -I

froth Equation [1071. Therefore Equation [106] holds for all n

-1

Ips

ps+l

;-0

n+1 n

and Equation [1.08] becomes

d i t J

n+in.

fl-8-4-1 by Equations [1061 and [104] 1

:

:

-

(n+ 8)!

I1_2

(na)!

This i proved by substituting for Vi - P,, using [971 and integrating, using formulas [104]

and [1O6] (2)

[fos+i'ps+iiL

f

(pS+1)2 39

(

1'

d1L I pSpSl.

J_1

n n+i

Substitute for 2 _{P,, using [98], and integrate, using [106] and [104L}

... 1_i 11d 2 (ns)!

- I pSpS+l.

'Z

J1-L

fl+l(n.sr.1)!

Substitute for $LP, using [98], and integrate, using [1021 and [111]. INTEGRAL IN _F1X

. .

This integral which appears in [29a], is the-following

[-108]

-0

[109] [1101 -[112]

_i:

:+1

P-'

d

-2

(n+ v'l

.2

-

a + 1-)!

40

jr

dP8 dPs _s I(n+1)2p:+1p: +_____.±1 --(1-2)+82P:+1P

₁₂

-1L d d dP3 dP5\ + (n+ 1)1 pS_' fl+1

-

p3 1(1 -d n+1 d, /

To reduce the integrand, we have from [99] and [100].

(1_2)

dp (n+1)p:=_(n_s+1)P3n+1

d p5

(1- 2)

11+1

+ (n + 1)zP1 = (n + 8 + 1)p

d

Multiplying these identities and reducing, we obtain,

n+1 S n-I-i

_pS

dPS dP5 2(fl+1)(P dP5 dPS d d d =82p5ps

_JL-n+i fl

Also, from [100] and 199],

dPS ps ' _{- [(n+s+1)(P,)2}

_(n+1)P:+1P:1

' dPs PS n

_-

_{1[(n1),zP,P,1 -(n.-s+1)(P,1)2]} n-I-i _d

1_,2

so dP3 pS n+1 ' _d 'n+i _d

1L2

+ _1)P:p:1]

Substituting [115] and [116] in the integrand above, we have

(n+ 1)P,1

(n₊ i)2J_1[(P:)2 ₊

+(n + .1)(1

T

by [98]. Using [1021, this reduces to 2(n + 1).

INTEGRALS iJs, s±1)

These integrals, which were needed to evaluate F1,. and were defined as

!(s,s

±1) =L11+l±1 [(

2)

_{(, 1)2}

_{+ s(s ± 1) (n + 1) (28}±1)] +(n+

1)(P,.'71

ps1!)(1

2)jd + 1)(n + S

+l)(P

+.(n+ 1)(n- s ₊ d

2(n+s+1)!

2[.82 _{- (n.+} 1)2]

_::+1

_IL}1. 2

(fl-8

using [104] nd 11061. This integration, breaks down when s =0, because a ppeãrs in the

deôminators of[104] and [106], but theresult is still Valid. The reduced form of the

inte-gral bra

= 0 becomes

2_, 2 =

(n+ 1)J.[(P_ pO

)2 (1 ₂₎ = (n -i-.1

.+ 82p:+1p:1.

r'r'

..2)

I

J1L (n+1)

(dP1

:+1 _d (p°)2 (1 ±1.dF'i+1)

dP,1 dP±1

d

42

In [113] replace a with a ± I and multiply with [114] as before. We obtain,

dp:±1 dPs dP5±1

(1

p2)2 _{+ (1 - 2)(n + 1)'P}

ps±1+1)

k n+1 dz dz d

1?(n+ 1)2P,'P,1 =(ns+11)(n

+s+1)Pps

72+1 72

Substituting in and reducing, we find that

[n

- S + 1)pS pS+1 _{- (n + 8 + l)pS+lpSi} d a + 1) (n -

s)J

_{n+1 n} n+1

J1

2(n + s + 1)! =

(nSi)!

a 1)

= (n + a + 1) [(n + S)Ps

_{n+l n}

_{P5' -(n - a} 2)P 2(n + a + 1)! (n - a + 1)!

using Equations [108], [109], [110], and [111].

INTEGRALS J72(s, s±1)

These integrals appeared in the expression for and Al1 They were defined as

1 dP,1 _dPsi - _{(n +} 1)p:±1 --I(1 2)

J72(s, sti) =J{[n:

dz d. +[n(8 ±1) + s(n + 1)] P p5±1 _ILj11 If we substitute for dP5 dp:±1 (1 2)_{____n_ and (1} 2) d d

using [99], this is immediately integrable, using [108], [109], [1111, and [112]. We find that

+ 1) (n - a - 1)! 2 (n -# a + 1)! [1191 (n + a)! J72(s,

-

(n - a)! [118]

Lagally -Force whéré 1T .4.3 APPENDIX 2 SUMMARY OF FORMULAS

In this appendix certain formulas which will be of use in applications are collected-to-gether for convenient reference For meaning of symbols and conditions of validity, reference

must be'mdè to th text.

Transformation from Moving Axes to Fixed Axes

-

dy

.+p..3Z

dt

M.= Mm + p(rg .

F F1(i) . [2OJ

'-p

=' 7(s)(a+i + .os+.

i+i) (i

T i.)!

L= .

-(ns,1

-+ b

50

13=0 I S0

L'fl's)

₃₀

(as

-

b

;+i).;

:

S

-

_bs+i s\ (n + s + 2)! n+i '.n)

_(ns)!

17(0) 2 'qC)= 1,

. >0

[30a1 [30b] .[30c1

Force due to Changih Flow Lagally Moment. F2() =-4iT

[

-F - 3

1g

_dt ,,(z)= [ri

x F1(i.

-

M(i)

=[rix

F1(i).

.44 F2 F2 M1(i) - S+1 S)(fl+ 8 + 1)! Ii fl fl _{- 1)!}

da°

gj) + t. +

-i1t dt p4Z. s ) (n+ 8)!.

'.

(n--s)!

+ bs41 p s s+i

-

+i\ (n + s + 1)! ' 'z 'z . '

/ 'n-.-s--1)!

Moment due to Changing FlOw,

Singularity Moving with Respect to Body

da°dA

F2(i)=_4n9[a(v_.r1.xc)

+Fj2'+_j

as p+1 n. n [32] [391. [481 [60a1 [60b1 [60c] [54b] [73]

Source

F1(i)= _47rpaq (r1) _{[88 '1}

where q(r) is Lhe velocity at r. due to the undisturbed stream

45

Coefficients in Expansion for Undisturbed Stream Potential

Iaa" 1

(){fl

(2 (4 ) ,'

-yyyz

-

4 ,'zzz) [78a] [78b] [82a] [82b] [87a] [8Th] [87c] [87d] [87e] fl+1 _{(2n + l)(2n + 2)}

_{' dy - az}

_/ 1

,(7)(fl

+\

n+1 _{(2 + 1) (2n + 2)} 1 aa a a

mt

2

a1

m+t+1 ax

1 ab

m+1m+tl

and in particular a0 =_j_ dx 6: = o " _{n! özTz} 2

as'

_(c)

(n + 1)! dz1

a2

a 2 n

(n + 1)! dx1

b2_ 2

a'2

_2(c77c6)

(n + 2)! 9z a 2 z3

-(n +

2)! dx"2

2 c3'3

(n.+3)! ax3

a 2 9'

_8c77

7z

(n+3)! dx3

2

(n4)!

ax n (n+4)! dz

46 F2(i) =

p[°(r.

x)

(89)

M1(i)=rxF1(i)

[901 Doublet

F1(i): -4

lTp[(A v)q

dA [911

-4ff

[921 F1(i) (93'I

47

REFERENCES

Kelv.iii, Lord, "On the Motion of Rigid Solids in a Liquid Circulating Irrotationally

Through Perforations in Them or.a Fixed Solid," Phil. Mag., 1873, Vol. 45.

Taylor, G.1., "The Force Acting on a Body Placed in a Curved and Converging Stream of Fluid," ARCR & M. 1166, 1928-1929.

3.. Taylor, 0.1., "The Forces on a Body-Placed in a Curved or Converging Steam of

Fluid," Proc. Roy. Soc. London, Series A, 1928, Vol. 120.

Tollmien, W., "Uber Kräfte und Momente in schwach gekrflmmten oder konvergenten Stthmungen," Ing. Arch., V. 9,. 1938, (Trans. by Stevens Inst., ETT Rep. 363, September 1950).

Pistoleul, E., "Forzè.e momenti in una corrente 1eggerment curva convergente," Cornmentations, Pont. Acad. Sci., 1944, Vol.8.

Munk, M., "Some New Aerodynaiiical Relations," NACA Rep. 114, 1921.

Lagally, M.,"Berechnung der Kräfte une Momente, die stramende Flussigkeiten auf

ihre Begrenzung ausilben," ZAMN4, 1922, Vol. 2.

Glauert, H., "The Effect, of the Static Pressure Gradient on the Drag of a Body Tested

ma Wind Tunnel," ARC R & M 1158, 1928-1929. .

B.etz, A., "Singularittenv8rfahren zur Ermittlung der Kräfte und Momente aur Korper in Potentialstromungen," Ing. Arch. 1932, Vol. 3 (TMB Translation 241)

Mohr, E., "Uber die Krfte und Momente, welche Singular.itten auf eine stationare

Flüss.igkeitsstromung iibertragen," Journal fur die reine und angewandte.Mathematik, (Crelle's

Jour.) 1940, Vol. 182. . . .

Brard, R., "Cas d'Equivaience entre Catenes et Distributions de Sources et de Puits," Bull. l'Assoc. Tech.Mar. et Aero., 1950, Vol. 49.

Mime-Thomson, L.M.,,"Theoretical Hydrodynamics,?' Scond Ed., Macmillan Co., New

York,.1950. .,

13 Baternan, II ,'Partial Differential Equations of Mathematical Physics," University

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