Vortex theory for bodies moving in water

ARCHIEF

-Biblio.heek van

d-Onderafdelugder Schpsbouwkunde

ecIin1sche Hoqeschool, DeRt

DOCUMENTAUE : DATU1: ( 9 UKT. 197 VORTEX THEORY FOR

BODIES MOVING IN WATER

by

Roger Brard

(Bassin d'Essais des Carènes,

Mithstère d'Etat chargé de la Defense Nationale) France Lab. v. Scheepsbouwkunde Technische Hogeschool Delfi DO (Ii H H _PREPRINT Paper to be presented at

Ninth Symposium on Naval Hydrodynamics

August 1972 Paris

Pages

ABSTRACT 1

NOMENCLATURE 2

INTRODUCTION 5

I - A BRIEF SURVEY ON VORTEX THEORY 7

II - POINCARE'S FORMULA

III - VORTEX DISTRIBUTIONS KINEMATICALLY EQUIVALENT

TO A HULL SURFACE 12

IV - FREE AND BOUND PARTS OF A VORTEX SHEET 17

- Euler's equation in a moving system of axes 17

- Free and bound vortices 18

- Behavior of the free part of a vortex sheet 19

- Definition of the force exerted by the flow on an

element of the bound vortex sheet adhering to a

moving body 22

V - THE STRUCTURE OF THE VORTEX SHEETS GENERATED

BYA MOVING BODY 23

T'

- The vortex family (D. ,

2X))+ ( !-

_, 23

T-T'

Tf

- The vortex family ( , , ) + (ZL , 25

VI - THE EQUATIONS DETERMINING THE TWO VORTEX

FAMILIES 33

VII - HYDRODYNAMIC FORCES EXERTED ON THE BOUND

VORTICES AND THE MOVING BODY - 36

- Dynamical equilibrium conditions 37

- Comparison between the system of hydrodynamic

forces on the bound vortices and the system

of hydrornamic forces

on the body 39 - Quasi-steady forces, deficiencies and forces due

to virtual masses 40

- The origin of the forces exerted by the flow on a

bound vortex distribution 42

- Case of a hull equipped with movable appendages 44

VIII - THE APPLICATION FIELD OF VORTEX THEORY IN

SHIP HYDRODYNAMICS - 45

motion 49 - Unsteady three-dimensional motion of a wing with

a finite aspect ratio 49

- Propellers theory SO

- Ship maneuverability theory 50

2ONCLUSION - 56

ABSTRACT

This paper presents in a synthesized form old and new results about the vortex theory for bodies moving in water. It is shown that the

hydrodynan-iic forces exerted on such a body can be derived from the

know-ledge of a vortex distribution kinematically equivalent to the body. A method is proposed for determining both the bound vortices and the free vortices when the bound vortex distribution is chosen to be adherent

to the hull surface. The total vortex distribution is divided into two parts. The first one consists of a volume distribution inside the hull and of a vortex sheet on the hull surface. The volume distribution is identical with the vortex distribution due to the angular velocity of the body. The sheet is determined by the condition that this first part of the total distribution induces outside

the hull a velocity null everywhere at every time. This first part may be calculated once for all. The second part consists of the free vortex sheets

shed by the body and of a bound vortex sheet on the hull. It is equivalent to a normal dipole distribution whose density is the solution of a Volterra equation. The determination of the hydrodynamic forces exerted on the body is derived from the dynamical equilibrium of the fluid outside the body, of the fluid inside the body and of the total bound vortex on the hull. This system can be subdivided into three systems : the quasi-steady system, the system due to the added masses and the system depending on the history of the

motion.

This paper has been written to suscitate new researches in

1. OPERATORS

-'V =(-_,

__

), (the axes x1 , x2 , x3 make a right-handed system)

= Laplacian operator

Symbol of vectorial multiplication.

2. - VORTEX FILAMENTS AND VORTEX SHEETS

-Vortex filament

d Element of area of a vortex sheet

The two sides of

Infinitely small thickness of the vortex sheet

Unit vector normal to the sheet in the direction from

towards

Z.

V Velocity of a fluid point

-*

CU Curl of V

-4 -4

-T Limit of Cc)

when .O ; T = (T - T') + T' in sections

V, VI, VII.

Unit vector tangent to

1

in the direction of T

-4 -4 -4 -4

V(P ) - V(P ), (or V(M ) - V(M ) ) : jump of

Vthroughi.

from P to P.

I e i e e 1

(or from M to M.), P

(or M ), P. (or M.) belonging to

e 1 e e 1 1

'-S

-

, respectively.

T=-n

A

_[

_(Me) (M.)]

Line on i orthogonal to the

r

Circulation of V in a closed circuit

d Intensity of a flat vortex tube on , the width of the tube being d

D.

, D,

Interior, exterior of

Z respectively

Domain occupied by the real fluide : D' D

e e e

V Absolute velocity of a fluid point

VR Relative velocity of a fluid point : velocity with respect to axes

VE

moving with 1

= curl

(with or without accents) : line on

I

along which Z is

attached to Z

Vd Velocity induced by the bound vortices Vf Velocity induced by the free vortices

Tf Vector T on

I

dr

Density of a normal doublet distribution over the hull surface

VI

cf

Velocity of a point fixed with respect to the moving axes Angular velocity of the moving axes

Surface supporting a free vortex sheet Incident velocity on

I

Velocity potentials : V = V if curl = 0

,

=

curl V

Density of a normal doublet distribution over the free vortex sheet

Vortex filament on

Velocity potentials generated by the normal doublet distribution

on , respectively

Velocity induced inside D. by the vortex family (D. , 2 _. )

1 1 _E

+ ( Z ,

) in Sections V, VI and VII

-V

0

4. DYNAMICS

-Mass density of the fluid,

Exterior force per unit mass : F =

.2

-Additional exterior force inside D. if g o 1

d

Hydrodynamic pressure

Hydrodynamic force exerted by the flow on an element

dZ

of

a bound vortex sheet or on an element of arc of a vortex filament

which does not move with the fluid

System hydrodynarnic forces exerted on the vortex sheet (

System of hydrody-namic forces exerted on the hull System of inertial forces inside D.

System of complementary forces p F' dD1 inside

System of forces _{- M} d (M) on

e

System of forces

_{- p}

T(M) A _R (M) d (M)

System of forces due to the added masses when there exists no vortex sheet

Estimate of when the deficiency affecting is neglected P F F'

dF

F' + Estimate of a0 are neglected.

The vortex theory in in oriiprt-ssible, inviscid and homogeneous fluids plays a role of importance in many chapters of Ship Hydrodynamics. However it is not systematically applied to all the problems where it should

be especially useful. This is the case of those related to maneuverability

and control of bodies which behave as rather poor lifting surfaces because

of their large displacement/length ratios, The research reported in the

present paper has thus been undertaken with the purpose of determining how much help one can expect from the theory when dealing with such

bodies in any given steady or unsteady motion.

Indeed the question w.s lint to dra up a new vortex theory, but rather to extend known reults relevant to fluid kinematics and dynamics and to increase their generality and effectiveness.

The joined table of contents suffices to show the writer's line of

thought,

The startpoixit is Puincar s formula which permits to determine the velocity in a closed donnnii et the vor tic ity inside that domain and the

velocity on its boundary are knuv.i.. This leads to a mathematical model

where the hull surface is repl ed hy a fluid surface moving without,

altera-tion of its shape. There exi5t an infinite class of vortex distributions kinematically equivalent to Ihe bly They oniy depend on the choice of the vorticity distribution inside the hull. The most interesting one is that which

permits the exterior fluid to be adherent to the hull. Inside the hull the

abso-lute fictitious fluid motion coincides with that of the body. From the point of view of kinematics, one of the features of the theory is that the total vortex distribution can be divided into two families almost independent of each other, One consists of a volume distribution inside the hull and of a vortex

sheet over the hull, it is su calculated u to induce outside the body a velocity which is null everywhere, it only (it pencis on the angular velocity of the body

and car be determined once for il! br any given hull. The second family is

the union of a vortex sheet distributed over the hull and of the free vortex

sheets shed by the hull. It is equiiIcnt to a normal dipole distribution

determinedby the cc;i.dition that the resulting relative velocity ins.de the hull be null everywhere, This suffices for the absolute velocity potential outside the hull to be identical with the real one.

Oiihy thit- st' utid vortex family has a physical meaning. But both tre necessary for deteruuruiiig the hydrodynamic pressure on the hull. This is

ot really surprising -iin both the Iluid inside the hull and the bound vortex Theet over the hull must be in dynamical equilibrium. A consequence is that the classical expression for the force exerted by the flow on an arc of vortex tilarnent which dus not move ithi the iluid cannot be readily extended to the case when this arc btIongs to the bound vortex sheet adherent to the hull.

incident flow may be.

The theory developed in the present paper is quite general. Its

application to practical ends does not seem to lead to insuperable diffiultics provided that reasonable assumptions can be made concerning the pusitiull

of the free vortex sheets with respect to the body and the possible

_viriaJun

of that position with time. In any case, it is shown in the last section h-t an older and less complete vortex theory is still useful in manoeuvring. Thus it is hoped that the present one can guide the experimental and theurtu

The vortex theory can be divided into four parts.

The first part is, in fact, a chapter of Vectorial Analysis. The

vector V is the velocity of the fluid points in a certain fluid motion at a fixed instant t and the vcrticity is defined as

(1.1)

_{Ct) =curlVVAV.}

The starting point is the Stokes Theorem, accordino which the flux of 0) through an open surface is equal to the circulation of V in the closed

circuit consisting of the edge of the surface. A consequence is that no vortex

filament can begin or end in the fluid. A vortex filament is therefore a closed

ring or its ends are located on the boundary of the fluid domain, or at infinity. A consequence is that the intensity of a vortex tube is a constant along the tube.

The intensity of a flat tube or of a tube whose all transverse dimensions are null can be finite. This is the case of the vortex tubes on a vortex sheet and of the is olated concentrated vortex filaments.

Equation (1.1) can be solved with respect to V. Poincaré's formula

ves V when the vorticity is known. A particular case is the Biot and Savart

formula which expresses the velocity "induced't by an isolated vortex filament. A consequence of Poincaré's formula is that the perturbation flow due to a body moving in an inviscid fluid can be regarded as generated by a vortex sheet distri-buted over the hull of the body and fulfilling the condition that the fluid sticks to the hull. There is a kinematical equivalence between the body and the vortex sheet.

The second part deals with the evolution of the vorticity with time under the assumptions that the fluid is inviscid and that the exterior force per

unit mass is the gradient of a certain potential. The basic theorems are due to

Cauchy and Helmholtz. The intensity of every vortex filament is independent of time and the vortex filaments move with the fluid. This means that every vortex filament is composed of an invariable set of fluid points. Lagrange's theorem follows according to which the fluid motion is irrotational if its Starts from rest under the effect of forces continuous with respect to time (shock-free motions).

This theorem seems to be contradicted by the possible existence of vorticity

in the motion of an inviscid fluid, but the difficulty can be overcome by considering such a motion as the limit of the motion of a real fluid when the viscosity goes to zero. Although the second part of the theory is based on the Euler equation, it only deals with fluid kinematics.

The concept of force exerted by the flow on every bound arc of a vortex

filament is now classical. Conversely, the set of fluid points belonging to

this arc exerts a force equal and opposite on the adjacent sets of fluid points which proceed with the general flow.

As it has been shown by Maurice Roy [iJ , the system of forces exerted by a steady flow on a body in a uniform motion can be

obtained in this way. This led to an important generalization of the Kutta-Joukowski theorem. Later von Ka'rmn and Sears have successfully solved the problem for wing profiles in a quasi-rectilinear non uniform motion L2] The pressure distribution on such profiles has been calculated by the

present writer [3] . There exist now powerful methods for computing the

pressure diatribution on a wing of finite aspect ratio: in the same Idnd of

motion (see, for instance {4J ).

The case of bodies of high displacement/length ratio in an uns-unsteady motion is sensibly more intricate than the case of the usual lifting

surfaces and there was a need for a general theory.

Poincarts formula

gives means for determining such a vortex distribution on the hull and

inside the hull that the fluid adhere to the hull. This vortex distribution is kinematically equivalent to the body. But it is not the only vortex distribution

with this property. Furthermore if the motion of the fluid about the body

is unsteady, any vortex distribution kinematically equivalent to the body varies with time. Lastly the theory would be without practical

interest if it

were not capable to take into account the effect of the free vortex sheets shed by the body and that of an arbitrarily given incident flow. This paper gives

an answer to the problems arising from the afore-mentioned needs. The fourth part of the theory concerns vorticity in viscous fluid motions, but it does not fall within the scope of the paper.

II

POINCARE 'S FORMULA

VORTEX SHEET

-Let be a part of a certain surface. The two sides , of

. e

are distinguished from each other . The unit vector normal to

Z is

uniquely defined at every point P of

Z

and in the direction from towards

We put PP. = n(O+)

i'

= - n(O+) P. and De being the intersections

of it with . and respectively.

Let _., Z denote the surfaces described by the points P.'

L 1 e

-

-- 1

_-a is continuously distributed between and c (M) is normal to n

at every point M of each segmentP.'P'. When E .i,'o, (M) has a finite

limit T (P) tangent to

!

. Let , denote two unit vectors tangent to

at P, such that the dir ections(i

9,

) rake a right-handed system,

with

in the direction of T. A line

tangent to Et at each of its points

is a vortex filament of the limiting distribution. Let , be two lines

close to each other. Let be a line orthogonal to the 's and containing P.

It intersects at P1. We may put PP1 = d &. For the limiting

distribu-tion, T

_ij

d is the flux of the vortex through the area of the infiriitely flat

rectangle P. P P

P. and is therefore equal to the circulation dr of the

i e ej 1

velocity V in the closed contour of this rectangle.

Consequently

(2.1)

_13()-

V(P)]

=T(P).d

=dr

or, equivalently

(Z.2)

V(P)-V(P)=T(P)t\n

_{i e}

-

. ..

---

.-,

such that PP.' = n

, PP' -

n , respectively. We suppose that a vortex

Conversely, if the velocity V is discontinuous through a

surface !

, and if the jump is tangential to , then the above formula

-.

r-defines a vortex sheet ( ! ,

I )

with T = n A

t

V(P.) - V (P )J.

& P P

1 e

The expression (2.1) is the local intensity of the "vortex ribbon"

located between and . It is a constant along the ribbon if no vorticity jy'

coming from the regions outside E joins the distribution .1 over . We

will see in the next Section that the opposite case is frequent.

POINCARE'S FORMULA

-We consider in the fluid a closed surface S with a tangent plane at every point. Let D.

_{, 1e}

be the interior and the exterior of S. The unit

vector normal to S is in the inward direction. The interior side S.1 of S is considered as included in D., and the exterior side S as included in D

1 e e

The time t is fixed.

The velocity V is supposed continuous and twice continuously differentiable within D. Let A be a vector function of its origin M and defined

-4 -,

-curl -curl A = V div A -

A,

where is the Laplace operator : = = , ,

and V the

opera-tor , -z--- , , the system of rectangular axes O(x1, x, x ) being_L.

right-handed. By Poisson 's formula

(M)ifM ID.

ifM 6 D

_e by (M)

=-V ()

I.. We use the classical formula

Furthermore it is easily seen that If

curl A =

-()

M* LcJJ 1NY div A = Consequently c ür 1 (2.3) +grad

J1(V)M

_Js db(M

Lj5

This is the Poincaré formula.

Let us suppose that the fluid is incompressible. The triple

integral in the second term is not necessarily null for sources with a density

= div V could be distributed through D.. The double integral in the second .

-term represents a source distribution over S with the density = (n . V)kA.

-p - ..- 1

There is a jump of V through S. Its normal component n (Ve - V.) = -(i V)M and according to (2. 2), its tangential

jump from Zto

Z is that due to a

vortex sheet over S, with the vorticity . (n A V)

M

Poincaré's formula solves the equation

- -p

CU= Vf\V

-p -b

with respect to V when the vorticity Co and div V are known inside D., if,

furthermore, V is known on

Biot's and Savart's formula

-Let us suppose that all the points of S are at infinity, that

-p

div V 0 and that U) is null everywhere except in a very thin tube with a transverse area . Let us suppose that the measure of

-goes to zero, while (A) O- -

r.

The vorticity reduces to a vortex filament

with the intensity

r

tangent to . Let ds denote the element of arc of Then Poincaré's formula gives

(N' _(NvJ MM' (dLV)M ,I,(

_'[

19 WM1) (V(M)ifM . I). 1

((dv)

_{d (M)]} 0 if M _e I.

(2.4) V (M) = curl

This is the Biot and Savart formula which gives the velocity induced at M by the vortex filament the intensity of which is

As 0 except on , there is a velocity potential except on . Let be an open surface whose edge coincides with

and n the unit vector normal to Z , oriented in the positive direction with

respect to the arc ds of . One has

(2.5) V = V with

_r

is therefore generated by a distribution of normal dipoles over , with the constant density 1 . Of course, c is not single-valued. If C is a closed

circuit intersecting at P and surrounding C one time, then the circulation

of V in that circuit is

(2.6)

jv=

r(c)=(p.)Application to vortex sheets

-Let us consider the part of a vortex sheet. The contribution

of Z to the velocity V is given by

tt-(2.7)

S=curiA

I! T(M')

iirJJ3 t1M

III

VORTEX DISTRIBUTIONS KINEMATICALLY EQUIVALENT TO A HULL SURFACE

Let denote the hull surface of a solid body completely

submerged in an unbounded, incompressible fluid at rest at infinity.

Let denote the velocity of the body and LI. its angular velocity. One has

LetD. -resp.D -b

_{i e}

the two sides of !

vector normal to

of the fluid inside D . One has

e

(3.1)

_{nVEE nVonZe.}

Poincaré's formula applied to VE inside De gives

curl[J

Adding these two equations and taking into account of the boundary condition

(3.1), we get

This shows that the vorticity distribution

(3.4) ( Z ,

--

) + (D. , ₎ , ( . = 0+),

is kinematically equivalent to the body inside De and generates inside D.

a fictitious motion identical with the motion of the body. The relative

velocity VR = - V' fulfils the condition

(3.5) _VR 0 on

I.

Therefore, the vortex sheet ( ,

I

allows the fluid to be adherent to the solid wall 2 of the body. This gives the image of a very thin boundary layer which the real boundary layer would reduce to if the fluid viscosity }k were going to zero.

e the interior -resp. the

exterior- of Z ,

and .

For convenience, . D. and C D . The unit

i e

is in:the inward direction. Let V denote the velocity

di-

1(1

E()

) (ii)] I4Tjjj MM1

VE)M(JE1fMDi

ifMD.

(h AVE) -4

The same formula applied to V inside D gives

r _IIT(M) (k') c

curlf I

LtJJZ \p.y '. .4 -4

T(M')=-n

V(M')-V

(M')]on.

M' e E VE(M) if MEDe v(M) if M E D. -a

(V)M

(0 if M D.

('f\V)M

.t

I NM' (M)]f d

(V(M)ifMD

_e curl ', (.4 .4

L.

P

+grad!

(3.2) V'=

where (3.3)

It follows that curl (3.7) curl J (M

j (M )

e = ( ₎

()

-

-

( . .2.

Equation (3.2) expresses that

(3. 8) curl 3 (M) = VE (M) - curl 3E (M) when M describes and therefore entails

-

A +

±

( M')

(i'

_(k).

2 _{N MN'}

It is easily seen that curl V = 0 inside De so that

V V . inside De

being the unique solution of the Neumann problem with the boundary condition

-

-nVE one.

For the sake of brevity, let us put

_

-*

L1L COF

(3.6) J(M)

--J(

()

_{o(M, E('jJJ}

The components of _E are continuous and continuously differen-tiable inside D. + De (and harmonic within D). Those of and of curl j are

discontinuous through because

(/\)

+ If

d(

2 N

JJ N MtI

D.,

1

curl [E (M) - curl

_E (M)] 0 within D.

div (M) - curl M)] 0 within D.,

-r

the difference VE - curl is within D. the gradient of a harmonic potential,

and, for (3. 8) to be satisfied everywhere inside D., it suffices that it be satisfied on . . Thus

I..

(3.8) < > (3.9

(3.9)

Consequently, T has to be determined by (3.9) and the comple-mentary condition

(3.9')

Oon !.

Equation (3. 9) is a vectorial Fredholm equation of the 2nd kind. It is singular -,

since T = A n , A being a constant, is a solution of the homogeneous

equation

(rA)

+ L

T(

1 \ ?-\

For (3. 9) to have solutions, it is necessary and sufficient that

its right side -say B (M) - be orthogonal to n on Z

(3.10)

jj

I

nBdZ. =0

(3.11) I -T = T' - m n-

-This requirement is fulfilled because VE and curl IEare diergenceless in.de D..

Hence, if T' is a particular solution of the complete equation (3. 9), the general solution is

- _{- S.}

T=T' + An.

But n T'= const. = m on and therefore

is the only solution which fulfils both (3.9) and (3. 9') (1)

-T being determined in that manner, the velocity V' generated by

-0 _{- _,}

-the vortex distribution L on Z and (A) = W inside D., is equal to V

1 E

within D.. Therefore, as the jump of V' through is purely tangential,

-4 - -+ -

-one has n V' = n VE °' and V' evidently coincides with the velocity V of the irrotational fluid motion inside D

e

The above results can be extended to the case when there exists inside De some incident flow of velocity It suffices to replace by (1) - An equation quite similar to (3.9) and (3.9') has been considered by.

-.

->

VI + V

0

VE - V0 into the right side of (3. 9). The resulting velocity is

V inside D.

E 1

V inside D_e

If De were bounded by solid walls and (or) a free surface, one would have to add singularities distributed beyond the boundaries. These

singularities would be linear functionals of T and therefore T would be found on the right side of (3. 9) too.

Furthermore, it is seen that the incident flow V could be due

totally or partly to free vortices shed by the hull itself. In such cases, the

right side of (3. 9) would depend upon the history of the motion of the body.

Various remarks

-It is to be noted that the vorticity inside D. could be chosen

arbitrarily.

E should be replaced by =

The possibility condition (3. 10) would be still fulfilled. And, for the same

reasons as above, T would still be determined uniquely by (3.9) and (3. 9'). But,

inside D. , the resulting velocity could no longer be identic&l with VE.

In all the cases, the resulting velocity V on - i. e. between

and - is given by

(3. 12) (M)=

When T is determined so that V (M.)

= E (M.), the jump of

V _{- VE} through Z is equal toVR (Me) and

V (M)= - V (M)on.

_R

2. R e

T is perpendicular to VR on Z and the edges of the vortex-ribbons on are orthogonal to the streamlines of the relative motion over

If, furthermore, no free vortices are shed by the hull, these

be the element of arc of a particular streamline . The intensity of the vortex ribbon between two rings ,

"

close to each other is (3. 13) d r (M) = VR (Me) d y (Me)

(iii) If CD 0 inside D., then d1 is a constant when M describes and the fluid motion inside De and inside D. depends on the velocity

potential

(R=MM')

This potential is generated by a normal dipole distributions. r is determined up to an additive constant.

If 0 inside D. ,

then, dr

is no longer a constant between and ; ds being the element of arc of , one has

-'

The same phenomenon happens in the case of the vortex distributions (ID. , 2 + ( ,

.I

) - see Section V, Fig. 5.1 and 5.2.

IV

FREE AND BOUND PARTS OF A VORTEX SHEET

EULERtS EQUATION IN A MOVING SYSTEM OF AXES

-The fluid is assumed to be inviscid, incompressible and

homo-geneous. Consequently, its mass density p is independent of position and

time.

Let 2. be the hull surface of a body moving in the fluid.

Let S', S denote two right-handed systems of axes. S' is at rest

and the fluid motion with respect to S is said the "absolute's motion of the

fluid. S moves with the body and the motion with respect to S is' said the

"relative " motion. The subscript R refers to the reltive

motion. F is the

dV

absolute exterior force ; V , CL) = curl V and - are the absolute velocity, vorticity and acceleration. The motion of S with respect to S' is the entrainment motion. If 0 and M are two points of S, the entrainment

velocity VE (M) of M can be expressed as

- -ø

-VE (M) = -VE (0)

+ 1A

6t,

where is the angular entrainment velocity.

The relative velocity of a fluid point P located at M at the

instant t is -VR (F) = V (F) - VE (M). One has

_-

₀

Y(P)=(

+

Av)

-0 k)

curlV

₌

-2_i

R

and the exterior force FR in the relative motion is given by

-FR =F

Let p be the pressure. The Euler equation in the system S can be written in the form (4.1) (4.z)

-,

7N

=

-ZAV.

We suppose thatF= vU.

, and put

PS = ç)

and

d are the hydrostatic pressure and the hydrodynamic pressure respectively.

FREE AND BOUND VORTICES

-From the Euler equation in the system 5t of axes, namely

1cL

it follows that

-0

This equation entails the following consequences

- If F and the boundary conditions are continuous with

respect to time, and if the fluid starts from rest, then

a)

0 everywhere at every t.

- Every vortex filament is made of an invariable set of

fluid points.

- The circulation of the velocity in any closed fluid circuit

is time-invariant.

According to consequence (i), the absolute fluid motion should be irrotational everywhere. This explains why the concept of velocity poten-tial is of importance in the motions of inviscid fluids. However, this conse-quence does not hold if there exist regions where F is not the gradient of a

potential . This is obviously the case when the fluid is submitted to the

condition of adherence to solid walls.

For this reason, we have to consider that the vortices existing

in the motion of an inviscid fluid about a set of solid bodies originate on the surface of some of them. Thus every vortex filament is made of two sets of

fluid points : one of these two sets is at rest with respect to the surface of

a solid body, the second one is free and moves with the fluid.

BEHAVIOR OF THE FREE PART OF A VORTEX SHEET

-Let dE be the set of fluid points belonging to an element dl of a vortex sheet. The sheet is attached to the surface

I

of a body moving in

(1) - This equation holds if the fluid is not incompressible, provided that Taking into account the continuity equation, one obtains the basic Helmholtz equation

(4.3)

the fluid. Its bound part is located between the interior side . and the

exterior side of and the relative velocity of every fluid set belonging to this domain is null. On the contrary if dE belongs to the free

part of the vortex sheet, it moves with the relative velocity _{VR inside}

the sheet. Let .! , denote the two sides of Z, _, be the unit vector

I

normal to in the direction from towards , and the

infinitelsmall thickness of the sheet. The vorticity inside the sheet is

=

j

with - -*

1-.

₇ Tf (M) = _{- nM} L VR (Me) - VR (M.)J, ME.

,MM.=

.

,MM=n

i M , e

M,,

and the relative velocity of the fluid point located at M is

VR (M)

= 2 fVR (Me) + R

(M./.

Since there is no exchange of matter between the sheet and the adjacent fluid sets

-p - - -*

(4.3) n_M V_R (M ) = n V (M.) = n V (M) = 0

e M R i

M R

Let I(dE) be the momentum of dE. One has

(4.4)

- I (dE) = n

d V _{(M )}

- V (M )

- /1

Jd(M)=0.

dt M R i R e

Furthermore, by the momentum theorem, one has

(4.5)

i(dE)

M LPd (Me) - d (M.)] dt Consequently (4.6) d (Me) = d (M).

Hence the pressure is continuous through the free vortex sheet.

For the sake of simplicity, let us suppose that we deal with

only one moving body.

The absolute velocity Vin the domain D' really occupied by the fluid can be divided into three components.

(4.10)

One has

- the velocity 17d induced by the vortex distribution (D. , 2 _CL)

+ (, T) which permits

the fluid to be adherent to the

side 1 of the hull

- the velocity V of some incident flow,

- the velocity due to one or several free vortex sheets shed

by the body.

According to Section III , we have

rn -i _1E(N

(I

(4.7) d = curlF T(H'

tt'

and T is determined by the Boundary conditions

- -

-Vd (M.) VE (M.) - V (M.) - Vf (M.) on

(4.8)

'1M (M) Oon

V is due to causes located outside

I

and one may consider that there exists a velocity potential such that

at least in the region of D'close to + Vf is irrotational

M. being given on and

1

one can find a path starting frcrx M1

tional everywhere along this path. Putting

(4.9)

= v,

we obtain, by integrating (4. 1) along the path d (Me) - d (M.) Hence (4.6) gives

! and inside D. +

e 1

{

V (Ne) v (Mt)] 0 1.

outside + .

Therefore, M and

e respectively -with MeMi = nM (0+),

-Consequently

(M)A VR(M) R(M)A

_{M LR}

Ar(M)

_e VR(Mi7J' =

M

1V2(M)

-

v(M.)J

Thus, when the relative motion is steady, Tf (M) and VR (M) are colinear,

and both are in the direction of the bissectrix of the angle (yR (Me) R(Mi If the relative motion is unsteady _, ) VR (Me) and IVR (M.)

are no longer equal to each other and therefore the direction of (M) is no longer that of the bissectrix.

Equation (4.10) expresses the dynamical equilibrium of any

part of a free vortex sheet.

DEFINITION OF THE FORCE EXERTED BY THE FLOW ON AN

ELE-MENT OF THE BOUND VORTEX SHEET ADHERING TO A MOVING BODY

-Let us now consider a set dE of fluid points belonging to the

element d of the bound vortex sheet.

We have

VR (M.) = 0,,

(4.11)

and

(4.12) 0= _dtd

Conversely, the equation

(4. 13) _{= [Pd (Me) - 1d} (M.)] _T1M

dZ (M)

shows that the fluid sets adjacent to dE exert the force d on the set dE.

7-d _{I (dE) = 0}

'

But the expression for

f1(dE) does not reduce to)Ipd(M

) _-

!Mi)I1M d2

for, because of the adherence of the fluid to ,

a force - d..

is exerted by the element d of the hull surface on dE. The equilibrium of dE requires

(5.1)

over and write that the two vortex distributions

-(5.2)

(D.,La...)+(

(53)

( + ( ,

sum up to the total vortex distribution

(DC1')+(

11\

/ THE VORTEX FAMILY (D., 2 (

_{) + (.}

velocity

-4

(5. 4) V'(M) = curl

1,t

We can introduce a new vortex distribution

-4

The new vortex distribution (5. 1) will be chosen so that the

V

THE STRUCTURE OF THE VORTEX SHEETS

GENERATED BY A MOVING BODY

is identically null inside D . Inside D., V' is the velocity of a fictitious fluid

e i

motion due to a certain force F + Ft per unit mass(determined in Section VI). One has

(5. 5) curl V' . 2 .TL inside D..

-

E 1

The condition

(5.6) V'(M) OifM D

is obviously equivalent to the condition

d(M)+

curi4jJ(

(M) J4Tt JD MM1 e

d"N.)

(5.6') (M ) 0 on e

-The fictitious fluid motion defined by V' is one of the fluid

motions which could exist inside the vessel bounded by Z if the body were

at rest.

(5.6') means that

ii

(']\

curl ±

)

(jon

curl LiT

T'(M

(N)

- _MM

Because of the discontinuity of the left-hand side through I , this equation

can be written in the form

(5.7)

AT(\

11

(t

_(M')

Ji 'M

where M is located on

This is a Fredholm vertorial equation of the 2nd kind which is quite analogous to (3.9) in spite of the fact that the condition to be verified

concerns the side _e

of I

instead of the side L

First one observes that this equation is singular, since the

left hand side vanishes when ' (M) = being a constant. The right-hand side must therefore satisfy a possibility conditions.

one must have

j

As *1

-*

(M

{,wi4

I

T'()

j7CJJ M

j

{ LJJ T' (M') d

()

_/

!

11 LL ( i) D J4ltJjjk N1N

This condition is obviously fulfilled since the latter expression is equal to

[11 E

') _M)]

e =

It follows, as in the case of (3. 9), that the solutions of (5. 7) are 'r'(M) = (M) + ' with n T = constant on

1'

and therefore that there exists one, and only one, vortex which is

tan-gent to 1. and satisfies (5. 7).

a system of orthogonal coordinates.

(5.2) are closed. The lines are of any segment of a vortex filament

line c? (fig. 5.1).

-.

-.

T-T'

THE VORTEX FAMILY ( Z

, & ) +

M'9t1'M

-Let us consider at each point M of

2

the unit vector

s(-*1

making a right-handed system, with V in the direction of T'(M). Let

-+ N

denote a line tangent to T' at every point, and ds' the element of arc

/ i

of Z in the direction of . Similarly let be the class of the lines located on 2 and orthogonal to the and the element of arc

of in the direction of . The variables a-", define on

The vortex filaments of the distribution closed too, since the two ends on

-*

necessarily belong to the same

We suppose that there exists at least one free vortex sheet Z

This vortex sheet is attached to the hull along a certain open line which

divides the hull into two parts denoted and . The exterior side of

Fig. 5.1

Fig. 5.2

One has -(5.8) '(M) =- A I V'(M ) - V1(M.)J = A \P(M.) on M _L e

Z is denoted .a and its interior side is denoted

ii,.

. A similar

notation is adopted for . Furthermore , and are the two sides

2

of , L.) being attached to

!,

and to Z, . On the unit

-vector normal to is in the direction from .! towards .!, (see fig. 5. 3).

The absolute velocity V on is

- -

-(5.11) V (M ) (V + V + V )

e

e o d

fM

where V is the contribution of ( Z ) and V that of ( : , .J ).

d f r &

We recall that (D. , 2 C1 _{) + (} T' ₎ contributes nothing inside D . On

the absolute velocity is

e (5.12) -

--. -, -ø -.

V(M)=V (M)=(v v'+V

+v)

I

El

o d

fM

I

The total jump of V through ., from M. to M , is

i e

-0 - -

-(5.13)

V(M ) -V(M.)V (M

_{e i R e}

_)=nMA

TM

-*

and, Vf being continuous throuuh , the part of the jump V(M ) - V(M.)

e

due to the distribution T _T' is

-p -, -0 (5.14)

v(M)-v(M)=v (M)+v'(M)=

A

_{(T_T')M.}

d e

di

Re

i M Conversely (5.14')

_(TT)M

flMA[VR (M)+v' (M.)]. /

Let B , B' be two points on , close to each other, and

the vortex filaments on which intersect at B, B', respectiviiy.

The part of the free vortex shet between cand

*j

is a free vortex

ribbon Lf. Let Bf B denote two points on , respectively, close

to B, B', the segment Bf B being orthogonal to . We suppose that Tf

is, for instance, in the direction of BBf , and that the three directions

( _nB , B1B , Tf) make a right-handed system. The intensity d of L

-r

is equal to the circulation of V in the closed cirduit Bf

B' B

Bf B , with

_____ 2 f2 1 1 f2

Bf2 Bf _{= nE} (0+) , BfBf

= B (0+), and so on. Thus

f 1 f

-p

This intensity remains constant along the vortex ribbon Lf(by Stokest theorem) According to the direction of Tf just assumed

(5.16)

df

₌

1BfB.J>O

where tp is the unit vector in the direction BB . Let

denote the unit

vector inthe direction of B

ff

, the three directions (- _{, ,} ) make

f a right-handed system. are (5.17)

dc:

dr

L

Let L denote the vortex ribbon located on between the

1 _r

'

points B, B', and let be its two edges. L2 , , are the

(I' L1 V'

analogous of L1

, ,

on

-. We choose B1 , Bj on

respectively, with in the direction ofa unit vector normal to

and take on L1 so that (- n, T4 ) make a right-handed system.

1. _, -

-,

B2 , B and (- n,

, 4t ₎ are on the analogous of B1 , B1 (- n,

c)

and is in the direction of B2B. The intensities d rj , d 1j of L1, L2

- --0

1-')i

-)jJ={(;e)-

V(B2.B,

As afore-mentioned, dt is a constant along Lf. We have also (5. 18) di = constant along L1 , d = constant along L2

-

-,

for , according to (5.11) and (5. 12) and to the properties of Vt , curl (Vf+Vd)

is null on both sides of Z

and of !

-V is not irrotational because the vortex filaments are not closed contours. Similarly at least one part of the are not closed

contours, and Vd also is not irrotational. But a non-closed % can be

considered as an arc of a closed vortex filament ( + _{' ) and Vd} Vf

sum up to an irrotational velocity(Vd + Vf).

It follows that there exists a velocity potential, which will be

denoted _{d +} with (Vd + Vf)M = _d

+ fM

As the discontinuity of

V through is tangent to . , that of V is tangent to and as is

d f f

continuous through and Vd continuous through ! , this velocity

potential may be written in the form

f)M7J

h)-

!(tN)

which expresses that _d + is due to a normal dipole distribution over

-Let , resp. , be the density of the distribution over

resp. . We put

dM)MwM

In the above formulae, represents the set of all the surfaces of the free

vortex sheets if there are several vortex sheets, for it can happen that L

which cannot end in the fluid have its two ends on two different lines

(example fig. 5.4).

Let us take on , at each point M, two orthogonal unit

vectors ( _'

_'

) such that the three directions

(-,i

j) make

a right-handed system, with _is in the direction of (T - T'). Similarly,

on , the system (-''f

_1r

'Sf) is right-handed and 1Sf is in the

direcion of Tf. The family

ofhe

tS and that of the curves tangent

everywhere to i make a system of orthogonal curvilinear coordinates. The direction of the increasing arcs d , ds are those of is

(5.19)

One has

d (Me) - d (M.) =1

(M2) _{- d} (Mf)

Let us consider the almost closed contour starting from Bf , arriving at

Bf2 , and intersecting at B1 and at B2. The total jump of +

+

'fB

- d + fBf1

[

so that, when Bf tends to on and B1 (5. 20) (B) = . (B2) - h-i. (B1).

Consequently, J- is discontinuous through any line along which a free

vortex sheet is attached.

and One has and similarly

p(MonZ

-e1)j on

iJ-be considered as iJ-belonging to are both directed towards

It follows that the two vortex ribbons L1 and L can never

-a s-ame vortex ribbon T1

- T' and T

- T'

1 2 2

if Tf is in the direction BBf on Lf. In this

while L1 and L2 arrive at . If there

arrives to a certain arc B B' of

on . Because the vortex filament

from B B'. This is sketched in

₀₀

case (fig. 5.3), Lf starts from

free vortex sheet, Lf

and BB' , BB' have opposite directions

+ is closed, both L1 and L2 start

Fig. 5.5.

is only one

When the relative motion is

LVR (Mf1) - VR(Mf)J and

ar =d

=

dr

I

?,

Fig. 5.5

steady, V

(M )-V (M )

= R f2 R f

(steady relative motion).

dc3 have opposite signs. Consequently, on

, d

(B2) = - d gS (B1), and

0

Consequently, L and r which are defined up to additive constants can be

considered as identical functions on On BBj and B2B da and

(5.21) df

=drj

+dlj

on BB'

leading edge

Fig. 5. 6 shows after Maurice Roy { 1

J the configuration of

the vortex filaments when one deals with a wing of finite aspect ratio in a uniform motion of translation when the relative motion is steady. The important point is that the vortex ribbons and L2 do not cover the whole

surface Z of the wing. The vortex ribbons L2 are located on the section

side of the wing. The vortex ribbons L1 are partly on the pressure side and partly on the suction side. There is on a vortex filament common

to the family of the and to that of the 's. It starts from the edge

of and arrives at the other edge. It is the second frontier between

and . The part _- ( + 2E, ) is covered by closed bound vortex ribbons

entirely located on

In the case of Fig. 5.4, it is clearly seen that the vortex ribbons L1 and L2 are distinct from each other.

trailing edge

/

The above description of the vortex distribution

The existence of the free vortex sheets, and,

conse-quently, the lines are fixed on the hull when t

varies,

(ii) - the non-occurrence of a true separation.

The first assumption is verified when the curvature of .! is considerably high in certain regions, for instance near the trailing

edge in the case of a wing, and in the case of a surface ship or submarine vehicle when there exist lines similar to chines,(even when the curvature

is not really infinit. If the curvature of the hull is everywhere moderate,

the assumption remains acceptable provided that the amplitudes of variation of VE (0) and ...i with time be not too large.

The second assumption is acceptable also when the separation

is due to a strong adverse pressure gradient behind the shoulder. However, when the hull is turning, the angle of attack varies along the hull (fig. 5.7.) the flow tends to skirt the relative flat surface of the hull and to pass from

the pressure side to the suction side. This phenomenon can generate strong eddies near the after body

T _{) involves two implicit assumptions, namely}

THE INTEGRAL EQUATIONS OF THE PROBLEM

-From the above Section it follows that the expression for the absolute velocity V of the fluid may be written in the form

V(d+)ifMED'.

f e

(6.1) v(M)

.- -0

-v

+vt+V(

+f)=vE (M)ifM E

D..

o d

In these formulae, V is the incident velocity due to causes other than the

body (effect of the boundary of D' , singularities not generated by the hull itself and so on). One has

(6.2) V = V in D' in the vicinity of ! and in D..

o o e e i

V' is determined by the condition

-. - 4 T'(M

i(M')+cwJ ±_

(

r1.(i')

curl (6.3) Vt = MM' MW with V' = 0 inside D e - - 0 .-. .-*

1'

And V + V due to the vortex distribution T-

+ (' ,

_L)

is such that

d f

Vd + Vf =

v

d + f) with VI

THE EQUATIONS DETERMINING THE TWO VORTEX FAMILIES

(M')

On , the two sides

1

is in the direction from

and

towards

are distinguished from each other,

. The choice of !j and

-, 11

determines the parts , of .

On Z

, n is

in the inward direction.

(6.4) _4f d

₍

₍

₎

On the lines , one has

(6.5) =

Since the normal derivative of is continuous through . the condition - -* V = V inside D. E entails (6.6)

+ V + V

+ _{f) = VE inside D..} This is satisfied if, and only if

(6.7)

d + o + °"

1

.

being such that

-

-4

(6. 8) V

- V' = V

_{inside D}

E i

Finally, there are two integral equations. ckie of them determines T' and

therefore Vt inside D.. This equation can be written inthe form

(6.9) _AT"M)+cw (

r(M')qMl)

CUJ ¶n.()

2. M _MM1

with n . T' 0. The properties of this equation have been explained in Section V. and therefore T' are functions of time.

The second equation is

2 nit iJ ' _M' t1P

If _(M'

_----

_f___

(M))

(6.10)

n)

_t1M'

with two conditions. The first one is alr ady known :

(6. 12)

Hence

(6.13)

The second one expresses that the circulation in any closed

fluid circuit is time invariant ; it can be written down immediately

(Mi, t) = . (B , t ) with

(6.11)

B Mf

=

V (F, t') dt' , t

where P is the fluid particle which is at time 't in B and is in Mf at time t.

THE EQUATION (6.9)

-Let uE. (j = 1, 2, 3) denote the components of VE(0) on the

moving axes 0 (x1 , x2 , x3) and (j = 4, 5, 6) those of .1 Let us put

cw

£

d(M'+i

'

M' _{_, -. MM'} 6

Vt is the velocity ' when -CL = , i. being the unit vector in the

j E,

j-3

j-direction of the axis x1 if j =

4...There exists a velocity potential

-b

-4

)

such that i A OM - =

3 - 3 3

.=

x.+z

1 E

When the three potentials (.j" (j = 4, 5, 6) are known,

obtained from (6. 13) for any motion of the body. For any hull, t he calculation of the (. h can be performed once for all.

EQUATION (6.10)

-(i) - If the fluid is unbounded, and if no free vortices are shed by the hull, then (6.10) reduces to

(6.14)

_{_.I(M)_4.}

L(M')

(fri1t)

This is the Fredholm equation for the Dirichlet interior problem, the given values of

d on being those of . The kernel of this equation is the sym-metric of the kernel corresponding to the Neumann exterior problem.

Hence equation (6. 14) is not more intricate than the equation of this latter

problem. However, if 0, the calculation of

i must be performed

before solving (6. 14). But, (6. 14) gives the possibility to obtain the velocity field and the pressure distribution on the hull in a much simpler manner than the usual one using a source distribution over the hull.

- If the fluid is not unbounded, is a functional of

o d

and therefore of . Consequently , (6. 14) is no longer identical with

the Fredholm equation related to the Dirichlet interior problem. In any case, the solution obviously exists and is unique.

- If one or several free vortex sheets are shed by the

hull, then (6. 14) becomes much more intricate in the most general case because the surfaces are unknown. But, in many cases of practical importance, a reasonable assumption can be made to obtain a first appro-ximate solution, which will be refined later if needed. This approximation

This gives B' and also 'C for every point M belonging to the sheet. Equation (6. 14) becomes an integral equation giving . on

Z

when

- _{)is supposed to be known on and finally reduces to a Volterra} integral equation giving on as a function of the history of the motion.

The procedure to be recommended depends both upon the hull shape and the nature of the motion of the body. Our purpose is not

to discuss this question, but, we shall recall in the last Section some

examples connected with the present theory.

VII

HYDRODYNAMIC FORCES EXERTED

ON THE BOUND VORTICES AND THE MOVING BODY

In the preceding sections, we have seen how to determine such a total vortex distribution inside D., over

and over the surfaces !

of

1

involves a reasonable choice of the lines and the replacement of

by

B'M= J V

'C

(P, t')dt'

(C I - VE (B', t') dt'

,

B'M =

the free vortex sheets shed by the body, that it generates inside the domain D' actually occupied by the fluid and inside the interior of the hull an

_,

-absolute velocity V - V satisfying the conditions

(7.1)

Consequently, to calculate the systems of forces exerted on the hull and on the bound parts of the vortex distributions we may replace the hull surface by a fluid surface which moves without shape alteration. But we have to express that the fluid is in dynamical equilibrium every-. where and at every time t.

DYNAMICAL EQUILIBRIUM CONDITIONS

-Inside D. , the fluid being at rest with respect to the surface

I

1 -*

'.4

-.

-,

the force per unit mass must be such that .±_ = - . Putting

-4 -. _, _P

= F + F', where F is the exterior force per u.nit mass, we obtain

(7.2)

1p

pd

= F' - inside D..

1

As curl = is different from zero when the angular acceleration

-,

of the body is not null, we must admit-that curl F' = curl . We shall

take (7.4) where -4 _-4

V=V insideD.

E n V n VE + n V on

I

where V' is the velocity generated inside D. by the vortex distribution

1

(D.₁ , 2 _A_ ) + ( , 3 ), because F'so defined is the only force per unitmass

which vanishes with

This leads to

p

p_d - L inside D

_ii

is the velocity potential defined by (6. 13).

-4

(7.3) F' = inside D.,

(7.7)

(7.7')

where (7.8)

Now let us consider a set of fluid points dE located between and and belonging to the bound-vortex sheet ( ,

I ).

It must be in dynamical equilibrium under the effects of the forces

- d due to the adherence to

I

I-- _Pd d exerted by the fluid filling up D.

+ _d dZ exerted by the adjacent sets of fluid points inside D'

e This gives

(M)

de - Pd1M M

dZ (M)

According to Euler's equation (4.1), and taking into account the fact that Ct.) = 0 in the vicinity of E , we have

(7.6)

I\

d

(M)=_p(-__,/_

e-

-to M of the velocity potptial

+

''

_'-

_,

T-T'

£ E

where is the total velocity potential existing at least in a certain part of D'_e including Z Thus we obtain on o r v

(r)J

d.

(M) -p -L fM M I-) fl 1 VR (M) =

_[R

_{(Me) + TR} (M.)J =

(M) °n

As shown in Section VI, _{(Me) -} .(M.) is the jump from M. generated by the vortex distribution

It is therefore possible to _express in term of the total vortex distributions. We immediately obtain

(7.9)

As pointed out at the end of Section III, d iL.the force

exerted by the fluid located inside D' on the element (d , ) of the bound

vortex sheet over

Lastly one observes that the fluid interior to D. is in equilibrium under the effect of the 8ystern of forces _d (M.)

M

dt exerted by

the

bound vortex sheet the system of inertial forces _- dD. and the system of complementary forces dD.. This follows from (7.2) by Ostrogradsky's

formula (7.10)

d (N)

TI _(F

'-'-'

)D#jj

It'

t

_?)

.(M+A

N)M(

D

COMPARISON BETWEEN THE SYSTEM OF HYDRODYNAMIC FORCES

ON THE BOUND VORTICES AND THE SYSTEM OF HYDRODYNAMIC FORCES

- ON THE BODY

-Let us define as follows the four system of forces which appeared from the preceding analysis

system of hydrodynamic forces_j.Pd(M d Z (M)on the body,

system of hydrodynamic forces d1 on the bound vortex sheet

I.

-system of inertial forces

withinD.

1

E

dDj

of the fluid located

system of complementary forces {PF! dD.J on the fluid located

witrun jj.

1

From (7. 10) it is seen that the system of forces

[

d (M.) M d! (M)] exerted on the vortex sheet by the fluid located within D. is equivalent to the system ( + ). Furthermore, the equation

-{Pd (Me) - d

(M./M d(M)

I

For the system of hydrodynamic forces exerted on the hull to be exactly equivalent to the system of hydrodynamic forces exerted on the vortex sheet ( Z ,

I

_), it is necessary that the sum +

I

be equivalent to zero. This is true only if 0 , = 0 fo ₉ only

depends on the derivative while depends on

() and on

-,

dt

Hence the equivalence

( requires

= 0, 92%= 0

(7.13)

does not require the steadiness of the relation motion of the fluid.

Condition (7. 13) is satisfied in two cases and in these two cases

only:

either a) the body is in a uniform motion of translation,

or b) the body is infinitely thin.

In particular, if the motion of the body is uniform but includes aconstant, non-null angular velocity, then = 0, but is the system

of the centrifugal forces on the fluid located inside D. , that is on the fluid

"displaced's by the body.

QUASI-STEADY FORCES, DEFICIENCIES AND FORCES DUE TO VIRTUAL

MASSES _-

_Letusput:

(7. 14) = system of

forces{

_e()M

aq)j

(7. 14t) = system of forces{

(A(M

(N)].

From the equality

shows that the system is connected with the other three systems by

(7.12) . = )

and from (7. 9) we get

and because of (7. 12) (7.15)

can be considered as due to virtual masses' which reduce

-to the so-called "added masses" when the fluid is unbounded, V is null,

0

and no free vortex-sheet is shed by the hull.

If the fluid is unbounded and null, and if the body motion becomes uniform at t , then the fluid motion still depends on t when t > t when

0 0

free vortex-sheets are shed by the hull. The differences

f =

()

- ()

t= t

and =

dt =' -

dt = t

0 0)

the "deficiencies". of and

d' respectively . The estimate

of J

obtained by neglecting these deficiencies is the "quasi-steady" system of hydrodynamic forces. The substitution of for is acceptable only

when the acceleration of the body is small. Even in this case, some compo-nents of the added masses effects are of the same order of magnitude as the corresponding components of the inertial forces of the body itself and cannot be ignored. Thus the estimate of the hydrodynamic system of forces exerted on the body in the case of slowly varying bcdy motion is

(7.16) -

/

q..

where is calculated by taking = 0. Hence is due only to the added masses effect.

In the general case, the substitution of for involves errors

affecting both and 2i 0

the domain

The sum of the last two system represents the deficiencies due to a delayed circulation

THE ORIGIN OF THE FORCES EXERTED BY THE FLOW ON A BOUND VORTEX DISTRIBUTION

-The above considerations started from the idea that a

dynamical relationship necessarily exists between the hull of a moving body and the vortex distribution satisfying the adherence condition on the hull surface.

Another viewpoint is that any vortex filament which does not move with the fluid is necessarily submitted to forces exerted by the adja-cent Sets of fluid points.

The proof is classical. It is sufficient to summarize it.

Let be a vortex filament, its bound part, ds1 an arc

of

4 .

_{Let 0 denote the middle of the arc ds1} , Oz an axis in the direction

of

ã,

and 9 z a system of semi-polar coordinates, Let D' denote

D' ₌

[ii

O

)Iz

dsJ.

7I

Hence D' does not include the arc ds1. We consider the relative motion of the fluid with respect to the system of axes just defined. The velocity may be

written in the form

V =

(v'- Sv')+ DV',

R o

where defines some incident flow, Vt is the velocity induced by and

Vt that induced by the part ds1 of . Let dE' be the set of fluid points inside Dt, and dE1 the set of fluid points belonging to ds1. One easily sees

that the momentum I (dE1) is null, and, therefore, thal 7' (dE') being the momentum of dE'

I('iEl)

I(+dE

( is in the outward

where S' is the boundary of (D' + ds1). One readily obtains

(7.17) urn

rt

where r is the intensity of

: r = r .

_{' is} being the unit vector

Si

tangent to ds1 and V. the finite incident velocity on the arc ds1 ...

v.=v+v'

3v'.

On the other hand, by the momentum theorem, one has

- 1(

I'd4j]

where dFT is the force exerted on dE' by dE1 . From Euler's equation

(4. 1), we have inside D' P

This gives, when R .-. 0, = V.P. ' V' + constant on S, and

(7.18) urn

J -r

-i(

_I -+

A ( ₊

_-#

By comparing (7. 17) and (7. 18), one finally obtains

-

-

_,,\ci

Li

(7.19)

[dT

prAV.ds1

dFT is the force exerted on the arc ds1 of the bound part

of by the adjacent sets of fluid points. The force vanishes with V.

that is when the arc ds moves with the fluid.

Formula (7. 19) is of practical interest when the vortex distri-bution equivalent to the hull is replaced by a unique concentrated vortex and a suitable distributions of sources or normal dipoles on the hull surface.

This formula does not imply that the fluid motion around the

bound arc of the vortex filament is steady. If E varies with time, one

in the time interval (t, t + at). is distinct from although their supports

have a common part ; the free part of does not coincide with the free

partof

Let us consider now a flat vortex tube inside the bound vortex sheet over a hull . This tube has a thickness ,

a width d'

Let. ds1 be the element of arc of the tube. Applying (7. 19), we have

(7.20)

for VR

t

p{T(M)

A(M)ds

=

(M) = lim V. (M).

d40

'

We know that, if the motion is unsteady, dFm must be replaced

by d _T However there exists no contradiction. The term -

= P

-which appears whenhe vortex tubes belong to a sheet comes from the integration of

_-?

through the sheet. When the vortex tube is isolated, there is no discontinuity of V on the surface St and the contribution of the

term p

in the integration of V on St is 0 (R' ds1) , thus negligibly

small. This is not the case when one deals with a sheet.

CASE OF A HULL EQUIPPED WITH MOVABLE APPENDAGES

-The treatment of the problem arising from the presence of such appendages obviously depends upon their position with respect to the hull. When the axis of the rudder coincides with the edge of the stern, this rudder

must be regarded as a part of the hull. The shape of the hull varies with time. At each instant t there is however a vortex sheet adhering to this hull.

The method of Section VI therefore applies in principle. But separation may occur at the leading edge of the rudder because of lack of continuity. Furthermore the effect of the viscous boundary layer is never negligible in this region.

When the rudder (or diving plane) is at some distance from the

hull, the rudder behaves as a lifting surface with a small aspect ratio.

GENERAL

-It has been seen in the preceding Sections that the vortex theory applies to any body moving in water whatever its motion may be. The methods to be used in practice may considerably vary with the shape of the hull,

the motion of the body, the boundaries of the fluid domain. To perform the calculations, it may be advantageous to substitute normal doublet

distribu-tions for vortex distribudistribu-tions because a regular scalar Fredholm equation

then replaces a vectorial singular Fredhoim equation. This is why the vortex distribution kinematically equivalent to the body has been divided in Sections V and VI into two parts, one of them being equivalent to a normal dipole distribution. When the motion of the body consists of a pure translation, the vortex theory leads to computations which are not more complicated than those involved when source distributions are used ; they are even simpler when the distribution of the pressure over the hull is needed. This can be of interest when there exists an incident unsteady flow. When one deals with a body completely submerged near the free surface, the waves

generated by the body and the vortex sheets attached to it can be determined by applying the linearized theor to the normal dipole distribution equivalent to the vortex distribution

TT'

+ T4 ). In the case of a hull piercing the free surface, the problem is of a known nature if the speed is small enough for the so-called Zero-Froude number approximation to be sufficient. But attention is to be paid to the fact that as shown by Kotik

and Morgan [ 6J and also by the writer [ 7] ,

the Zero-Froud-number

approximation yielded by a normal dipole distribution has to be corrected to become acceptable.

rudder becomes meaningless and the thin wing theory is to be used.

VIII

THE APPLICATION FIELD OF VORTEX THEORY

One of the main features of the theory developed in the present paper is that it includes the case of bodies which are neither thin, flat nor slender. However, to the knowledge of the writer, the vortex theory is still usal

only in cases of thin lifting surfaces. There is thus _{a need for more general}

methods and one of the purposes of this paper is to give means to extend the field of applications.

In this Section the present field of application is briefly out-lined. Yet the problem of manoeuvrability and control of marine vehicles is examined in a more detailed mariner, for progresses in that domain seem to be strongly needed.

D'ALEMBERT'S PARADOX

-There exist many proofs of this theorem. The following one

may be of interest for it clearly explains the physical meaning of the

hypo-thesis required for its validity.

The body moves with a constant speed VE in an unbounded,

inviscid fluid at rest at infinity. One supposes that no separation occurs and that no vortex sheet is shed by the body. Let ( ,

I )

be the vortex sheet which allows the fluid to adhere to the side of the hull. The

votex filaments are closed rings on . The; are orthogonal to

the relative streamlines

. Leti

be the unit vector tangent to a line

.- _{-. 1T}

in the direction of T, and the unit vector tangent to a line ' in the

direction of the relative velocity . A vortex filament is defined

by the curvilinear abscissa _{of its intersection with a line chosen}

once for all. The intensity dr

of the vortex ribbon located between two vortex filaments ( _çj- ) and ( c% + is a constant. One has

(8.1) dE

and the absolute velocity of the fluid is

(8.2) _V (M')

0(V)

_I

_f

d5 ()

Fig. 8.1

The relative velocity is VR V _{- VE. Thus the hydrodynamic}

force on the part (d

, -

) of the vortex distribution is

,with

d

=-p T (M)

A

_(VE)d!.(M),

i.

4

=-

T(M)f\v(M)dZ (M).

The system of forces dEr. is that of the forces exerted by the vortex distribution on itself and is therefore equivalent to zero. Also the

D n)

systems of forces of the general theory are equivalent to zero. Consequently the system of the hydrodynamic forces exerted on the body reduces to that of the forces d . This gives the general resultant

= p

T(

A

()

and the resulting moment with respect to a given point 0

Hence (8.3)

Al

sie f,

.E JLO( 0

Pj{T(M)

A -:-

-=eVEI

°J

.d(pJ

d0/

-

7o (8.4) 1

Y(0)J

(

)d(M.

0

This term is not null, at least in general.

Equation (8.3) expresses the d'Alembert paradox. If free vortices are shed by the body , then has to be divided at least into two parts, on each of which is constant but with a different value, and (8. 3) is no longer verified. has two components ; one of them is a lift and the

other one is the induced resistance".

KUTTAJOUKOWSKI'S THEOREM

-The first version of this theorem concerned wing profiles in a uniform motion of translation, with V = 0. The wing is an infinite cylinder and its profile is its intersection by a plane normal to the generatrices. The problem can be considered as the limiting case of that

of a wing, the aspect ratio of which tends to infinity. When the aspect ratio

is finite, the relative velocities on the two sides

I

, of the wing near

the trailing edge are equal and opposite. This follows from the continuity of

the flow between and and between and . Hence, in the

'i

-,

case of a wing profile, one must have VRE = 0 at the trailing edge 13. This is the Kutta condition which determines the density of the vortex sheet on

that is the ratio on the contour of the profile.

The Kutta condition holds when the motion is unsteady.

The theory developed in the preceding Sections applies to wing profiles. But because one deals with two-dimensional motions, the