Forces in oblique towing of a model of a cargo liner and a divided double body geosim

ME DDELANDEN

FRAN Tel.: di

7,-;2,232e co Den

STATENS

SKEPPSPROVNINGSANSfrolsMtele

(PUBLICATIONS OF THE SWEDISH STATE SHIPBUILDING EXPERIMENTAL TANK)

Nr 57 GOTEBORG 1965

FORCES IN OBLIQUE TOWING

OF A MODEL OF A CARGO LINER AND

A DIVIDED DOUBLE-BODY GEOSIM

BY

NILS H. NORRBIN

Also published in "Sehiff und Hafen" Hamburg 1964

SCANDINAVIAN UNIVERSITY BOOKS AKADEMIFoRLAGET-GUMPERTS.G6TEBORG TECH, ...__NISCHE

uersonar

uxinuonunt roar

scheepehydromeabanke Archlef

Icov mootrociv,)

MaregOlamovireiscsect

*OW

MO 00

asgks rs.

vow,

mew

sit

%If. ereetrc

SCANDINAVIAN UNIVERSITY BOOKS

Denmark: 1KUNKSGAARD, Copenhagen Norway: IINIVERSITETSFORLAGET, Oslo, Bergen Sweden: AKADEMIFoRLAGET-GTIMPERTS, Goteborg

SVENSKA BoxFoRLAGET/NorstedtsBonniers, Stockholm

PRINTED IN SWEDEN BY

In this report are described the experiments, results and analysis of force measurements in stationary oblique towing tests with an

1:25 scale model of a single screw cargo liner, and with an 1:50 scale submerged double-body geosim.

The larger model is identical to that "model of MANDALAY"

pre-viously used in open lake manoeuvring tests, thecircle tests part of which was presented in a publication from SSPA two years ago.

In

the captive tests the model was free to heave and trim, and the total forces were measured by an internal six-component balance, rudder forces by a three-component balance. The model was fitted with two

alternative stern arrangements, and it was run with and without rudder, propeller, and bilge keels, and in the full load and ballast

conditions. It was also towed normal to the flow.

The image or double model was towed submerged, and in addition to three-component total forces lateral force distribution was studied

by means of two separate shear transducers at

sections near the

forward and after shoulders. This model, corresponding to the full

load form, was tested with and without rudder andbilge keels, but

only without a propeller.

The object of the captive tests was chiefly

to supply experimental information on the influence of the free surface in relation to the application of slender body theory for

predicting the lateral forces on the hull;

to supply first and higher order stiffness and control derivatives

for these particular model forms and to show how these derivatives were influenced by ballast conditions and screw loading, and by

the presence of bilge keels.

The shear force and moment measurements are compared with

calculations from the momentum theory. The results are presented

in graphs and tables, and there is also a comparison with data

published for similar forms. The main conclusions are summarized in

Section 8.

The present report originally appeared in the German journal Schiff

This report is the second one presenting the results of an experi-mental investigation of manoeuvring qualities of ships within the

special program set up at the Swedish State S

_{hip buil}

d-ing Experiment al Tank (SSPA). The first

publication in

this series was concerned with circle tests with a radio-controlled

model of a cargo liner in an open lake [I].')

An analysis of these circle tests, and of the zig zag tests also

per-formed with the same model, has given numerical

_{values to the}

effective hydrodynamic coefficients appearing in first-order approxi-mations to the linear and non-linear equations describing these

mo-tions [2, 3]. From such tests it will also be possible to deduce the

hydrodynamic coefficients associated with a higher approximation to

the physical problem recently suggested in ref. [29]. Even better

approximations, however, will require such an exceptionally accurate record of the motion, that is seldom to be obtained in out-door_model

testing or full-scale trials. For this reason, again, it is _{no surprise to}

find that the motion of the model or ship usually will be "sufficiently well" predicted from a suitable approximation, provided the hydro-dynamic coefficients can be estimated from earlier manoeuvring tests with a similar form, or can be calculated with some confidence.

All these effective coefficients are functions of hydrodynamic forces

in coupled motions, and are conveniently described by first and higher

order force and moment derivatives; see [4], e.g., and further

dis-cussion in Section 4. Only a few of these derivativesmay be calculated

by purely theoretical methods and another few by

use of

semi-empirical formulas, but most of them can only be obtained from

experiments, using one or more of the special facilities now often found

in model testing laboratories.

The majority of experiments in this field has been directed towards

naval projects, and more specially to submersibles, but during the

course of years a good deal of data has been published on commercial

forms as well. Part of this literature, up to 1959, was reviewed in

ref. [4]. Among more recent publications may be mentioned those by TSAKONAS [5], BENDEL [6], BtiLow [7], and SUAREZ [8] On results of

steady-motion force measurements in towing tank and rotating arm facility tests, and the reports by MOTORA and COACH [9], and by

PAULLING and WOOD [10], in which are included results from force

measurements using two different types of the lateral "planar motion

mechanism". (-P 1 an ar Motion Me c hanis m",

of course,

was the name originally attached to the unique facility designed by

GERTLER and GOODMAN, [11], but it seems to have been widely

adopted for all the new mechanisms capable of producing the pure

small-motion yawing (or pitching) and swaying (or heaving).') All the captive tests performed with the Swedish cargo liner form

here referred to were confined to straight-line steady towing in the main basin, first investigating the effects of alternative stern and rudder arrangements, screw loading, and bilge keels, in loaded and

ballast conditions.

Whenever experimental results are obtained they shall be analyzed for a better understanding of the hydrodynamic phenomena involved and to support the ultimate formulation of theoretical or semi-theo-retical methods. For this reason, the oblique towing tests were supp-lemented by athwartship towing of the surface model, and by oblique

towing of a submerged, divided, double-body geosim of the parent

model.

The athwartship towing tests furnish data on cross-flow drag

coefficients, which may be related to the large-value non-linearities

of the lateral forces in oblique towing. Similar tests with different

models have previously been reported by THIEME [12].

The image model tests were performed to provide data on the influence of the free surface upon the stationary lateral forces and

moments at small angles of drift, which are more fully discussed in Section 3. The author does not know of any earlier tests of this kind,

although GAWN did publish the results of total force component

measurements on a semi-submerged ellipsoid, which were compared with theoretical values, [13].

The transverse shear force measurements on the divided model were included to give an idea of the longitudinal distribution of the

lateral forces, and, in particular, to make possible an analysis of the fore-body lift not obscured by viscous effects.

i) Since the original publication of this paper there have appeared two new reports

on oscillator tests, one by VAN LEEUWEN covering a wide range of frequencies, [31], and the other by CHISLETT and STROAI-TEJSEN, [32], in which non-linear characteris-tics are also deduced from the experiments.

2a.

0,

Fig. 2. Body of revolution in oblique flow. Systems of axes fixed in body Distribution of transverse forces

-PLANE )"- PLANE ')'-PLANE

22 2'

Fig. 1. Orientation of body axes.

22 21

Fig. 3. Coordinate systems in conformal transformations. 6 (BEN I (50) 2 zz? z, y, PLANE

2. Symbols and Units

When applicable, the symbols and abbreviations here used have

been chosen in accordance with the nomenclature suggested by the

ITTC Manoeuvrability

Committee, and the general

practice in related fields touched upon.

The symbols used in defining the orientation of the model are also

shown in Fig. 1, and separate systems of coordinates are found in

Figs. 2 and 3.

Dimensional numbers are given in metric units; wherever so is the case the units are clearly stated. The results are given in non-dimen-sional forms, where a prime of a force or moment symbol is used to indicate that the figure is based on the reference area LT. Other non-dimensional forms are presented by C-symbols.

Primes are also used for symbols in a new complex plane obtained by a conformal transformation, etc.

Symbol Definition Physical

Dimension Non-Dimensional Form as Used IA A,. A,., Ax AxH Aw B Bx C CO D F .FN FT F,,, EST H x

J

Reference area for non-dimensional

coefficients; A= LT

Total proj. area of rudder Moveable proj. area of rudder

Transverse section area of hull, corresp.

to section draught Hx

Half the added mass unit length of

L2 L2 L2 L2 ML-' ML-1. L2 L L

-L L2T -I-MLT -2 MLT-2

-L

-2Ax, per

two-dim. double-body element in hori-zontal oscillation in unbounded fluid Added mass per unit length of two-dim. body element in horizontal oscill. in a

free surface, neglecting gravity Area of slender wing

Beam of hull

Breadth of hull at station x

Position of force balance centre

Centre of gravity

Diameter of propeller Complex potential Normal force on rudder

Tangential force on rudder Froude number based on length Froude number based on draught Draught of transverse section

Propeller advance coefficient

Cx,

-ir pHxz

c

2,1;H xH ,pilx,

-FnL= Virg-L-Fn T = Vag T

-J --= V alnD r I --,

Symbol Definition Physical Dimension Non-Dimensional Form as Used K K K L L_PP N 0 P R Rn1 T T T U V Ks X Y Yll Y_PPP Yr Yrrt, Y fifir yR yRR ax

Static loop gain of linear equation for

steering

Rolling moment about the x-axis

Rolling moment about propeller axis

Length of reference (Here L==L)

Length of hull between perpendiculars

Yawing moment about the axis of z

Origin of reference system xyz, Mean pitch of propeller

Maximum radius of body of revolution

Reynolds number based on a length / Draught of hull

Kinetic energy of fluid Time lag of first order steering

equation

Total local velocity of flow

Velocity of origin of body axes relative

to the fluid; speed of ship (Vs) or model

(Vm)

Propeller speed of advance

Hydrodynamic force on body along the x-axis

Hydrodynamic force on body along the

y-axis

Typical first order stiffness derivative

Typical third order stiffness derivative

Typical first order rotary derivative

Typical third order rotary derivative

Typical third order coupling derivative

Y-force due to rudder Y-force on rudder

Section radius of body of revolution

T-1 ML2T -2 ML2T-2 L L ML2T-2 L L L ML2T-2 T LT-1 LT-1

LT'

MLT -2 MLT-2 MLT-2 MLT-2 MLT-1 MLT MLT-1 MLT-2 MLT-2 L K' = KLIV P K' = KI PAL 2 K'R= KRI7P V2AL

-N'=.1s7/7P V2AL R,= 1711v T' =TVIL X' =XI--P PA Y' ---- Y/P PA 2 Y p=-Y

RI PA

P " 2

*ft= Y

_PfiP/-PL- V2A₂ Y;= Yr/-2-P V.4L Yr' rr= P 7 V - 2 A L3 P

Yfiftr= Yr/ 7 VAL

Y'R =YRI

PA

P 2 CYRR= yRRI_P2v2A, I ii

-Symbol 1 Definition, . I T Physical Dimension , Non-Dimensional . Form as Used a b bn, b xn en,. g ii P q r i .x, y, z [ 1" u, v, w A Ar Ar 4) OP 'If g 8, 74 V 0 e;, n P cr a) 11,

of series for the

transforma-tion of a contour

Height of rudder at

stock-Half span of a wing

Coefficients of series for the complex

potential

Mean chord of rudder Acceleration of gravity Mass of body

Number of revs, of propeller in unit time Pressure in fluid

Stagnation pressure in undisturbed flow;

q =

2 2

Angular velocity of yaw; r =t Radius vector of polar coordinate

Time

Orthogonal coordinates ,of .a righthanded

system of body axes

Surface depression

Components of V along body axes x y, z

Aspect ratio in general

Geometrical aspect ratio of rudder Aspect ratio of rudder reflected in a

plane mirror

Velocity potential;, also crossflow

poten-tial in special Perturbation potential Stream function

Angle of drift or sideslip of body

Rudder angle; So is neutral angle for zero Y and N at fl= 0

Free surface elevation in en,-plane Angular coordinate

Angle of pitch

Kinematic viscosity

Orthogonal coordinates in crossf low plane

Mass density of fluid Section-area ratio Angle of heel Angular frequency Change of heading angle

L"+1 L, L L!7- 21-1 L LT-2 ra l' T-1 1 ML -1T -2. ML -yr -2 T-1 , L T 'L Il L LT-1 I

.

--L2T-1 L2T =1 L2T -1 --, L DT -1 L ML-34 T-1 = I i - ! m'= m/ P AL 2 1 ' '=rLIT ' .=.-.

Iv

t' =

fdt

L --=-4%,2/Aw A, =b21A , --=262/Ar, i a= Ax1I3xHx I --m r Hr

3. Forces on a Ship in Steady Sideslip

For the theoretical prediction of lateral forces on a bare hull

moving oblique an analogy may be supposed to exist either between the hull form and a low-aspect-ratio wing, or between the hull form

and a slender body, such as a boat-tailed body of revolution. From a pure geometrical consideration the latter analogy may look the

more appropriate, although the narrow stem and forebody may appear

as the leading edge and forward area of a wing. In any case some assumption has to be made about the way in which viscous effects

will modify the "effective geometry" in a potential flow. Of primary

importance is also the influence of the free surface, even in those cases, for FL<0.2 say, where gravity or wavemaking effects are

often ignored. Will it be permissible to look upon the water surface as upon a simple rigid wall, in which the underwater hull form may

be mirrored?

In the first-order theories of slender bodies and low-aspect-ratio (or more correctly zero-aspect-ratio) wings inclined to the main flow it

is assumed that the flow potential around the frame contour in a transverse plane is dependent on the two-dimensional cross-flow

component only, as first shown by MuNK [14] and JONES [15] re-spectively. A similar assumption is often made for the viscous flow, by which the distribution of cross-flow section drag enters as a non-linear expression into the lateral force. (See below).

The slender body theory has been applied to the calculation of

lateral forces on submerged and surface ship forms in waves by

KAPLAN [16], and on surface ships in calm water manoeuvring by

FEDYAEVSKY and SOBOLEV [17]. In view of the complex nature of

his work KAPLAN did expressively neglect the primary effects of the free surface, for which he substituted a semi-empirical correction to the results, whereas in [17] the free surface is treated as a rigid wall, without comments. In an analysis of oblique towing and rotating arm tests with surface models MARTIN [18] included some calculations based on this slender body theory, using strip values for added mass coefficients, which were taken from calculations for high frequency

vertical vibrations; these values will correspond to the horizontal vibration in an unbounded fluid, although the frame contour is less

adequately approximated.

A more stringent approach to the study of forward speed effect on

PIING NIEN flu [19], finding a. doublet distribution on the centre

plane of the ship and its wake to represent the

circulation on the

hull and in the trailing vortices. In the limiting case of zero forward

speed the free surface even here was regarded as a rigid wall; he

found a rapid increase of all the four stationary lateral derivatives

with speed, throughout the interesting range 0<FL<101, 3.

Below some concepts of simple theory will be used to justify the

,application of a simple analogy with the unbounded fluid case for the surface model at low speed as well as for the submergeddouble model.

First consider a body of revolution, moving forward with velocity

V and drift angle 13 through an unbounded ideal fluid at rest

at

infinity. The relative cross-flow velocity is constant along the length of the body, and equal to

V sin p vp=r, parallel to the positive

axis y fixed in the body. At a certain instant the frame contour at

station x, with radius ax=a(x), meets the cross-flow in a

certain

plane en located in the fluid and parallel to yz. (See Fig. 2 a.) The

complex potential F=O+iT of this flow is directly obtained from

the complex potential F'=vC of a parallel flow past a thin

stre-amwise lamina of length 4 ax by the well-known transformation

.-I-ax2g, so that

ax2

a2\

a2\

F= v = - 113

(r

+

cos 0 =Jug (sr sin 0 ,( 3.1)

where e+in,=rele. On the contour r=-ax, Which is now the stream-line 0, the surface potential is

7i _(3.2)

ax

where the lower sign is valid on the leeward side of the body. For

gI >ax the e-axis forms the rest of the

dividing streamline W=0,

here again equivalent to a solid wall with the boundary condition

ao

ao

an an- °

(3.3)

Now as the body moves through this plane 6, the contour varies,, necessitating an instantaneous change of flow, which can only be

generated by a certain distribution of impulsive pressure increments .4,

LIP 17) et

(3.4)

ao,P dx

0,

uga, cos 0=±

ugaxil

With a constant dxIdt=V cos g

the ideal side force on a

small cylindrical element of unit length is given by

da

dY axdp cos 6d0=4qaxdax13 cos 206/0=-477-qax--- t3 (3.5)

dx dx

or, in non-dimensional derivative form on base of product 2LRq,

a

dY A'=27r da

(x

dx

L)

where R is the radius of the largest section of the body. A parallel

portion of the body is seen to carry no force. Integration over the total

length of a boat-tailed body gives the well-known results for the lateral

force and moment

IT; = 0

(3.7)

RD

or, if the quotient 2171L is chosen as an alternative area of reference,

a N

(3.8)

'?)3 \ qv)

This is the well-known free instability moment, for which MUNK [14] introduced a three-dimensional end effect correction by use of LAMB'S coefficients of accession to inertia, k, and lc,. Rotating the body either way against this "broaching" moment means increasing

the total energy of the system of body and surrounding fluid by an

amount equal to fNd6; if the rotation is sufficiently slow the moment

depends on the angular position only, and all the work performed

will add to the kinetic energy of the flow. When thisenergy is a

maxi-mum the resulting moment is zero, and the body moves along its

unstable axis of equilibrium left to itself it will broach again. As

above the yawed translation may be treated as composed of

transla-tions along the longitudinal and transverse main axis, the apparent

mass in these motions being (1+ k,)p17 and (1+1c2)p17 respectively. Thus the kinetic energy of the flow will be

P T= --V2 V (k, cos2 g+k2 sin213) 2 (3.6) (3.9) a d

aax/ax 24p(77)= Sq

11 (7cE2)

and hence the moment due to the impulsive presstres on the body

surface is N=aT lag, or

/ N

ag q17 )

For the infinitely long body of revolution 1c2-1c1=1, and for finite bodies this factor may therefore be interpreted as a correction factor.

When applying equation (3.4) for calculating the impulsive pressures

on body sections, which are not circular, it is .convenient to map the region outside the contour into the region outside a circle. Thus

ii

a2\

the function transforms the circle element gi;---ax into 2

a plane wing element of same height normal to the flow, and on the wing surface the potential is still given by equation (3.2). The span-wise distribution of impulsive pressure difference should now follow from (3.4), so that

(3,10)

(3.11)

but of course the infinite pressures so indicated for +ax are not realized in a real fluid, where small vortices will form at the edges.

The growth of a separating boundary-layer always requires some finite

time, however, and in the frontal portion of the wing the lift will

mainly be due to the longitudinal changes of additional section-mass momentum. For the rear portion of a thin wing with narrowing span

the theory predicts infinite negative pressures along the edges. In a real fluid, again, the flow breaks away aft of the maximum width,,

and the theoretical requirements are of small interest. By considering the boundary conditions of a free surface of discontinuity behind the widest section JONES was able to show that no lift was developed aft of that section [15]. The lateral force on the wing therefore depends

on the span only, and with aspect ratio A as the square of

maximum span over total wing area A, there is

a Y

77-=

(3.12)

ap kgilw 2

i.e., exactly half the value predicted by the PRANDTL high-aspect-ratio lifting line theory putting 11,0. JONES originally developed his

Fig. 4. Chordwise distribution of normal force on a rectangular wing of small aspect ratio.

theory for pointed wings only, but it has later been shown to apply

to the rectangular planform of low aspect-ratio as well. Theoretically

all the wing load is then carried at the leading edge, and the simple analogy between the wing and the submerged double hull of a thin ship suggests that the lateral force on that hull is concentrated to

the bow portion, and that for instance the deadwood of normal type

has only a small influence on Y p' . (Cl. Fig. 4.)

Unlike the tapered rear of the thin wing the boat-tail of an elon-gated body does carry a negative lateral force, but not of the full

magnitude inherent in the ideal-case formulas (3.7) and (3.10). Both axial and cross-wise flow components are modified by the presence

of a viscous boundary layer. Working against the pressure gradient in the leeward regions of body surface this layer will lose much of its energy in vorticity, first on the tail already at small angles of

side-slip. As often experienced this gives rise to a "viscous" positive

lateral force on the afterbody, reducing the ideal free instability

moment by some 15 to 30 per cent for bodies of revolution; due to the large free moment the apparent centre of pressure will then be

situated from 2.5 to 1 body lengths forward of the bow. (See bottom

of Fig. 2.)

The results of the oblique flow tests with the double model form being towed submerged should help to explain the extent to which

the simple analogies now mentioned may be applied. As already

stated the first-order slender body theory may be approved, however, by using separate conformal transformations for each frame contour.

Such a strip method was first introduced by F. M. LEWIS in

connec-tion with his work on the flexural vibraconnec-tions of surface ships [20]. For a class of special two-dimensional forms he thus obtained the

added mass in vertical oscillations of high frequency. In 1957

LAND-WEBER and DE MACAGNO presented calculations for the added masses

of more general forms oscillating horisontally as well as vertically in

a free surface, with very high or very low frequency [21]. It is of

interest, if possible, to apply their results here and therefore to study the free surface condition pertinent to the present aperiodic problem.

Again consider the ship advancing with speed V at a small angle of drift equal to p. Relative to the body axes the water is flowing essentially aft, and at some distance from the hull the undisturbed velocity potential is equal to uxvy=V(x+gy). The presence of the hull gives rise to small perturbation velocities corresponding to a

a0P

potential OP (x; y, z); of these may be safely ignored compared ax

with u, whereas their squares are small compared with the velocities

themselves. The square of the local water velocity U therefore is

approximately

aoP

ao-)

U2=P+2V

+

ax ay

This expression can now be inserted into the general equation for

the pressure in a perfect flow with a gravity potential, gz,

p

0

1

=

gz 2 U2+F(t)

(3.14)

p

On the surface the pressure is constant all over, so that the relative depression of the water, :IL, is given by

z 1 ZO V (For' aoP)

+

(3.15)

L

+

gL ax _ay

For the stationary motion there is

aoP aor' a(I)a=u ax ay .17 ax

+P

ay ) (3.13) (3.16) + v

and therefore :z 2V 180' asPP \ v2 r

ao,'

L gL ax + fi ay j

_gL[a(xIL)+fi

a(yIL)1 1 (3.17) 20P

where OP'= -- is a non-dimensional form of the potential, following VL

VOSSERS [23].

The flow on the surface also defines the kinematical condition for the relative depression velocity,

and combining (3.17) and (3.18) the approximate free-surface condi-tion yields

ao,/

a2op,

=2,32 _(3.19)

2(z/L) _c2(Y/L)

where the superscript P may of course be dropped. Thus it is seen

that for low FROUDE numbers or for small drift angles alike it is aoi

a(IL)z =0, i.e. the free surface z=0 may be treated as a rigid wall,

along which W=const.=0.

This condition may be compared with that for the free surface in

case of a two-dimensional section oscillating in the yz-plane,

80

+g

-0

az

as given by L klUB for horizontal as well as vertical oscillations.

For large values of w there is 0=0 along the free surface, upon

which the water particles move up and down, and in the limitw>oo

there are no transverse waves even at a distance from the body.

Physically this case postulates that gravity forces are neglected

when compared to inertia forces. The boundary conditions at the

juncture of the horizontally oscillating section contour and free

surface may be complied with by assuming the image contour above

the surface at each instant to move in the opposite direction. The added mass in this case therefore turns out to be lower than in the unbounded fluid case. (For elliptic sections the reduction factor is

known to be 4/772; cf. Fig. 29.)

(3.20)

1 V

EV

== 1

ay.'

aczyL) 1 a0P' 1 a(- . IL)

az 2 8 (zIL) =_- a(x1L)H

g+

2 a(y1L) j

[

For small values of co there is found again aolaz=0 along the free

surface, corresponding to the unbounded fluid case, in which the

section and its image form a single rigid form.

If a double-body section is moving as a single rigid form in the

77-direction the boundary condition on the contour must satisfy

ao

v

(3.21)

so that by use of the CAUCRY-RIEMANN condition there is

= v77 (3.22)

The general frame section at station x and its image in the free surface will be supposed to form such a closed contour in the

r-plane, symmetric with respect to both the axes 7/"=0 and e"=0.

The exterior of the contour may then be transformed into the exterior

of the circle gl---ax in the -plane by a LAURENT series with odd

terms,

a

- (3.23)

The contours obtained by retaining these three terms only are the

"LEwis forms". The more general forms have been studied by

LAND-WEBER and DE MACAGNO [21], who introduced a second series for the

complex potential in the "-plane,

b xi bx,

(3.24)

where the b-coefficients are to be chosen so as to satisfy the bound-ary conditions on the frame contour and the free surface.

They also derived a formula for the area inside the contour in terms of the a-coefficients, by means of which they were able to

present the coefficients of added masses as functions of two simple

parameters, the section area coefficient and the draught-to-beam

ratio.

These results are of immediate interest to the present investigation

because of the equivalence of fluid impulse and momentum. In

equations (3.4) and (3.5) the side force element was found from an integration of impulsive pressure increments on the contour, but of course it can also be directly obtained as the time rate of change of

momentum of the virtual added mass known for that section.

2

For the cylindrical double-body element of unit length along the x-axis, at station x and with transverse section area 2Ax, the added

mass in horizontal oscillation of high frequency will be 2Ax, in an

unbounded fluid. (The x in the indices have been added to the nota-tion taken from ref. [21].) For the two-dimensional flow outside this

cylindrical element the total kinetic energy may be proved to be

for integration in positive direction round the fluid; the triple primes

have been omitted, as this integration can be performed in the

plane. It follows from either free surface condition 0=0 or W=const. that the integrand vanishes there, and the integral may be evaluated over the frame contour only. (Turning in positive direction round the

contour instead of round the fluid adds a negative sign to the right

member of equation (3.25).)

The general form of the added mass as given in ref. [21] for

hori-zontal oscillation is

A'-

7TP

0..12

2v2 ct,2"

and the coefficient of added mass in an unbounded fluid,

2A'

"

7rpld

(3.26)

(3.27)

is given in suitable formulas involving the form parameters only. By use of these values in the strip method the lateral force on the

forebody is easily seen to be TF' fig as obtained from

n

T d h

(3.28)

fl

L j

dx H

The ideal flow will break down on the afterbody and on the

lee-ward side along most of the body. The cross-flow resistance concept was early used to explain the lift curve non-linearities of

low-aspect-ratio wings, and adopted for the ship-and- wing analogy by SUER

[24]. ALLEN and PERKINS studied these effects on bodies of

revolu-tion [25], and KELLY pointed out the importance of the finite time

required for the development of the cross-flow boundary layer flow

[26]. On a ship form the viscous separation is likely to be more or

T=

-2 0c/

p f

dO (3.25) ao n=1

less spontaneous, especially in the presence of bilge keels. This suggests

that an estimate of the total lateral force on a hull in sideslip might be obtained from the complete analogy with the low-aspect-ratio

wing according to the FLAX-LAWRENCE formula, [27],

Y'()== TA13±P

(3.29)

where and Where the eross-flow drag coefficient equals 1.

(Cf. Section 8.)

4. Formal Representation of Forces and Moments

Although the tests to be reported here did not include rotating

arm measurements, this more general case will be made the basis when formulating one suitable way of evaluating captive test data

for future computer work.

From the discussion in the previous Section it was seen that the transverse force and yawing moment on the hull (with zero rudder,

if fitted) will be essentially proportional to the angle of drift or side-slip, but that mostly viscous effects will be responsible for large-value

non-linearities, especially in the transverse force, where a second-order term is to be expected. For a model without propeller these

effects may be formally represented by an odd function in f3, however,

symmetric in the origin. (See also Section 8.)

Similarly, the transverse force distribution on the turning hull will

give rise to a transverse force and a damping moment, which are

odd functions of the rate of turning, AP' =r' . For merchant ship forms the linear terms are dominating.

A rudder deflection 8, effected when f3=r'=0, causes a transverse force on rudder and after body, again associated with odd functions

for Y'(8) and N'(8), and, often best described by saturation type

approximations.

An ideal actuator propeller in front of the rudder will raise the velocity of water past the rudder, and in yawed motion it will

con-tribute a fin effect by itself, but it will not detract symmetry from the

picture so far treated. A single-rotating screw propeller imparts a,

rotation of the race, however, thereby effectively twisting the rudder, and in the unequal velocity field behind the hull it also has a tendency to throw the stern towards one side.

To keep to a straight course in a calm sea it is therefore. necessary

to give a certain small helm, 80, and the hull then takes up just that angle of drift, go, by which there is a balance of transverse forces

and moments. Now the hydrodynamic unsymmetry referred to above is usually confined to the aftermost part of the hull, and to the rudder,

and g, therefore is small enough to be ignored. The approximation will also be accepted, that the helm angle 8, is that angle for which

the rudder experiences no resultant transverse force.. (These neutral

angles are not to be mixed up with the mean values of drift angle and helm experienced in straight running in normal sea conditions;

cf. [29].)

It may always be disputed that a mere shift of neutral rudder

position will suffice to characterize the unsymmetry at larger helm,

but at present there seems to be no use of a more complex

represen-tation. (See also below.)

In the common general case with only a small heel the stationary mode of motion is described by the three variables g, r', and 8 to-gether, and there are the odd functions Y'([3, r', 8-8,) and N'(3, r',

8-80). In addition the motion is characterized by the relative drop of speed and r.p.m., and the actual loading of the screw must be known. By use of the TAYLOR expansion around the neutral point (0, 0, So)

the first and higher order force and moment derivatives are

intro-duced; thus for Y'

a a a gk_i

r(g, r, 8)= Y+ E

1

(8-8,)

Y'

(2k-1)! ag a?, as _o,o,00

(4.1)

and. a similar expression for Ar_ According to above the constant term is zero in the free-running case, but it will be retained in the analysis of captive test data, where a systematic error in symmetry

May appear due to experimental technique especially this is the

case for the image model, which was towed in a strut to one side of

the model.

Provided there are experimental data points enough an analysis

by the method of least squares will supply higher derivatives of any order to the expression (4.1). To facilitate the use of the results, and to avoid a snaking of curves, which is physically untrue, n will here

be taken equal to 2, however, i.e. only first and third order

deriva-tives will remain in the analysis. Even so this representation contains no less than 26 hydrodynamic derivatives for the stationary transverse

force and moment only, and it is not yet attractive for use with a

medium size analogue computer. (Note, again, that the unsymmetry

due to propeller action here is included by one single constant, 8,, whereas in ref. [8] it is expressed by 12 more derivatives of second

order. The introduction of 8, requires a special step in the data

anal-ysis, however, as will be explained below.)

If interest is focused on steady turning conditions with unstalled

rudder a few of the cubic coupling derivatives may be dropped in the

evaluation process, but not so when the result will also be used for studies of complex model or ship manoeuvres, where there is no

simple relation between momentary helm, rate of turning and sideslip

angle. It is only after a careful examination of the data actually

obtained that the number of terms may then be reduced.

As is well known there are at least five major causes for discrep-ancies between model and full scale behaviour at large helm: the maximum lift of a model rudder is sensitivily reduced as an effect

of a low REYNOLDS number, the surface roughness of the full scale rudder is responsible for a similar reduction of unstalled section lift

below the values obtained in high speed wind tunnel testing, the friction wake of the model is larger than that of the ship, the screw

loadings may differ in absence of a frictional allowance for the model, and the engine characteristics usually are different in the two cases. Fortunately the effects upon steering of the first and second of these

acuses tend to cancel. If, still, maximum lift is estimated to be say

20 per cent higher in the full scale case this would mean a reduction

of Y," by 30 per cent. The application of this reduction to results for model rudders in the turbulent race behind propellers may be queried, [29]. The effects of friction wake and screw loading also cancel, whereas the differencies of engine characteristics will vary

from case to case. Anyhow, it is much upon accepting this coincidence

of a cancelation of errors that the experimental control derivatives

may be successfully adopted for computer studies of ship behaviour.

Turning back to (4.1) this equation is seen to be non-linear in the unknown variables. It is therefore first linearized by selecting an

approximate value for 8,, which will be denoted 8,, and which differs

from the "correct" value by the small amount AS,. With 8, used in the equations a number of first approximations to the constant term and the derivatives are now obtained, denoted by single bars above

will be accompanied by small and unknown 'corrections to all the

derivatives to be deduced by a second application of the linear

method of least squares;, all products of small quantities are therefore

ignored.

In the oblique towing tests there is r'=.0, and the conditional

equations then may be written in the form

rp-H5i-80) Yo-Fie Y;flp-E-iffl(8i'8.0) Yinsa

7--+ilPit8i-50)2 SS-Fi(8i-50)3 no+

AYO,

Y+ ()J

kk),61 Va-P37A Y;flp+ 17:gpa

±

Or 41 755+ i(8 iSo) Lino

ra-l- -07 rppa-Pgi(8-6)Y So+ /0i-1'02 YL,516180= 1"(8i,

(4.2)

and for N' in analogy. The right hand member is the experimental

value obtained for drift angle gi and rudder Si, whereas the first

seven terms of the left member are included in the first

approxima-tion. In the second approximation these seven terms appear as

constants transferred to the right member,, and the eight unknown

corrections are deduced using the full equations. At least nine

equa-tions are required, but in the present case up to seventy equaequa-tions

were available. The evaluation was performed on a Facit EDB

computer of F a cit Electronics AB in Goteborg.

The same technique was used for results of tests with model with or without propeller, in which latter case 8o-=0 and the correction AS, then found was probably equal to a small error in rudder zero*

position.

The discussion above was considering the effect of heel to be

ignorable in normal cargo ship manoeuvring; in ref. [1] an initial static heel was shown to call upon a small helm and drift to balance the model in straight course motion, and the effect at large helms (where this model did not experience an additional heel of any ap-preciable magnitude) was such as to be expected by a mere shift of

this straight motion rudder angle. The captive tests did include force

measurements at different heels, but at zero drift and helm only

and the result furnishes the uncoupled forces and moments.

The hydrodynamic rolling moment Kc, measured at the balande, :is% an odd function of g, r, 8 and 0. Note that the results of rolling

moment measurements in oblique or athwartship towing can not be presented by K'-derivatives, or in terms of lateral centre of pressure position below the still water level, because the distribution of pres-sures on the bottom is still unknown.

The X-force, positive in the direction of propeller axial thrust,

was measured on models with and without propeller, in the former

case as a disturbance in the balance with the pulling force exerted

by the propeller working at constant r.p.m. X(J3,8, 0)

is an even

function, which is dominated by the increased resistance due to a rudder deflected from the neutral position. There is, of course, a

large amount of induced drag due to lift on the hull, but the

compo-nents of lift and drag almost cancel in the axial speed (or surge) equation of the moving ship. (The decrease of speed in turning is mainly caused by the rudder resistance and the axial component of

the centripetal force, and by the change of thrust due to engine

characteristics; see [1].)

5. Model Tested

The full scale sister ship series related to these model tests may be represented by the MANDALAY, for which manoeuvring trials have

been reported in [1] and [28]. This ship is a modern 9000 tons d.w.

cargo liner with a 4-bladed right-turning screw, a SIMPLEX type rud-der, and bilge keels of normal depth.

Two different models were used in the captive tests, one being

the 1:25 scale fibre glass ship model No. 958 described in ref. [1], the other being a 1:50 scale double-body geosim model No. 1152 made from paraffin wax.

Both models were run in naked condition as well as fitted with rudder and bilge keels. In addition, the surface ship model No. 958

was tested with a MARINER type rudder in an alternative rudder

arrangement also used in the free-running model experiments.

Hull Models

A drawing showing contour, waterlines and body plan for the ship and models Nos. 958-A and 1152-A (underwater portion only) is seen

in Fig. 5. Configuration No. 958-B had part of the deadwood removed.

(Fig. 6.) The main particulars of the ship and models are given in

"

Iit

1

14

av WIIINIUMINMIlmi

II WM IIIWIIIII

KIL %.11011111ffM11111

It\.\\WHIMMNII

WW1\ IIIIMEWIII

litWIIM11111111WIDII

WI KM I I 1111 1111/MNAC

\

\\MIL FHA VMA NW

i

.\-\\.-Nlv _AV/.

.. 'Zdi

=I

IIIIIUMLIMIIIW

=IV

1111MIll

Mir

TABLE I

A full description of model No. 958 was included in ref. [1]. The form of the double-body model, No. 1152-A, was obtained by

reflecting that of No. 958-A in the LWL still water plane. As men-tioned in the introduction this model was to be devided for shear force measurements at two stations, frames nos. 6 and 14. The hull

was therefore cast in one solid block from aristo wax, which was then cut off in four parts see Section 7 hollowed and planed on inner

surfaces, and bolted together before cutting the waterlines in the

shaping machine. The longitudinal gap between the separate portions

of the hull was equal to 2 mm at the surface of the model. From duplicated measurements with this gap open and sealed (by use of foam plastic sheets) it proved not to influence the hydrodynamic

forces recorded. (See Section 7.)

Rudder Models

Rudder models and skegs were made from teak, which was shaped, wetted, and grinded by fine sandpaper. The MARINER type rudder

used for model No. 958-B had a total projected area equal to that

of the SIMPLEX rudder for No. 958-A. (Cf. Fig. 6.) The full particulars

of the rudders are found in Table IL

On models Nos. 958-A and 958-B the rudders could be manoeuvred,

during a test run, by a handle on the steering gear mechanism, re-placing the DC shunt motor used in the free-running experiments.

On model No. 1152-A the rudder position could be adjusted when the model was raised to the surface only.

Spec. Ship Model No. 958-A Model No. 1152-A

Length between perpendiculars L in 135.6 5.424 2.712

PP

Length in LWL in 139.0 5.560 2.780

Breadth, moulded. B in 18.90 0.756 0.378

Mean height of topsides on LWI F in 7.5 0.30

Draught on LWL, moulded T in 7.80 0.312 (0.156)

Lateral area (incl. rudder) AL m2 1036 1.658 (0.415)

Reference subm. area Wetted surface (i ncl. rudder)

LT

_PP

s

m2m2 1058 3484 1.692 5.57 0.846 2.785 Displacement v no.3 13 170 0.843 0.211 Block coefficient _sPP 0.66 0.66 0.66

Position of LCB forw. of Lppl2 tlipp 0.01 0.01 0.01

TABLE II. Rudders Spec. Ship Model Rudder R,

No. 958-A No. 958-B No. 1152-A

R, 114 Type

-Simplex Simplex "Mariner" Simplex Section

-Aerofoil Aerofoil Aerofoil Aerofoil Height at stock b m 5.600 0.2240 0.2300

Gap above rudder at stock

g m 0.310 0.0120 0.0065 Mean chord Cmean m 2.94 0.1174 0.1195

Chord at mean height of stock

Cb12

in

2.94

0.1174

0.1188

Thickness at mean height of stock

t612

m

0.587

0.0236

0.0238

Total proj. area (one rudder)

Ar

m2

17.10.0274

0.0274

Moveable proj. area (one rudder)

Arm m2

17.10.0274

0.0241 Gap ratio gib b2

-0.0554 0.0554 0.0283

Aspect ratio (geometric)

Ar=

-1.84 1.84 L93 Ar Taper ratio cilcr 1.000 1.000 0.591 Thickness-chord ratio tIc

-0.200 0.200 0.200

Ratio of reference area to total proj. area

L TIAr PP

-61.9 61.9 61.9 61.9

Ratio of moveable proj. area to total proj. area

AnJAr

-1.00

1.00

0.88

Moveable area balance ratio

-0.257 0.257 0.151 . r

-I

-.41e.

Fig. 6. Model stern and rudder arrangements.

Bilge Keel Models

The bilge keel models were made from flat brass bars, positioned

in the planes of bilge diagonals, symmetrically about the midship

section. In ref. 91 the presence of the keels was seen to have a major effect on the manoeuvring properties, whereas their actual depth was less critical. In the captive tests, therefore, only one set of keels was used. For dimensions, see Table III.

TABLE III. Bilge Keels

w 4 LWL , k IV S 2 1 ',If 11, 11 I _ I, I, t il 11

STILL WATER LEVEL BALLAST CONS

ii .147 "R7.and."R"2 Ii .k I 11 I L I 11 A , 1 A I, I. "Friand "Ri ii I I 4 I ,Ir 4 4 t I JI I STERN 0" II

r

I I, k - -'-_____________,._ -Spec. Model Ship'

No. 958-A&B No 1152-A

(S1) s1 1, ,,01

Type of bilge keel Length of bilge keel Depth of bilge keel

' Bulb plate ' 30 m , 0.33 ra Flat plate 1216 ram 12 mm Flat plate, 608 ram 6 mm

Fig. 7.. Model screw

P829.-TABLE IV. Propeller

= 5300

FULL SCALE

DIMENSIONS IN,MM

Propeller Model

Propeller model No. P829 for models Nos. 958-A and 958-B was cast in white metal. The design is shown in Fig. 7, and the open water

'characteristics in Fig. 8. Further particulars are given in Table IV.

,Spee--_

Ship I Model No. 958

'Mandalay" I to ship scale Mod. P 829

(P) P

Number of blades

Direct, of turning Diameter, D 1 Pitch (0.7 R), P

Blade area ratio, ADIA0

1 Pitch ratio', PID

4 Right. 5.00 m 4.93 m 0.59 0.985 4 Right 1 5.10 m j 5.30 m 0.55 1.04 4 Right 204 mm 212 mm 0.55 1.04 500 1700 300 615 590 635 750 PITCH,P 5300 5300

10 KT 100 ? 8 80 7 70 6 60 5 50

4

40-3 30--2

20

1

10--0 0 0 0,2 0,4 0,6 0,8 1,0

Fig. 8. Model screw characteristics from open water tests.

6. Static Tests with the Ship Model

The static tests with this model chiefly comprised oblique towing

of the various configurations at five drift angles within the range from 3° to 4-9°, and athwartship towing of two configurations at

several speeds. The oblique towing program also included total and rudder force measurements at several rudder angles and at different screw loadings. The majority of the tests were run at a speed

corresp-onding to 15 knots and 105 r.p.m. on full draght. The full program is summarized in Table V, were the model configurations are iden-tified by the same symbols as were used in ref. [1]. In all some 200

different sets of complete measurements were recorded.

Test arrangement

The model was towed from the towing carriage in the main basin by means of a rigid linkage cradle with roller bearings, designed for this purpose and giving freedom in heave and pitch; see Fig. 9. The

TABLE V. Model No. 958, Static test program.

total forces and moments on the model were taken up by a beam-type strain gauge balance joining linkage and model, whereas the rudder forces were recorded by a separate strain gauge balance

in-corporated in the steering mechanism.

The cradle was clamped to the carriage at the nominal sideslip angle desired, which was checked against centre line markings at

stem and stern. Due to a small flexibility of linkage and balance the true angle had to be observed during each run. (The maximum angular

deflection recorded was zip. + O.° 15 at a nominal 13=4 +9.'00, all

figures given within .±0.0 05.)

The whole linkage system could be lowered or raised by use of

spindle gear, so that the towing force was always applied in line with the axis of the propeller. The vertical leg of the linkage was locked in. all tests, and the "self-propelled" conditions were defined from r.p.m. adjustments to present figures; in these cases the (small) axial force

between carriage and model was still recorded, of course. (See also

below.). Series Coidig. , Constant parameters _ Variables yV kp Trim Heel Speed m/s R.p.m. g 5 Heel Speed m/s ' /3 I AR/S/ 843 0 0 1.18 ' v I 0 . ± II AR1S1 843 0 0 1.54 v v + + III IVa AR,S, AR/SIP 843 843 0 10 0 0 1.75 1.54 525 A v 0 v 1 ± -r + IVb AR,S,P 843 0 0 1.54 560 0 v ₊ lye AR/S/P 843 0 0 1.54 625 0 v ₊ V VI VII AS, ARIS, ARISI 843 843 843 0 0

68

0 v 0 1.54 1.54 1.54 v 0 v 0 0 + + + VIII IX AR,S, AR1S1 692 422 0 10 0 0 1.54 1.54 v v 0 0 ± ± X AR,S/ 422 68 0 1.54 --- v v + -I-XI AR,S,P 422 68 0 1.54 475 v v ± ± XII XIII Em,s, BR4S1 P 843 843 0 0 I 0 0 1.54 1.54 525 0 0 v V + + XIV XV AR /S, AR/ 843 843 0 0 0 0 v v 900 90° 0' 0 XVI AR, 843 0 +0 1.54 v 0 . a ±

The SSPA six-component strain gauge balance

This balance, which is seen on the arrangement drawing in Fig. 9, was designed and built in 1955 for use in single-strut static testing of submersibles; the main features are similar to those of earlier designs for high speed wind tunnel balances, although the inner beam here is

immersed in an oil container. The small cylinder seen on the after

gable end of the container is an expansion unit.

The inner beam is suitably shaped for sensing in each of two planes

the bending moments in two points along the x-axis, each moment

recorded by the change in distribution of voltages over a series of four variable-resistance wire strain gauges connected to a low voltage DC supply. The forces Y and Z are essentially proportional to the

differ-ences of the moment signals thus obtained, whereas the moments

NC and Mc (about the centre of the balance) are essentially propor-tional to the sum of these signals.

The axial force X and the rolling moment Kc are each one

essen-tially proportional to the single signals from special strain gauge

bridges.

The balance is carefully calibrated in a special test rig together

with the potentiometer recorders used in each series of measurements

to obtain a reliable formula for each force component as a function of the main and interfering signals. With the possible exception of

the axial force signal the linearities are extremely good. As is obvious

the accuracy of Y, say, will be lowered if this force is applied far ahead

or abaft the balance.

The maximum uncoupled loadings permitted on the balance, when

orientated as shown in Fig. 10, are X=50 kp, Y=150 kp, Z=35 kp,

Kc=7.5 kpm, -12-c=25 kpm, and Nc=50 kpm; the figures are slightly reduced in the normal case of simultaneous loadings.

TOWING CARR AG

z,

POINT 'V IN SIX - COMPONENT TOTAL FORCE BALANCE

Fig. 10. Model No. 958-A. Total schematic view of test arrangement.

BEE- COMPONENT RUDDER -FORCE BALANCE

-3

The three-component strain gauge rudder balance

This balance was designed in 1958 for use onboard radio-controlled models, and was therefore built into the steering mechanism shown

in Figs. 11(a) and (b). The balance housing turns about an vertical

axis in line with the rudder stock; inside the hull the stock has a knee,

however, and the normal (FN) and tangential (Fr) forces

on the

rudder are taken up by the deformation of four preloaded steel tube

elements, between which the rudder bearing is "floating" in a rigid

linkage with ball bearings. The torque on the upper part of the stock, somewhat ahead of the leading edge of the rudder, is taken up by a lever fixed between another two tube elements. On each one of all

these elastic elements are four strain gauges in full bridges.

The balance is designed for a normal force of 8 kp, and for a torque

on the upper stock equal to 0.50 kpm. The amount of rudder load

to be permitted usually is governed by the stock clearance in the hull aperture. There is some mechanical interaction between the

deforma-tion signals, and in a test all three components must always be

recorded.

Test procedure and data reduction

The model was towed at constant speed with a certain nominal

angle of drift down the basin, while the rudder position was changed

by hand in steps of 7.°2, corresponding to one turn of the handle.

During the "self-propulsion" runs (without frictional allowance) the r.p.m. of the propeller was adjusted to correspond to the model self

propulsion point with zero drift and helm; the exact r.p.m. as well

as the X-force obtained were both recorded. At a certain sideslip and helm the advance coefficient then was lower than that corresponding

to the true model self propulsion, and the test series _{were therefore}

repeated with at higher r.p.m., so that it will be possible to interpolate for the forces due to rudder at any desired loading.

At the lowest forward speed, corresponding to _{a FROUDE number}

F,,0.16, the model was only towed without propeller, for

compari-sons with the tests with the submerged double model. The

photo-graphs in Figs. 12-14 give some indication of the wave formation

around the hull at a speed corresponding to F.L=0.21, _{or 15 knots}

for the ship, and at different angles of drift. Note, of course, that

the local angle of drift at the bow is extremely small in normal

a 410-41,t _ 1

r;:Z

- - "" .mallb if' . 14-7'ci, i t K1571 it :1; c- Ifill 41 64.11.11-1LA k ik *141)pf,,Lato ft ...,, rtfl e:r.f..! .._ , - 4 -..1

Fig. 11.. Top ,(a) and bottom (b) views)of three-component rudder balance.

11010WW: TO.

je re

1-4 a)

0°,

-MeV:

_ 829Yons V Ieg °.t. aie4 17:77.6". _ h54 niis f3 _AdS 831 899Y 9 I e? N1'44

-;

328V118 19; -Miiii.0111111 i2 virMas., ersnme 40.10 611 -4 mis = +0°

Fig. 12, Wave formation along model NO. 958-AR1S1-000 at a speed corresponding to 15 knots.

At zero drift

At a drift angle p= +6''

Fig. 13. Bow waves of model No. 958-AR181-000 at a speed corresponding to 15 knots.

4

1 958.._6123 15163 11 an e,s,- iiie3 15A3 958....6126 i8183 ei5esie2 458...6127 le b , 325-Q45_l

e?

nr...

Fig. 14. Bow waves of model No. 958-AR1 S, P-530 at a speed corresponding tn 15 knots. v.=1.51 ti/s =-1-3° = . 5 4 nits = +6° = 0 /3 =0° 1 1.5 in/s 922-"Ildi 121e3 =1.54 an/s 328,7 els1 le? _=0° =

lic

(a

Fig. 15. Wave formation and wake of model No. 958-ARIS1-000 in athwartship towing.

a) At zero speed b) At F.,=0.282 c) At F.T=0.342

The athwartship towing tests were performed at speeds up to

0.6 m/s, corresponding to FflT 0.32. It is not possible to simulate the

complex boundary layer flow characterising the cross-flow in oblique

towing, but the frame contours of normal ships probably promote a

leeward separation more or less independent of REYNOLDS number effects. (See Section 3, and results in Section 8.) The wave formation is small enough to support an approximation neglecting gravity in the free-surface conditions. (Cf. photographs in Fig. 15.)

The primary results of the static tests have been evaluated and recalculated to non-dimensional forms in a system of body axes having its origin below the LWL displacement centre of buoyancy

and on the extended axis of the propeller. (Cf. Fig. 10.) Some of the results are presented in graphs in Section 8, and it is hoped that other results can be reported elsewhere.

7. Static Tests with the Double Model

In these tests the model was towed submerged at four to seven different sideslip angles within the range from 6° to +12°, for the

TABLE VI. Double model No. 1152-A. Static test program

stations. The program, Table VI, included test with and without

rudder and bilge keels, and the influence of speed, tripwire, and gap

sealings was also studied.

Test arrangement

The double model was rolled through 90 degrees and towed in a single streamlined strut at different angles to the horisontal, on a

depth of 1.8 m (or L) below the surface. (Figs. 16 and 18.)

The model was joined to the strut via the total force six-component

balance described in Section 6. The two midship parts of the hull shaping, described in Section 5, were bolted to a special steel box forming the main structural member of the model, to the gables of this box were mounted the variable-reluctance transducer elements

for shear forces, and these elements carried the fore- and afterbodies

of the hull. (Cl. Figs. 16 and 17.) Due to the flexural deflection of the six-component balance and, to a less extent, the wooden strut,

there was again a correction Agto be applied to the nominal sideslip

angle; hereAgwas automatically recorded by use of an

accelerometer-type transducer housed in the model. As the water now had free

access to the six-component balance compartment the X-force must

be corrected for differences in the total pressures due to forward g (degrees)

Configuration Series Speed a,

6 3 1

0 1 3 1 6 1 9 12

m/s degrees 1152-AS1-000 I 4.0

Ox

x x x x x 3.5 0 x 1152-AR151-000 II 4.0

Ox

x x x x x 3.5 0 x ___. 3.0 0 x x x x 1152-AR1S1-000 III 4.0 10 x x x 4.0

10

x x x 4.0 20 x x x 1152-AR1-000

with gap sealings

IV 4.0 0 x x x x x 1152-AR1-000 V 4.0 0 x x x x x 1152-AR,-000 with tripwire VI 4.0 0 x x x x .

Fig. 16. Photographs showing arrange-ments for oblique towing tests with the

double-body model No. 1152-A. Above: Close view of divided model. Left: Spray and wake formation of surface-piercing strut at 4 m/s.1 Below: View of recording instruments on towing carriage.

iffejA

1.AWAI

46,

MCP

"rt

111

_A'14-.1011

FLANGE BOLTED TO STRUT

TRANSDUCER

FOR

ANGULAR DEFLECTIONS

SIX-_,COO PONE

NT ,'.AN

SECTION

A -A

Fig. 17. Arrangement of instruments within double-body model No. 1152-A.

A

Fig. 18. Double-body model No. 1152-A. Schematic arrangement for oblique towing tests.

speed, and these differences were therefore recorded by a transducer on the carriage, connected to the model.

Variable-reluctance transducers

The design of these one-component (modular) force gauges is based on the principles described by GERTLER and GOODMAN [11]. In Fig. 17 there is a sectional view of one of the cubic element, from which

is seen that unlike the T M B samples it is made in two pieces bolted together; great care was exercised in assuring a plane fitting, and no changes of output signals are caused by any additional setting of the

bolts.

The transducer has two coils in series, connected to a 400 cycles current supply, and the midpoint output is balanced by a

potentio-meter. When the core is deflected from this null-positionthere is an error signal which is amplified andfed into the servo system of the recording equipment. The spring constants of the elements used were

Test procedure and data reduction

The model was towed at constant speed and (nominal) sideslip

angle, and with the preset angle of helm; using same helm for several consecutive runs the sideslip was now changed between each of them.

Most tests with this smaller geosim were performed at a REYNOLDS

number 30 per cent higher than that valid for the surface model at

its normal speed, but still without any special devices for stimulating

the turbulent transition. Additional tests were run at a somewhat

lower REYNOLDS number as well as with a 0.5 mm tripwire round the contour 0.05 L aft of the forward perpendicular.

The positions of the gauges, at frame stations Nos. 6 and 14 equally

distant from the midship section, were chosen so that the forward

gap would be aft of that area, in which a marked difference in

pres-sures was to be expected on the two sides of the model. (See next

Section.) Nevertheless it was felt that a flow through the slits might

interact with the main flow, and one series of tests was therefore

duplicated using sealings of foam plastics.

S. Results and Conclusions

This Section includes a short discussion of the results of the total

force measurements on the ship and double-body models, and of the shear force measurements on the latter one.

In Section 3 it was seen that the distribution of the lateral force

on the ship at low speed as well as on the submerged body could be predicted by use of the section values of virtual mass in an unbounded

fluid, but that the results of such a calculation would not hold on the afterbody, due to the viscous effects. In Fig. 19 is shown the

distribution of the lateral force along the forward portion of the

ship, and it may be noticed that this ideal-theory loading almost

vanishes aft of frame no. 14, which is the position of the forward slit

on the double model. The integrated lateral force is compared with the experimental curve of shear force T, in Fig. 20. The agreement

is believed to be satisfactory, but it must be realized that the simple low-aspect-ratio wing analogy 17;=-

=

_{supplies an almost} equally good approximation, 17;3= 0.184. The slope of the Trcurve

as calculated in Fig. 20 is np=0.193 and that of the cubic curve

Fig 19. 'Theoretical distribution of transverse force on forebody of double-body modet. 0,04

[

LT 0,03 0,02 I,0.00 qa '0* CUBIC FIT TO EXPERIMENTAL VALUES

zz

CALCULATED 'VALUES, 007,71p 197 11 V

-Fig. 20. Transverse force on forebody of double-body model from

theory and experiments.

0,6

0,4

0, 2

0

For small angles of sideslip some negativ lift was actually carried

on the after body of model No. 1152-AS1-000, and the figures just mentioned should be compared with the derivative Y=0. 172

ob-tained from the cubic fit to the simultaneous total-force measurements.

Neglecting the small difference in lift distribution the wing-analogy

does estimate the total lift of the submerged double-body model

without rudder within 7 per cent.

With rudder (but without propeller) the total-force derivative is increased to Yfi=0. 232, the added lift of course being carried on

rudder and after body. In Fig. 21 experimental results for Y', T;

and T, are collated to display the change of load distribution on model

AR1S1-000 within the sideslip angles of interest. For very small angles

almost 90 per cent of the total force is carried on the fore body. For angles above 60 the cross-flow drag on the middle portion is seen to

contribute to the non-linearity of the total transverse force, but

there is also a very marked increase of the transverse force on the

after-body.

p 12°

Fig. 21. Experimental distribution of total transverse force on double-body configura-tion AR1S,-000 as a funcconfigura-tion of drift angle.

1,0

0,5

0 NOUN° 958- AR, 5,000 FLAT PLATE (DEEPLY SUBMERGED, INTERPOLATED1

958-AR, -000 PRISME OF SQUARE SECTION -

-0 0,02 0,04 0,06 0,08 0,10 Fr,T2 0,12 Fig. 22. Cross-flow drag coefficients for ship model configurations

AR1-000 and AR1S,-000.

The cross-flow drag coefficient [r]p=900 as obtained for the surface

ship models AR1-000 and AR1S1-000 at different speeds are shown in

Fig. 22, together with the deeply-submerged values for a flat plate and a square-section prism with rectangular lateral areas of same

effective aspect-ratio, the latter interpolated from data in ref. [30].

There is a small influence on cross-flow drag of speed, i.e. of FROUDE and REYNOLDS numbers. The REYNOLDS number effect is likely to

be largely off-set by the fitting of bilge keels. The cross-flow drag coefficients found are quite well approximated by the value 1

sug-gested for the low-aspect-ratio wing case by FLAX and LAWRENCE [27].

In Fig. 23 the prediction of r(g) from eq. (3.29) is compared with

preliminary spots as well as with the square and cubic fits for

AR5S1-000 at FL----0.21. The non-linear term of (3.29) appears to be

appropriate, whereas the linear term now overestimates the normal force at small angles. The conclusions to be drawn from this fact,

and from some other comparisons of force derivatives of the surface

and double-body models are not evident. In general the moment

measurements are more accurate, of reasons given in Section 6.

Hge

1,5

--"-

-0

V 0

aos Y'{(3), IPA 003 CI 02 0.0t -oat V((3)=a bB .c.133(b = 0.931;c = 4.10) Y'([3) +b (3- c (34(3/ (b = 0,117, C

\Pun = -T-A- .)3.431 (Eq13 29))

mOD, No 959 -ARSi-000

(4.4117/

e 31- 4 5

5`/°

e

Fig. 23. Total normal force 1".(13) on configuration AR1S1-000 according to formula

(3.29) and as obtained from tests at FnL=0.21.

The total, force and moment measurements on model

configura-tions AR1S1-000 and AR1S1-000 are compared in Figs. 24 and 25, and

for configurations AS1-000 and AS1-000 in Figs. 26 and 27. In the condition with rudder the surface model was run at several speeds,

and the results are plotted on base of FROUDE numbers in Fig. 28.

For the linear moment derivative the trend is fully clear, showing a

steady raise above the zero-speed or unbounded-fluid value obtained

for the double model, in consistency with the theoretical results due

to PUNG NIEN Hu [19]. A diagram originally published by GAWN

shows the unstable moment of a semi-submerged ellipsoid being

below the ideal unbounded-fluid values for all FROUDE. numbers [13];

it is given. in a new form here as Fig. 29. The ideal values would not have been realized in a test with a deeply submerged ellipsoid, how-ever, due to the presence of a viscous afterbody lift. (Cf. Fig. 2 b.)

_ e -2°

AC.2

a,.01

o 1,__ I I, g 4

--I

0° 2° 4° . 6° sr 100' p 120

Fig. 24. Model configurations AR1S1-0'00 and AR1S1-000,. Yawing moment NT3),

MOO. No. 958 AT, 11-ni.= 0,24_0,21

CONF. AR,Si,-000 _0,16

/7;-'

MoriNo, 1152 AR151,-ooa

Fig. 25. Model configurations AR1S1-000 and AR1SI-0001. Transverse forceY'(51.

0,08

0,04

0,02

yi( 0,07 DO6 0,05 0,04 0,03 0,02 0,01 0,03 0,02 MOO No. 956 - A -000 AT FnL 0,21 2° 6" 6° (3 12°

Fig. 26. Model configurations AS1-000 and AS1-000. Yawing moment N'($).

/7

1 1

6° 8° 10° 12°

Fig. 27. Model configurations AS1-000 and AS1-000. Transverse force 1-'().

0,01 \moo mo.as2-As,-cia

o° 2°

0,03 N'(?, 0,02 0,01 Yp,'CAR151 AR, 5. - 000) IA R, 5, - 000) V , R, 5, -000) 0 0 0,1 0,2 _{Fn L} 0,3

Fig. 28. Comparison of stiffness derivatives for double model and ship model at different speeds.

N )

ap2,40

0,05 0,10 020 FnL 0,30

Fig. 29. Yawing moment stiffness derivatives for a semi-submerged ellipsoid at differ-ent speeds. (From measuremdiffer-ents published by GAwN.)

1,0 ON 0,8 0,7 0,0 0,4 0,3 0,2 0,1

IDEAL FREE INSTABILITY MOMENT COEFFICIENT kr k IN UNBOUNDED FLUID F EQ 3101/

/

ZONE OF EMPIRICAL

INSTABILITY MOMENTS IN UNBOUNDED FLUID

IDEAL INSTA5ILI1'Y MOMENT COEFFICIENT

°

IAT FREE SURFACE " NEGL EC TI NG GRAVITY'

0,5

I

-6O - 40' (SCALE OF 40 )

CONF AR,S, P- 000

Fig. 30. Model No. 958-A. Configuration AR/ S1P-000. Yawing moment N'($, 5). Example of results with Taylor function.