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KUNUL. TIKNISKA HOUSKOLANS

HANDLINGAR

TRANSACTIONS OF: THE ROYAL INSTITUTE OF TECHNOLOGY

STOCKHOLM, SWEDEN

N! 64 1953

GOTEBORG 1953

ELANDERS BOKTRYCKERI AKTIEBOLAG

UDC 629. 12.072.5.001. 57

PUBLICATION NO 2 OF THE SHIP TESTINO LABORATORY

ON THE POSSIBILITIES OF. ESTIMATING

THE. TOWING RESISTANCE OF SHIPS

BY TESTS WITH SMALL MODELS. I

BY

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1.

Introduction

The resolution into components Of the total resistance to motion of models of ships is discussed. In particular, the influence of the form on the frictional resistance and the separation phenomena at

the stern is dealt with on the basis of previous investigations.

Experiments at low values of the Reynolds number (5 l0

2,5 106) have been made with 1 -metre and 2-metre models In these experiments, the extent and the shape of those parts of the wetted

surface where lamjnar flow is present have been determined by

means of the soluble film technique. A description is given of a method

for routine applications of this technique.

Comparison is made with earlier tests with models of about 6 metres in length. The results seem to indicate a possibility of using such

small model sizes for transforming test data to full scale conditions,

if corrections are made for the laminar flow portions of the hull's surface.

This investigation has been carried out at the Ship Testing

Labora-tory of the Royal Institute of Technology, Stockholm. In order to

secure good comparative conditions referring to results of tests with two greater models nos. 107 a and 266 (see p. 22) earlier run in the

Swedish State Shipbiilding Experimental Tank (SSPA), Gothenburg, the very towing tests of the actual small modelswere made - under

the conduct of the author - in the Gothenburg Tank. 1950.

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4

2. Resolution of the Towing Resistance into

Components

The forces exerted by the water on the surface of the hull can be resolved into- components which are tangential and normal to the

surface elements The projections of the tangential forces in the longitudmal direction of the ship form the skm or frictional resistance

The projections of the normal forces form the pressure resistance. The total resistance 'will be dependent on the size, the form, the velocity, and the, surface. finish of the model, on the temperature,

the density, the initial turbulence of the water, and on the acceleration due to gravity. Accordingly, we can write

= Cu (form;

Re, --, -i-,

F, ,

where

U

represents the imtial turbulence, U0

1

T

represents the roughness of the surface of the model, '1,

/ Weber's number, which represents the influence of

OC. the surface tensjon (o = the constant of capifiarity),

V eLy '

p0e

a -

2 - the cavitation number (e = the pressure of

saturat-ed vapour).

The above formula can be approximated by

= C1 (Re

Rer, form, -) + C,

(form, F, Re) +

+ C (form,Re, F)

where the resolution of the resistance into components in accordance

with the abOve is supposed to be possible. It is assumed that a

cannot infittence the resistance to motion of models towed in an open U

tahk, that '--- mfluences Rer only, i, e, the position of the transition U0

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5 point, that the frictional resistance is independent of the influence

of wave-making, and, finally, that the influence of the surface tension is negligible.

The method which is commonly used for transforming towing test results to full size data is founded on the approximation

Cir(Re

-i-) + [C + C] (fon, F)

where [C ± C] - C is the coefficient of residuary resistance and

= C,tb.

It has been found that this approximation yields practically

acceptable values for Re > 3.106 in those cases where it is required

to predist the results of trial trips. In such cases howëvër, it is

necessary to apply corrections to the basic value of the ship resistance

obtained form the model test. These corrections, which are more or

less subjectively estimated,. allow for the effects of the resistance of the air, steering, appandages etc.

In addition to the above-mentioned assumptions, this

approxi-mation implies the following .presuppositions:

The influence of the laminar boundary layer on friction is so slight that it can be disregarded.

The variation of the frictional resistance with the form of the

hull can be neglected.

-The effect of the variations m the extent and the character of the boundary layer on the pressure distribution around the hull

is so slight that this effect can be disregarded.

Accordingly, the dependence of the frictional resistance on the form of the hull is in this instance as an approximation included, in

the coefficient of residual resistance Cr.

For Re < 2.106, the assumption (a) is no longer admissible as large parts of the hull are exposed to laminar flow. Furthermore, the approximation involved in the assumption (c) may possibly be too rough in the interval Re = 3.io to 3.106.

In what follows, these questions will be investigated in detail. (a) by means of tests made with models 1 and 2 metres in length with the aid of the soluble film method, (b) and (c) by studying previous

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3.

Coefficient of Skin Resistance

fr

The transition from laminar to turbulent flow takes place gradually in.a transition region whose boundariss must be determined statisti-cally.. In calculations, it is customary to replace the transition region

by a line at which the transition from completely laminar flow to completely developed turbulent flow is assumed to take place

in-stantaneously.

The position of the transition line in a given case cannot be

prede-termmed with any great accuracy It is known that this position is dependent on the initial turbulence, the pressure distribution, and

the surface finish of the hull. For two-dimensional profiles,

Schlich-tmg and Ulnch (1942) have evolved a method of calculation per-mitting: an approximate determination of that point at which the

laminar velocity profile becomes unstable Tins implies that a

distur-bance of some kind or other can be increased so as to bring about the transition to turbulent flow. Ahead of such a point, turbulence

can therefore occur in local areas only, e. g. in the neighbourhood of discontinuities in the form of the hull surface. Fage and Preston (1941)

have made observations of flow past radially symmetrical solids by means of an ultramicroscope. Their experiments have led to certain conclusions regarding the magnitude of a dimensionless

number (comprising the scale and the intensity of turbulence, the thickness of the boundary layer, and the mean velocity outside the boundary layer) at the transition point.

For certain simple geometrical solids, some calculations can thus be made in order to determine the position of the transition region,

whereas this is completely impossible in the case of ship models. The extent and the shape of the laminar surfaces can be determined

experimentally by means of the modified sublimation technique using a thin film of a sparingly-soluble solid which diffuses at different rates in laminar and turbuleht flow.

When the flow is two-dimensional or radially symmetrical, the coefficient of total frictional resistance can be calculated if the position of the transition point is known. In such cases it is often possible

to find exact solutions of the boundary layer equations for the laminar

part of the body, whereas the calcilations for the turbulent surfaces

can be made on the assumption of a definite velocity distribution in the boundary layer. Milhikan (1932) and Young (1939) have carried out Such calCulations for solids of revolution, while Prandtl (1927,

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7

1932) and Faiktier. (1943) have made corresponding calculations for

smooth plane surfaces. These calculations are based, on variOus assumptions regarding the behaviour of the quantities which are characteristic of the boundary layer at the predetermined transition line.

In reality, all quantities relating to the boundary layer may be

considered to vary continuously from the values for the completely

laminar region to the values fOr the region of completely developed

turbulence. When the transition region is replaced by a line, some

of the quantities relating to the boundary layer become discontinuous.

Thus, if the impulse thickness of the boundary 1aer is made

continuous, a slight discontinuity will be obtained in the thickness of the boundary layer (defined by a velocity which is 1 per cent lower than the velocity of external flow), whereas the increase in shearing stress will be greater than that observed m experiments Prandtl's assumption that the turbulent boundary layer behaves as if it were turbulent from the stem leads to a considerable disconti

nuity in the thickness of the boundary layer, while the discontinuity in the shearing stress under this assumption is slight, and more closely

in agreement with experiments (Goldstein, Modern Development

of Fluid Dynamics, 1943, p. 329). The function assumed by Prandtl

can be used for any formula representing the turbulent friction of plane surfaces, without any further assumptions as to the velocity

profile, etc. Consequently, this function should preferably be used in

applying corrections for the laminar surfaces to experimentally

determined friction formulae for plane surfaces in turbulnt boundary

layers.

Since the frictional resistance of a hull cannot be determined

directly from the boundary layer equations, it is assumed, as a rule, that the friction of the hull is equal to the friction of a rectangilar

plane surface having the same length as the length on the water

line of the ship and the same area as the wetted surface of the ship.

(at rest) when the plane surface is moved at the same velocity as the ship Of course, this assumptionis equally justified when the plane

surface partly moves in laminar flow. The ratio of the actual fric-tional resistance of the hull to that of the plane surface as defined above will be called sthe form influence ratio>> in what follows

The study of the form influence ratio is rendered difficult by the

lack of agreement between the approximate formulae for determining the friction of smooth plane surfaces with turbulent boundary. layers.

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At the Fifth International Conference of Ship Tank Superintendents,

London, 1948, it was recommended to use Schonherr's empirical formula, which had previously been adopted in the United States

(Amer. Towing Tank Conf. 1948).

Fig. 11 shows Schönherr's curve in comparison with curves repre-senting several other formulae.

It is seen from this diagram that the agreement between these

formulae in the interval Re -- 4 106 to 10 is good within the limits

from 3 to 5 per cent Schonherr's curve is m the lowest position

In the interval from 7 108 to l0 (for displacement ship models,

6 m in length), the curves due to Schönherr, Schultz-Grunow, and Blasius are in agreement within the limits of accuracy from 0 to

2 per cent. At Re < 4 106, their mutrial deviations increase to a

considerable extent, and reaches abOut 20 per cent in the neigh-bonrhood of Re = 10g. The logarithmic formula C1 = 0,455 (log Re)2'58 represents a mean curve through the whole interval.

In the Reynolds number range of the 6 m models, all curves are in good agreement, but Schönherr's curve has been recommended for international use, and should therefore preferably be used for

these models.

An idea of the form influence can be given by calculations and experiments dealing with solids of revolution and double ship

models. Graff (1934) has used the method evolved by Millikan (.1932), and has found that the form influence is of the same

order of magnitude (8,9 and 9,6 per cent respectively) when the solid of revolution is moved with a completely laminar or a com-pletely turbulent boundary layer Graff iS of the opmion that this form effect

can be regarded as independent of the Reynolds

number. Subsequently, Amtsberg (1937) has' made a similar in-'vestigation. However, such calculations are extremely complicated

and time-wasting, even though they are, in themselves,

approx-iniations of the boundary layer equations. Of course, these cal-culations are not suitable for routine work in transforming towing test results to full size data F Weinig has summed up his investiga-tions of flow past, solids' of revolution ' and their comparison with.

double ship models in simple approximate formulae.

For the coefficient of friction of a solid of revolution, he puts

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Qe -0,4 0,2 where = prismatic coefficient B = gieatest radius, L = length, K (see Fig. 1),

Q = the wetted surface,

the periphery of the midship section.

'9

4 6 10

2R

Fig. I. The coefficient K for the calculation of the excess velocity for solids of revolution in accordance with F. Wéiriig's formulae.

I

2R

n

p O,54

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In choosing the values of the coefficients, Weinig has partly taken

into account Amtsberg's calculations made in accordance with

Millikan's method. The agreement between Weinig's approximate formula and Amtsberg's detailed calculations is remarkably close. (It. is to be noted that the value of n. used by Weinig does not cor-respond to our definitiOn of the form influence ratio.)

In. the case of a double ship model, the fOrm influence is greater than ii the case of a solid of revolution having the same curve of

sectional areas. This is partly due to the velocity distribution along the periphery of the section. This distribution is assumed to be

dependent on the ratio BIT only. Then

1/ l

.2RK

n=l+(_Oo7)VBI9Tl+T/B

L D

2(1K)

= 1

B/2.T + 2T/B

where is the midship section area.

Young (1939) has applied Tomotika's (1935) method to a series

of sclids of revolution varying in the fineness of displacement

and m the position of the transition pomt He has been able

to divide the resistance into friction and pressure components by applying momentum considerations to the wake which was

calculated from the boundary layer &juations. His calculations

agree closely with experimental results. It follows from his paper

that the ratio 2 R/L is the principal factor determining the form influence ratio, so that the variation of the form at a constant

slenderness ratio produces only a slight change in that coefficient

of resistance which is based on the total surface of the solid. Young's

results are represented in curves which permit interpolation for any value of the slenderness ratio from 0 to 0,3, and for any position of the transition, point between the nose and 0,6 L. As Tomotika's method is a better approximation than' Millikan's approximation concerning the velocity distribution in the boundary layer, it may be of interest to compare Weinig's approximation formulae with

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11

TABLE 1. Comparison of Young's and Weinig's Values of Forn't Influence Ratio.

This comparison is shown in Table 1

where the values of

Cr obtained by Young were computed from the relation G1 =

/ 2R\

= C1 - 04--) deduced by him.

If Schönherr's mean curve is used as a basis, the difference in

the form influence ratio between Young's and Weinig's methods varies from 0 to 50 per cent. The absolute difference in the values of friction is about 12 per cent. The closest agreement is obtained

for the most slender solids, and Weinig's formula is adjusted to

them in the comparison with Amtsberg's results.

Fig. 2 representing the form influence ratio, shows the agreement

between Young's and Amtsberg's methods, and the results of wind tunnel tests on solids which are reported to have been tested in a completely turbulent boundary layer.

The curves in Fig. 2 do not exhibit any consistent variation with the Reynolds number. It is to be borne in mind that an extremely

slight change in the total resistance or a very small error in the

value of the friction of a smooth plane surface produces a

consider-able effect on the form influence ratio. These curves can scarcely be checked by experimental means since, in that case, the whole diagram falls within a region of high accuracy in measurements.')

The correctness of the approximation C/C9 = 0,4 . 2 RIL, which has

been used for the representation, is of course hnnted

The results are summarized as follows:

(a) Young's calculations correspond to the actual conditions with the greatest accuracy which has so far beeri obtained in computations.

I) Because of the necessary measurement of the pressure resistance.

Form Form

Solid Re g' L/2 R CI,. C17 C, Influence

Ratio

Influence

Ratio

106 Young Weinig Schonherr Young Weiriig

Akron 1,0 0,676 5,92 4,76 . 10- 4,77 . 10 4,41 10 1,079 1,082 10,0 0,676 5,92 3,21 3,ié 2,93 1,096 1,078 100,0 0,676 5,92 2,28 2,22 2,07 1,101 1,072 a 1,58 0,622 5,44 4,46 4,40 4,04 1,104 1,089 B, b 3,16 1,58 0,627 0,631 -3,00 3,00 4,52 5,39 4,19 4,75 3,56 4,04 1,270 1,334 1,176 1,177

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r2 f,000. Young's coves R8M 1874 o Test points o Mil//kan's method applied by Arntsberg 6 - -- lôgRe

Fig. 2. Form influence ratio (Young's curves, R & M 1874).

The inThience of the form on the frictional resistance is to a

gOod approximation independent of the Reynolds number.

For those solids of revolution whose slenderness corresponds to

that of ships, the magnitude of the form influence ratio varies from 8 to 14 per cent when a comparison is made with Schon

herr's mean curve.

For estimating the amount by which the form influence ratio of a hull exceeds that of a solid of revolution having the same ratio L/2 R, use can be made of Weimg's method See Table 2

TABLE 2. Comparison of Form Influence Ratio for Double Model of Ship and Solid

of Revolution Having the Same Curve of Sectional Areas. = 0,7.5, L/2 B = 6,0.

'00

BIT LJB 7ship fr aol. of rev. 1,0 9,5 1,016 1,5 7,8 1,002 2,0 6,7 1,000 2,5 6,1 1,002 3,0- . 5,6 1,006 4,0 4,8 1,016 ioa

\L4R5O

- - 5,00 L_L- 66 /2R=800 .10,00 V2R=548 '/2R=IQOO L/2R=8,00 - Iaoo.

I.

1,200 \L/

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13

BIT = 2,0 corresponds to a minimum since the cross section of the double model in that case is square, and the effect of the local

excess velocity may be assumed to be slight.

In towing tests on models at the water surface (surface models) it is moreover necessary to take into account the effects of Velocity gradients and prolónged path curves due to wave making. These effects are inaccessible to theoretical and experimental analysis in an even higher. degree than the influences dealt with in the above.

For practical use in cakulations, some values of the form

in-fluence ratio are given in Table 3. In this table, the form inin-fluence ratio of a single model with BIT = 2,0 was computed according to

Young (mean values in Fig. 2). At other values of BIT, this value

was increased in conformity with Table 2.

TABLE 3. Form In fluence Ratio over and above Schönherr's Mean Curve for Ship8

of Approximately Rectangular Midship Section (Based on Calculated Values in the

Interval Re = 5 . 106 to 108).

As a result of the discussion in this chapter, we can postulate

Cir (Re, Rer, form, i/ic) -.- n. C1 (Re, Rer, i/k)

and we may attempt to put

iam

C1 =

- Q

(C1

-

C1 Iam)ReT

4.

Coefficient of Complementary Pressure

Resistance C

The coefficient is determined by the difference of the

compo-nents of the actual pressure in the longitudinal direction of the ship

in actual flow and in potential flow. In practice it is possible to

. - BIT L/]/SBT

IV

1,0 1,5 2,0 2,5 3,0 3,5 7 1,10 1.09 1,08 1,08 1,09 1,10 6 1,12 1,11 1,10 1,10 1,11 1,12 5 - 1,13 1,12 1,ii 1,11 1,12 1,iS 4 1,17 1,16 1,15 1,15 1,16 1,17 3 1,24 1,23 1,22 1,22 1,23 1,24

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lit 05 048 040 032 0 30 o 14 CD o t6 0 08 0 38 40 42 44 46 48 50 52 54 56 58 60 62. 64"66 Lo910R

Fig. 3; Coefficients of resiBtance for spheres in flbw, of varying initial turbulence (GoLrsrnn' 1938, p. 495).

calculate the potential flow for a wave-making ship model only in connection with restrictive approximations. The experimental de termination of the pressure distribution in towing tests on surface models is extremely difficult, but has nevertheless been carried out in some cases (LAu'rE 1933). In tests on completely submerged models,' just as in wind tunnel experiments, 'these problems are

somewhat simplified.

If the flow does not separate, so that' the wake can be regarded

as a coutmuous extension of the turbulent boundary layer then

the pressure resistance. of such solids can approximately be calculated

by means of purely theoretical methods (YoulcG 1939).

In routine towing tests oil ship models, it is desired to avoid these

difficulties by measuring the total resistance directly; In order to transform the test results to full size scale, it is thennecessary that

the variations in the coefficient C with Re are. 'known, or that the

magnitude of the coefficient C, itself is so small that these variations are of secondary importance;

At very low values of Re all solids move in subcritical flow. This

implies that the laminar boundary layer separates from the solid,

with the result that the separation resistance becomes high At

comparatively high values of Re, separation is followed by renewed contact. The transition from laminar separation to supercritical

flow with a turbulent boundary layer takes place in a Re mterval where the flow phenomena are very complex. Towing tests, at

ub--I

-Joropping Lunnon I I tests ,, -. *

t1

-

V 0 T{ '4 ____________ . frIdattO5Crn -480 Galcit

.

I' '

(14)

-C 0 irnforin W//701 ,thzddwt Wino' 15

_____

I". ___________

-

---'---

--&u---0 f05 2f0 & 5i0

Fig. 4. Coefficients of resistance for bodies of varying slenderness. (Modified

extract from PRANDTL - TIETJENS 1934, p. 101.)

critical values of Re are probably not possible; the variations in C are entirely beyOnd control. Separation phenomena can also occur

m supercritical flow if the pressure gradient can overcome the kinetic energy of the turbulent boundary layer (e g if the aft edge is sharply

cut off), but this type, of separation causes a regular variation in

resistance with Re, and most frequently does not preclude the

extension to full size scale of the tesults obtained from model tests. For those solids which are bounded by surfaces of continuous

curvature, the variations in C within the critical Re interval are

more strongly marked, and take place at higher values of Re as. the coefficient of fineness decreases. These phenomena are in a high degree dependent on. the imtial turbulence It is seen from Fig 3

that the critical interval for a sphere is passed at Re = 106.

Fig: 4 shows the changes which occur when the slenderness of the solids increases. (This diagram represents the total resistance C.

The frictional resistance is only a few per cent for a sphere, but

constitutes the greatest part of the resistance for an airship model.) The variations in the coefficient of resistance (Jr, are reflected in the pressure distribution curves for the stern.

(15)

TABLE 4. Comparison between Solids of Revolution Subjected to Pre8sure

Measure-ments at Low Values of Re and Solids of Revolution Equal in Slenderness to Ship Models in Series A and B (p. 21).

rotationally sythrnetrical bodies having a slenderness within the Re-interval correspondixg to 1- and 2-metre ship models has been studied

(Zmvi-SiuTR-LoUtEN, 1928, LocK-Jo sEic, 1929, 1933

Owsn-Hurro*, 1929 BATEMAN-JOHASEN, 1930). The forms of these solids of revolution are shown in the three upper rows of Table 4.. The pressure distribution curves for different values of Re are in close

agreement for these mOdels. No definite tendency to variation with Re is to be observed. The same can also be said of Amtsberg's solid

of revolution R 1257 (L/2 B = 8,0, ô = 0,628, = 0,8) which was tested iii water at Re = 1,7 106, 4,0 . 106, and 6,2 108.

In or4er to render possible dirct observation of separation pheno-mena in the vicinity of sterns of varicius shapes, a series of solids

of revolution was made, and these solids were tested m the lowspeed wind tunnel of the Aerodynamic Department (Roy. Inst. Techn.,

Stockholm) by mea;ns of the titanium tetrachioride smoke filaments technique and by the aid of the china-clay technique (cf. PRESTON

1946). It proved possible to make smoke filament tests up to wind

velocities of 1.5 rn/sec (corresponding 'to Re = 106), while the

*china-Fo.'i,àfthiSb/id. or f,'evolution Ref i, . , Notes -L-f 9 . t452 5,5Q75c?57Q5.9Q45OnsQso

The of the relative pesswe measwwne* have en found to ague at

i.97'1O 55lO8.oetO

inawind tunnel.

Nec.

4.67O.7öQ60055t?4?Q64Q5O

'

The rem/ti of the re/.Me aeisure ineasu,en,ents have been Mood te agree at

Re-MtOftC/l.og/O6

IsufO. 1,6.g 10d',0 a wind tunnel

a

6Q59c?52l?47Q62c49

The ,vsotts of the ,elatne pressure me,suren,enfr have been Mood to agree at

pgc,5 7io

iee'O

2.w1O92 iO ma wino' tunnet

a ±

- This

-Qis? Q52 Q

Solid of revolution having the same cave of section areas as the mode/s in Series 8 Drag tests made within the ,vnge Re-5'IO-t.2-tO7

-<

: .

.>

-p4oer9 Q Q' Q Q Qne

Solid of revolution having the same con, of section areas er the models rn SeiresA. Drag tests mad. within time range

Re..41O

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C

'cOca

Fig. 5. A Subcritical flOw. B Transitional type. C Supercritical flow.

clays method was used at. higher velocities in order to determine the

position of the transition point.

On the whole, three different types of flow, which are marked A, B, and C in Fig. 5, were observed in these tests.

Table 5 shows the shapes of the solids of revolution and the

approxi-mate limits of the types of flow A, B and C based on subjective

estimates. It is seen that, according to these tests, the supercritical

type of flow past the slender solids is obtained at values of Re which

are comparatively high, and higher than in the case of less slender solids.

The relative magnitude of the pressure resistance determines the scale-effect upon the expansion of. the test results to the full-size scale. This question is diffiult to answer experimentally since the pressure resistance is obtained as the difference between two large

values of the pressure resultants at the forebody and at the afterbody, which are integrated from the observed pressure distribution curves. The results will be considerably influenced by the necessary

correc-tion.s for the pressure drop in the wind tunnel, etc. Consequently the data to be found in the literature are extremely contradictory.

2

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18

V

TABLE 5.

Fuhrmann's ballon models (Fumu&&N 1912) exhibit a pressure

resistance which amounts to about 50 per cent of the total resistance

(the value of Re in these tests was about 8 10), but several subsequent measurements show a pressure resistance varying in magnitude from

0 to 10 per cent. In this connection, reference is made to Zahm's and Millikan's discussion of a paper by Millikan (1932), to Amts

berg's results (AMTSBERG 1937 p. 213), and to Laute's pressure

measurements on a surface model (1933).

Young's theoretical calculations for solids of revolution lead to the approximation Cp/Cg 0,4 2R/L, which corresponds to 6-10

per cent fOr normal airship htills. The value of C, for ship models

can. be expected to be greater than that for solids of revolution of the same slenderness on account of the .monosymmetrical form and

because of the wave motion, as the wave motion produces pressure gradients which can correspond to a curvature of the surface of the

model that can be greater than the curvature resniting from the form

Solid ofRevolution - Type of How

4 A Z50 066066052 Q52 Re<

27/0

o/06

-czE--.+----5O 6 52q52 2,ef0

af0

50o Oss 0s 052 052 50 ,O 50 10

-500055079 QfaQo2

50 10'

500072 5562 049 SolO'

500097 0,55 0,68 0,49 5o10 5,00055 0,870,490,69

4of

0'

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19

of the hull in the absence of a free surface of the fluid. The German

investigations (GRAFF 1934), (AMTSBERG 1937), and (LAUTE 1933) include experiments with solids of revolution, double ship-models, and surface models having the same curves of sectional areas but the data published in these investigations are not sufficient for the plotting and comparison of dimensionless pressure distributiOn curves.

The question of the variation in C, with the scale and its niagnitude in towing tests can be Summarized as follows.

The present investigation is not sufficient to allow any final

conclu-sions, but this investigation indicates that the pressure resistance varies with Re in a slight degree, and its order of magnitude is so small that these variations are of secondary importance for normal forms of hulls at Re > 5 10. This implies that

C (form, Re, F) C, (form, F)

5.

Coefficient of Resistance Ca,,

By comparing calculated and measured values of thewave resistance,

Wigley (1944) found an empirical correction for the viscosity. This

correctcon

is a function of F independent of Be. With direct

intent to check this property of Wigley's correction, Shiells (1948) made experiments with 4-foot models which were geometrically

similar to Wigley's .16-foot models. These experiments seem to have

been made with special regard to the surface finish and to laminar flow. They show close agreement between the wave resistances on

the different scales (from 0,4 to 4 per cent according to the direction

of motion). In particular, Shiells states that humps and hollows are obtained on both scales at the same value of F, and that the

difference between the wave resistances in forward and backward

motion becomes zero at approximately the same value of F for both sizes of models. Hence, these experiments indicate that the coefficient

.C,,,

is independent of the scale. On the other hand, Havelock

(1948) has demonstrated by theoretical calculations that a rela-tively slight change in the lines of the hull at the stern causes that modification of the wave resistance which is expressed by

Wigley's empirical formula. Havelock considers that the viscosity

is responsible for this change of the lines, whiCh is. therefore

charac-tensed by the displacement thickness of the boundary layer. This

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20 015

r4

VhUUU

V4L1WU

b2v2 0,35 0,30 0,25 0,20 0,95 0,10 0,05 018 .cn draug/,t o

-, -

-5 -w Ihf,rnte a'eJQM_ 025 0 -0,25 -0,50 -0,75 -too 034 oso

A

017 ' 0,79 021 023 o25,Z 027

Fig. 6. From INA 1948 (Figures on pp. 262 and 264).

the displacement thickness of the boundary layer varies as

R5.

Quantilatively speaking, the reduction of the scale to a tenth

corres-ponds to the doubling of the relative displacem&it thickness of the boundary layer at the same value of F. Havelock's assumed curves

BC and FE, see Fig. 6, correspond to an increase of 1,8 and 2,2

times respectively in the displacement thickness of the boundary

layer at the stern post. It is seen from Fig 6 that great errors in wave

resistance can occur particularly at humps In view of the siz of b2 V2 0,30, 0,25. 0,2o 0.15. 0,10 oos

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21 modern freighters, the scale used in model tests on large (6-metre) models most frequently lies within the range from 1: 10 to. 1: 25.

Havelock's mvestigation can therefore be regarded as a serious

objection to the common model test methods in which C is a jriori

considered to be measured to scale,, and the towing test itself is made

for determining this coeffiCient, since the other Components of the resistance are calculated by means of a .simple formula. From this point of view, it is necessary to make extensive investigations in order to carry out further calculations of this kind, which should be based on other shapes of hulls and on assumptions of different

modifica-tions due to the boundary layer.. Before the results of such

investiga-tions are avaiable, it is not possible to form any reliable estimates regarding the quntitative magnitude of the scale effect on C,,, for

normal forms of hulls.

The agreement between the wave contours in model tests made on

different scales ought to be a dependable criterion of the scale

conformity of the wave resistance. Such comparisons can be made experimentally between 1 metre and 6 metre models For these

models, the relative displacement thicknesses of the, boundary layers are in the ratio of 1,7 to 1, and. should therefore reflect a considerable

effect according to Havelock. For the -present, we assume that

C,L, (/orm, F, Re). C (tom, F)

-

EXPERIMENTS

6. Models and Measuring Equipment

Two series of models were tested at the Swedish State Shipbtfflditig

Experimental Tank (SSPA) in Gothenburg. Each series comprised

three geometrically similar models.. The pincipal dimensions of the

models are given in Table 6. Model No. 266 was available in two specimens, wood and paraffm wax, while the other models were made of wood.

The magmtude of the total resistance vanes from 10 to 100 gr

forthe 1-metre models and from 80 to 800 gr for the 2-metre models

at F = 0,16-0,29 Accordmgly, a reasonable accuracy of measure

ments should be ± 0,5 gr for the resistance and . 0,003 wfsec for the velocity. The resistance was measured by means of an extremely

(21)

TABLE 6. Principal Dimen8ion8 of Model,s.

simple device (see Fig. 7) which had previously been designed and

tried out at the SSPA. This device consisted of a bicycle wheel

which was loaded with weights at different radii. The model was

towed by means of a thin piano wire which passed over the

circum-ference of the wheel. The deviation corresponding to the resistance was read directly on a scale attached to the rim. No dampers were used, but the desired accuracy could be maintained because of the smooth running of the carriage.

7.

Surface Finish of Models

The models Nos. 183 and 184 were painted with a spray gun and then wet-ground with the finest emery paper, whereas the models Nos. 189 and 190 had the lustrous finish of the lacquer coating, in which

local irregularities due to grains of dust and the like were unavoidable.

r

I

Fig. 7. Photograph of set-up for towing tests. Series Model No. Scale L m B m T m A kg £2 m2 Appendages A 107 a 1 12,5 6,426 1,000 0,464 2 209 9,805 None 189 1 : 40 2,008 0,312 0,145 66,02 0,9464 None 190 1 : 80 l,00 0,156 0,073 8,252 0,2366 None 6 = 0,74, 0,75 L/B = 6,43 BIT = 2,16 B 266 1 14 5,774 0,893 0,382 1 265 7,026 Rudder 183 1: 40 2,021 0,312 0,134 54,23 0,8606 Rudder 184 1 80 1,011 0,156 0,067 6,778 0,2152 Rudder 6 = 0,64, = 0,66, L/B = 6,47 BIT = 2,34

(22)

rnr77 Ti

If

'I:

Tt

-i--1i. -. 1 \ \ \ I'

v\-\\\ \\

r J-f! j L4

Fig. .8. Top, left: Non-ground, glossy model. Top, right: Local disturbance on the surface of a non-ground model. Bottom, left: Ground model, parallel to the grinding scratches. Bottom, right: Ground model, at right angles to the grinding scratches.

(23)

By examination in a surface-indicating apparatus with the brand-name Talysuxf it was possible to determine the surface roughness

A graph obtained for the ground models, in which the main direction

of the grinding scratches coincided with the longitudinal dlirection of the shij, is reproduced in Fig. 8. As is seen from the graph, the

depth of the grinding scratches was of the order of 4 u, and the

width, was about 100 t. In some parts of the surface the grinding

scratches were superimposed on small waves, 6 u in height and 4 to 5 mm in length. The glossy models exhibit, according to Fig. 8 (left), a surface withlocal irregularities of the order of 10 u (right).

In addition to these micro faults, there must also be other

macro-scopic surface defects, e g. at the frames. The models were made by means of external moulds, which were used for carving the wood to

the contours of the correct form. After that, the interjacent wood

was removed by grinding. When the surface of the models was examined in reflected light, this surface appeared to be smooth.

Consequently, these defects must be small. The models were made within a tolerance of 0,5 mm on the dimensions of the body plan.

Check measurements showed accordingly this deviation at the mdi vidual gauge points.

In order: to find out whether the surfaces of the hulls were altered by the action of the water, use was made of a specially prepared test specimen of wood in several directions of the fibres. This specimen was given the same surface finish as the models, immerged in water

for a week, and then examined by means of the surface indicator.

This examination did not show effects of the water on the microscopic

surface roughness. After the test specimen had been exposed to the action of the air for hail a year, it was examined again by the aid

of the surface indicator, and the result was the same.

The 6.metre models were painted by hand, and were ground in accordance with the tandard practice for routine tests at the .SSPA.

8. Measurements

The tests on the largest models in each series consisted only in

determining a towing resistance curve without any stimulated

turbu-lence, whereas the 1-metre and 2'metre models were towed with

and without trip wires. These latter models were further subjected to

(24)

(PRINGLu-MAn-TABLE 7. Summary ofTe8tB.

25

S1VIITH 1945), so as to determine the extent and form of the laminar

region and the wave contour along the side of the model. The trip wires consisted of 1 mm. nylon fishing line which was attached to

the model with adhesive tape, chiefly at those points where the

transition from laminar to turbulent flow was expected to take place

in conformity with the results of the soluble film experiments. The models were provided with a single trip wire each time. The model No. 190 was aLso towed with a 8strut3 consisting of a piano wire, 3 mm

in thickness, which was placed about 100 mm ahead of the stem, and

was inclined at an angle of 20 degrees with respect to the vertical in accordance with a recommendation due to Dr Davidson (Ship Tank Superintendents, Fifth mt. Conf. of. 1949). In this case no

trip wiie was used; and no test was rna4e by means of the soluble

film method. The tests are summarized in Table 7.

Series ModelNo. L m Test To. Limits of F in Towing Tests Stimulated Turbulence.

-Position of Trip Wire (Section at Value of F)

Soluble Film

Method Test Made at Value of F

A 107a 8,426 1 F = 0,16-0,23 Nàne Not made

189 2,008 2 F = 0,16.--0,23 None F = 0,16, 0,20 0,24, 0,27 3 F = 0,16-0,27 Sect. l9 F = 0,16-0,25 F = 0,16, 0,24 Sect. 17 F = 0,21-0,27 F = 0,24, 0,27 190 1,004 4 F = 0,16-0,27 None F = 0,16, 0,20 0,24, 0,27 5 F = 0,16-0,27 Sect. 10 F = 0,16-0,17 F = 0,16 Sect. 12 F = 0,17-0,21 F = 0,20 Sect. 13 F = 0,21-0,24 F = 0,24 Sect. 14 F = 0,24-0,27 F = 0,27

8 F = 0,16-0,26 Strut Not made

B 266 5,774 7 F = 0,18O,27 None Not made

183 2,021 8 F = 0,16-0,29 None F 0,16, 0,20 0,23, 0,27 9 F 0,16-0,29 Sect. 16 F = 0,16-0,18 F = 0,16 Sect. 17 F = 0,18-0,20 F = 0,23, 0,26, 0,27 184 1,011 10 F = 0,16-0,9 None F = 0,16, 0,20 0,23, 0,27 11 F = 0,16-0,29 Sect. 6 F = 0,16-0,18 F = 0,16 Sect. 11 F = 0,18-0,21 F = 0,20 Sect. 12 F = 0,21-0,24 F = 0,23 Sect. 13 F = 0,24-0,29 F = 0,27

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9. Analysis of Test Results

The purpose of the experiments is to examine the reliability of the approximation

= n c,

(Re ReT, + Cr (form, F) (I)

where

Qblm

C1 = CItUrb1

-

(C - C, lam)ReT (II)

and

= C + C

in accordance with the discussion in Chapter 1 to 5.

The experiments furnish information only on the magnitude of C8 for all models and on Q for the 1-metre and 2-metre models. Therefore, it is necessary to make several assumptions, namely:

The value of n iii Table 3 is independent of Re. iam = 0 for the 6.metre models.

C/turb for the 6-metre wood models is in accordance with

Schön-herrn's mean curve.

-

=

If the above approximation is correct, it must be possible to use it for computing the values of C1 tbBe' which must lie on probable

mean curves for all models, irrespective of the shape of the models. The calculations are therefore made so that the values of C,. are

first computed from the observed total resistance of the models Nos.

107 a and 266 (made of wood)

C,. = C8 - 1,10 C/Schonhe,.

(n = 1,10 for both series)

This value of C,. is subtracted from the observed total resistance C8 of the 1-metre and 2-metre models. After division by the value of n, we obtain the values of the coefficient of friction for a plane surface

C, in flow of mixed type.

The results of these calculations are represented in Figs. 9 and 10. The correction for the resistance of the trip wire was made in accor-, dance with Appendix 1.

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C Z0 5,0 5,0 4,0 3.0 2o 7 9f0

Model Series 107a - 789- f9

3

45

.7

Fig. 9. Model series A in dimensionless representation.

Model Series 265-188-184

with tnpw,re (corrcto/i is made for the

rasistar,ee of the t,7p$'5) withoit t,7I*7/ -.021R4 o 1184 10 1,81 ,,o 0'2S6woorP fScho.nhe,r

11UiluiiL.

ScI,oenharr.meen cun (mvot

255paraI%,

23 4'S

Fig. 10. Model series B in dimensionless rejresentation.

wood 7 9f0 Re es) 27 1° -- with

-

without tnow,re (corrsctioil tnpwi 0Q/Q0r rsisfance -.iO0 is of /07. Shonherr

made for the

the trøwire) OgCr 1,9O .-Cg Cf/Q7 i0 ' /07# 110 1 / 09190 -____

____.

707. 10 1 ff89

I

20 - - -- f07a -2 . 4 5 270° 2 3- 4 5 7 210 Re

(27)

The calculated values of the coefficient of frictiOn of a plane surface.

are individual since the area and the form of the laminar surfaces vary

with Re, the fOrm of the model, and F. However, the tests made by

means of the soluble film method (see Plate 1) can give a fairly

accu-rate idea of the laminar flow. The laminar region assumes the form

of trinngles which are bounded in the backward direction by a diffuse

transition range from an eddy region and the wave line, and by the

eddy region itseli This region at the fore end of the ship, mcluding

the eddies, has been traced on a sheet of transparent paper. It is

therefore possible to measure the areas of the surfaces and to estimate their mean length After that, the correction for the laminar regions gives the approximate values of the coefficient of friction of a plane surface corresponding to the models in conformity with the formula

C1 turb

±

(/ turb - C, lam )Rer

Since C,tUb enters into this equation implicitly, it must be solved by means of a trial-and-error method. At first, it was assumed that C,tUb followed Schönherr's mean curve, but the results obtained

from the tests on the 1-metre and 2metre models showed that the coefficient of friction was much higher. The result of the final

calculations is shown in Fig. 11. The plate friction values computed

in this manner, which correspond to completely turbulent flow, are

scattered around mean curves which have preliminarily been drawn parallel to Schouherr s mean curve

The difference between the mean curves fOr the models and the mean curves for smooth plates can be explained as an effect due to

the surface tensIon, the roughness of the model surface, the resistance

of the air, friction in steering devices, etc. For 1-metre models, this

difference should therefore be greater than m the case of the 2-metre

models. Furthermore, it is obvious that the value of Re based on

the length on the water line of the model at rest cannot be

representa-tive of a mean value above the surface of the model The mean

length (XT) of the laminar surfaces is difficult to estimate, but the influenceof an error in this length is not great, since the difference between the coefficients of friction of smooth plane surfaces in laminar

and turbulent flow varies but slightly within the velocity range of an

individual model. In Appendix 2 it is shown how the value of X

shall be chosen in order to yield a correct value of the coefficient of friction for a triangular surface. Further corrections for these

(28)

influen-Fig. 11. Comparison of several formulae for the friction of smooth plane surfaces in turbulent flow. The

friction of a plane surface in turbulent flow corresponding to the models in the series A and 13 (test values)

qt

)i,thu/enl of Piene Su,face Corréspond,n to Mode/s //os.

7 'V. Mean curve ,

''

;

"!.

III

o with 0 withaW trip sqre

V

h

j

,y69,7 (2metms mode/

V a boundary layer C,.- V V V Coefficient of ft/cl/on V 2.

U

of Smooth Manes Surfaces

0-0,oao6 V -V V

---,lb

Young RIM 1874 f_ Afrcraft E, 1949 --°--B/Is/us ID/-Poithw,gsheft Schonhern t (tm. 0ç.Qo74Re.2 (11TG V V S/lAME1.92 911/1-2) Q242C-1o9.(Re4) V * Ldo91n'7 1940 60427(1ogRe-Ci40)" V V

AW-ScNithtfng-Hb,ni, AE4nst Ooft.JY CçQ455(/og

.0

II

(29)

ces are, of course, imaginable, but this is to be regarded as

over-elaboration of the test results.

The deviation of the observed values from the mean curves, in Fig. 11 is shown in Table 8.

The mean curve corresponds to

C1Schonherr + 1,35 i0 for the 1-metre .models and CtschSnheff + 0,73 iO for the. 2metre models.

It is seen from Table 8 that the per cent deviation is moderate.

All observed values for the model No. 184 with trip wire are situated

above the mean curves whereas all observed values for the model

No. 190 without trip wire, and, the model No. 183 with trip wire are located below the mean curves. Each, towing resistance curve

corres-ponds to a separate mean curve in Fig. 11, the deviation of the

observed values being very small. This implies that the resistance

measurements can be made with a greater accuracy than the following calculations. This accuracy of measurements is ifiustrated below.

An error of 5 per cent in the plate friction value corresponds to an absolute error m measurements of about 11 gr at Re = 5,0 10

rhe spread of the observed values is to be attributed, chiefly to

the lack of routine in the use of the soluble film technique. Therefore,

a very accurate deterimnation of the laminar surfaces was not possible

Originally, it was intended to measure these surfaces on photographs, but this procedure was soon found to be too laborious on account of the aberration of the camera and the poor quality of the photographs

TABLE 8. Deviation of Ob8erved Values from Mean CurveB.

1) Only two observed values.

Model No. 184 190 183 189 Without Trip Wire With Trip Wire Without Trip Wire With Trip Wire Without Trip Wire With Trip Wire Without Trip Wire With') Trip Wire Positive deviation,

mean value, per cent 1,9 1,6 3,0 4,6 3,0

-5,7

-1,0

-3,2 1,6 2,0 4,1 4,3 2,8 1,5 4,3 1,7

-3,8

-5,9 2,5

-5,3

-0 0 0 0 Negative deviation,. mean value, per cent

MaximUm deviation,

positive, per cent ...

Maximum deviation,

(30)

31

(reflections). Consequently, the lamirar stirfaces should have been traced on transparent paper in all tests, but this was unfortunately not the case About 25 per cent of the total number of the tests

made by means of the soluble film method had t be evaluated by

the aid of photographs. Thus, the accuracy of this experimental

method has not been utilized to tie full in these preliminary tests. It is therefore to be expected tht it will subsequently be possible to obtain more reliable empirical mean curves of turbulent friction

for smooth plane surfaces corresponding to small models.

The model No; 266 made of paraffin wax exhibits a higher towing

resistance than the same model made of wood. If the values of the

coefficient Cr for the paraffin wax model are assumed to be the same

as for the wood model, then the value of the coefficient C1 for the paraffin wax model will be approximately constant C1 3,10 (the

maximum deviation is 2,2 per cent). See Fig. 10.

Weitbrecht (1944) has investigated the gram size

and the

surface finish of paraffin wax models and the coefficient of friction

which is dependent on them. The results of his investigations,

includmg those obtamed from experiments with a series of banana freighter models which were towed at various testing institutions,

show that the values of the residual resistance can be caused to agree if the coefficient of friction is rendered mdependent of Re, but varies

with the institutions according tO the method of treating paraffin wax. This confirms in some measure the correctness of the

above-mentioned results relating to the model No. 266, and hence also the calculation of Cr based on Schönherr's values for wood models.

In the test No. 6 (see Table 7), )>strut was used for producing

turbu-lence. The increase in resistance caused by this device was smaller

than that due to thetrip wires in the test No. 5, but a distinct difference was observed in comparison with the test No. 4. This type of stimu'

lated turbulence will not be subjected to any further investigations.

10.

Special Results of Tests Made by Means of Soluble

Film Method

a. Practical ApplicatiOn of Soluble Film Method

The chemical composition of the soluble solid use4 fOr producing

a film on the test surface was roughly the same as that specified by

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is initially colourless, can conveniently be sprayed on to the models by means of a pneuhiatic spray gun having an adjustable oval spray field and sexternal dispersion.

The soluble solid forms a white film havmg an appropriate rate

of solution in water This film dissolves more rapidly in the turbulent boundary layer. Consequently, the laminar regions will be represented

by. areas of lighter colour shade The surface of the model should therefore be pamted black This pamt shall be highly resistant to

the solvent. (acetone) of the soluble solid. In order to prevent

distur-bances during the periods of acceleration and deceleration of the

carriage, the model should be hoisted above the surface of the water

until the motion of the carriage has become steady. For the same

reason, the model shall be hoisted before the beginning of the braking

period.

It is, of course,. rather. difficult to handle in this manner a 6-metre model exceeding 1 000 kg in displacement, but even a 2-metre model hv&rig a displacement of 50 to 70 kg requires special arrangements.

The mechanical strength of normally constructed wood models is not sufficient for carrying the ballast directly on the bottom plank

while the model is m the hoisted position The ballast was therefore

placed on specially designed cradles which rested on the bulwarks and were attached to the model by means of pivoted clamps. The hooks of the tackles were fastened to the cradles, so that the model

in the hoisted position was suspended by the cradles.

In order that the quantity of soluble solid dissolved in the laminar

region, shall be the same at different velocities of. the model, the lengths of rim used m the towing shall be determined by means of

1

the fprmula constant.i) The corresponding formula for the VRS

turbulent region is ,, - = cdnstanti), where n = 4-8. In practice,

the lengths of run will be determined by the conditions m the tur bulent boundary layer It is seen from the above formula that the variation with the velocity is slight within the velocity range of an

individual model. Since the results are estimated by ocular

examina-tion, they will not ,be influenced by this variation. Therefore, the same length of run can be used at all velocities w'ithm the towing

(32)

33

range. From this point of view, it is far more important, tO apply films of equal thickness every time and to avoid double films on any part of the surface of the hull.

b.. Transition from Laminar to 'Ttirbulent FlOw

Accordmg to Schlichtmg and Ulrich (1942) the transition from

laminar to turbulent flow cannot take place before the laminar velocity profile becomes unstable, so that the disturbances of a.

certain defimte frequency are increased These disturbances can cOnsist of turbulence in the tank or eddIes due to local

irregulari-ties of the surface.. The turbulence in the test tank is probably áubject to wide variations from one day to another. The

cha-racter of this turbulence may be taken to be a Uflction of the

velocity used in the preceding run, the form of the model,, and the

length of the time interval between the runs. If the towing tests are made in accordance with a routine procedure, so that these intervals are relatively regular, then the model will probably be

towed in a turbulent flow of a uniformly variable form.

It follows from: the tests that the eddies starting from the bow

wave and the fore end of the keel bound characteristic lammar areas

at the forebody See Fig 12 (Note, m particular, the eddies ex tending from the fore end of the keel on the lOwer photograph!)

:In the case of the slenderer' models, these disturbances meet at

the bilge, with the result that they will completely mfluence the

flow at the stern (See Fig 13 top) In the tests on the models of

smaller fineness, the eddy starting from the bow wave is closer .to the surface of the water without meetmg the eddies extendmg from the :fore end of the keel. Therefore, a part of the flo* will not be influenced by the eddy system. .(See Fig. .13, bottom).

It is to be regarded as probable that the eddies control the

transition from, laminar to turbulent flow ,in a certain definite:

man-ner, and that they act as natural devices stimulating turbulence.

A model of smaller fineness ought therefore to be more sensitive to trip wires, scratches on the stem, and the like. This is also confirmed

by previous experience and by a comparison between the values of C for the models Nos. 183 and 189 (see Figs 9 and 10).

In the tests on the 1-metre and 2-metre models, laminar flow was also observed behind the eddies, and the transition from 'laminar to

turbulent flow was diffuse, so that 30 to 40 per cent and 10 to 20

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Fig.. 12. Disturbing eddies.

per cent of the surface of the hull are to be considered to be laminar. Consequently, the eddies are situated in a region with laminar

velo-city profiles, or cannot act as sufficient initiators of transition for

some other reasons; If a trip wire is placed at a point where the

(34)

I

Fig. 13. Comparison of disturbing eddies in tests on two models differing in slenderness. - .35

Fig. 14. A trip wire which is too close to th1 stern causes a local disturbance, after which the laminar flow recommences.

(35)

36

.instantaneous and well-defined transition. Plate 2 shows how such trip wires bound characteristic laminar areas at the stem.

If a trip 'wire is too close to the stem post (see Fig. 14), it gives

rise to a local disturbance. After that, the laminar flow recOmmences.

Under such conditions, it is very difficult to determine the region

where the transition;takes place. For this reason, it was not possible

to evaluate the results of the test No. 3 in which the trip wire was placed at the section 19'/2.

c. Wawe Contour

The wave contour along the side of the model stands out ith

extraordinary sharpness in the film, and can readiJy be measured by placmg strips of paper at the different stations and by markmg

the intersection of the wave contour and these strips. The tests

made by means of the soluble film method were carried out at

cor-responding speeds, and the wave contour was measured in this

manner after each test. The values observed in two different tests made on the same model and at the same velocity agree within + 0,5 mm. Accordingly, this figure can be regarded as the tolerance in the expansiOn procedure. (The lengths obtained from the strips of paper must :be laid off along the ribs in the bbdy plan so as to

enable the determination of the vertical distance from a certain

definile water line to the wave contour.)

A comparison of all wave contour measurements made on the 1-metre and 2metre models leads to a result which is surprising at

first sight. The wave contours on the 1-metre models are at a

relatively higher level, and the heights of the bow and stem waves are slightly greater, whereas the agreement in respect to the

wave-length and the general character of the waves is extraordinarily close

On an average fOr all tsts the wave contour on the 2-metre models

on the scale of 1: 2 is abOut 1,5 mm below the wave contour On the 1-metre models along the vertical side of the model, and up to ,

mm below the latter contour fore and aft, where the side, of the

model bulges, outwards.

This difference in wave contour is chiefly to be. attributed to;the

capifiary elevation of the water surface.. On measurement, this

elevation was found to vary from 2 to 4 mm according to the slope

of the side of the model. The wave contours observed on the

(36)

where ie the constant of capillarity.

1) Computed from the formula

g;

2r,

37 measuring the wave contour on the model No. 266, use was made of a small-scale photograph. Naturally, this caused some

diffi-culties. For instance, a correction had to be made for the distortion of the side contour of the model by the camera. The measurements were made in two sets by different persons, and the results for the

smaller models were not yet available when the first set of

measure-ments was carried out. The wave contour for the model No. 266 shown in Fig. 15 is a mean curve averaging these two sets of ob-servations. They have proved- to be in remarkably close agreement

(with the exception of the values observed On the afterbody of the

model). The difference between these curves corresponds to a

capil-lary elevation of 3-5 mm

The effect of the surface tension is slight at those velocities which

are involved in this comparison. At a velocity of 0,4 rn/see, the

wave length of 100 mm decreases by 2,8 per cent, but the reduction corresponding to a velocity of 1,25 m/sec (wave length of 1 000 mm) is only 0,03 per cent.1)

The agreement of the wave contours indicates close agreement

between the coefficients of wave resistance in the model series.

If the flow at the stern undergoes break-away, this will influence tie stern wave. This influence manifests itself in a lowerirg of the stern wave contours on the 1-metre models in relation to those on the 2-metre models. However, the uncertainty in the determination

cf the capifiary elevation is too great to allow an7 reliable conclusiOns

to be drawn from the observed wave contours. In the meantime, tere are no results indicating that the stern wave was suppressed iii. the tests on the smaller models.

d. Stream Lines

Fig.. 16 shows that an exceptionally distinct image of the velocity

field can be obtained by applying an appropriate soluble film to the test surface. The disturbances on the surface of the hull are grains of dust and other particles (order of magnitude 0,01 mm) which occur in the non-ground lacquer coating of the model No. 190.

(37)

- 266 (ôinefres), mean cww

266'. fl,E measurement

o 266, seGond measurement

188 (2metres)

184 (1 metre)

(38)

4

Fig. 16. Stream lines on the forebody of the model No. 190.

11. Conclusions Regarding the possibilities of using

Small Models for Estimating the Towing

Resistance of Ships

The tests described in this paper are too few to admit any

far-reaching conclusions. Nevertheless, it seems to be possible to make

a correction for the laminar regions determined by means of the soluble film. In this manner we obtain mean curves for the friction

in turbulent flow of a plane surface corresponding to that of the

models, and these curves are independent of the form of the model. The mean curves shown in Fig. 11 are to be regarded as preliminary.

(Further tests on three model series are planned in order to im-prove the quantitative determination of the mean curves and to

afford additional observations for verifying the method in question.)

Nor did the study of the literature bring forth any evidence proving

the existence of an uncontrollable variation with Re in the flow

past the afterbodies of normally designed displacement vessels (for Re > 5 . 1O). In principle, it ought therefore to be possible to

(39)

2,o

012 Q?4 Q16 Qie 0,20 (222 (224 (226 (228 a3o

F

Fig. 17. Comparison of coefficients of residual resistance for 1-metre and 6-metre

models.

expand to full size the data on towing resistance obtained from

1-metre and 2-metre mod1 by means of a method involving the resolution into frictional resistance and residual resistance. This method should comprise a correction for the variation in the Mc-tional resistance with the form of the model. Table 9 shows how such calculations are carried out, and Figs. 17 and 18 represent the results obtained for the model series A and B. The mean curves of friction for a plane surface in turbulent flow fitted in Fig. 11 were

used for all models.

It follows from the calculations that the correction for the laminar surfaces is considerable in comparison with the coefficient of residual resistanCe to be determined. The soluble film method, in its present

form, does not provide any sharply outlined laminar areas. For

io

to

184 £ f94 without with without wth tripw/re tripw/re fripwire fripw/i 2 2 S

-ST

Cr 4,0 1,

a

(40)

41

0

Q12 0,f4 0,15 0,10 (7,20 0,22 0,24 0,26 0,28 0,30

Fig. 18. Comparison of coefficients of residual resistance for 2-metre and 6-metre

models.

instance, there can be doubt as to the friction in those parts of the hull which are affected by disturbing eddies, as to the extent of the transition region, the mean length of the areas, etc. The experience

gained in these tests shows that sharper outlines of the laminar areas

are obtained if a trip wire is introduced at the fore edge of the

transition region, Since the position of the transition region varies with the velocity, a consistent use of the trip wire will cause

con-siderable difficulties in the experiments and in the extension of

their results to full size. For the models emplpyed in these towing

tests, the coefficients of residual resistance are more closely in

agree-ment if no trip wire is used. The discontinuities occurring in Figs. 17 and 18 are due to changes in the correction term which must

take place when the trip wire is moved stepwise. The corresponding

10,9 with tr,pwire 0 189 without tripwire 183 with tr,pwire ft o 183 without tr/pwire 0 ft 0 ft A A 0 ft

I;

.

at O . 0 0 S. 0 5 S S S S 5 0 - --4,0 3,5 3,o 20 1,5

(41)

TABLE 9. Extension of Towing Te8t Data to Full Size Conditions by Means. of Method De8cribed in This Paper. (Model No. 183, without Trip Wire.)

Column 4. Cj tb computed from the mean curve in Fig. 11.

Colu±nn 6 Regarding XT, se. Appendix 2.

Column 8. Correction calculated from soluble film tests

(Corr. Qiam (C turb C/lam)R) Column 9. Correction used fOr evaluation.

Column 10. 'C, = C,b_Corr.

Column 11. C9 obtained from a fitted resistance curve. Column 12.. C = C9 . C, where n = 1,10.

discontinuities in the observed curves of tOtal resistance are much smaller, relatively speaking, and can hardly be determined since they are too close to the limits of accuracy in 'measurement for the

individual observations.

1 2 3 4 . 5 6 .7 8 9 10 ii 12

v Re F C/ turb X ReT Corr. Corr. C1 C9 Cr

rn/sec 106 10 0,58 1,030 0,130 5,13 0,68 4,45 4,62 0,62 1,101 0,139 5,07 0,68 4,39 4,75 0,66 .1,172 0,148 5,02 0,68 4,34 4,82 0,05 0,70 1,243 0,is 4,97 0,68 4,29 4,94 0,22 0,705 0,150 0,36 2,19 0,678 0,68 0,78 1,385 0,175 .4,88 .0,68 4,20 4,96 0,36 0,82 1,456 0,184 4,84 0,68 4,16 4,92 0,34 0,86 1,527 0,193 4,80 0,68 4,12 4,94 0,41 0,879 0,149 0,34 2,63 0,678 0,68 0,90 1,598 0,202 4,77 0,88 4,09 4,96 0,46 0,94 1,670 0,2i1 4,73 0,68 4,05 5,01 0,55 0,98 1,740 0,220 4,70 O68 4,02 5;04 0,62 1,02 1,045 1,812 0,229 4,67 0,148 0,32 . 2,89 O,67 0,67 0,67 4,00 5,09 0,69, 1,06 1,883 0,238 4,64 0,67 3,97 5,20 0,83 1,10 1,954 0,250 4,62 0,67 3,95 5,34 1,02 1,14 2,025 0,256 . 4,58 0,67 3,91 5,61 1,32 1,18 2,096 0,263 4,58 - 0,67 3,89 6,12 ' 1,84 1,192 1,22 2,167 0,274 4,54 0,146 . 0,30 3,18 0,887, 0,67 0,67 3,87 6,81 2,55 1,26 2,218 0,282 4,52 . 0,67 3,85 7,80 3,37 1,30 2,309 0,292' 4,49 0,67 3,82 8,41 ' 4,21

(42)

VRe

43

Considering the low towing resistance of the 1-metre models, which

requires highly sensitive balances, in view of the large share of the frictional resistance in the total resistance, and taking into account the magmtude of the lammar surfaces and their variation with Re, these models should not be used otherwise than possibly for com-parative tests.

The laminar areas on the 2-metre models are relatively smaller, and their boundaries are sharper on account of the greater intensity of the disturbing eddies. It seems that these models can be used

m practice for extension test data to full size conditions on condi tion that the testers are thoroughly trained in the soluble film tech-nique, and that the total resistance is measured with high precision.

Appendix 1

Correction for Trip Wire Resistance

The correction for the resistance of the trip wire cannot be made with any notable accuracy since the local velocity both outside and inside the boundary layer is unknown. Furthermore, the coefficient of resistance for a cylinder is unknown when velocity of the water

varies along the diameter. However, it is possible to make an

approxi-mate calculation in which jt is assumed that the boundary layer is the same as in the case of a smooth plane surface having the same length and the same velocity as the model, and that the resistance

of the trip wire can be put equal to

U2

B = 1,0 .1 . d

where

1 = the length of the trip wire,

d the diameter of the trip wire,

u2 = the mean value of the squares of the velocities of incidence.

1. Trip Wire in Completely Laminar Boundary Layer.

The thickness of the boundary layer at the trip wire is

(43)

where x = the distance from the trip wire to the stem,

Re

-V

(U = the outer velocity of the water).

The velocity profile can approximately be assumed to be

For the model No. 183, with a trip wire at the section 16, if v =

= 0,704 m/sec, this corresponds to a trip wire resistance of7,26 1'0

kg (0,7 per cent of the model resistance).

2. Trip Wire in Turbulent Boundary Layer.

If the turbulent boundary layer begins at the stem, we obtain

If we assume the velocity profile u/U

(_-)7,

the resistance of

the trip wire is

R=1,olt'2 7/9-_)

If we assume that the thickness of the turbulent boundary layer corresponds to -i-, then the resistance of the trip wire at the section 16 of the model No. 183 is about 4,3 10' kgs at v = 0,704 rn/sec.

The resistances observed in the tests were corrected for, the

resi-stance of the trip wire as follows: For each model and each position of the trip wire, its resistance in a laminar boundary layer was calcu-lated at three velocities. Since some parts of the trip wire are exposed

= 2 y/

-and

I

d3 d5 d4

R=

(44)

or

R = A . f x112 dy = 2/3 A b . c'12

45 to turbulent disturbances, these values were doubled. The curves

drawn through the three points obtained in this manner give an

approximate idea of the correction at .the intermediate velocities.

The correction varies from 1,3 to 3,4 per cent of the total resistance. The results of the towing tests on the model No. 183, with and without

trip wire, show a slight change in the area of the laminar surfaces. It may therefore be presumed that the difference in resistance is mostly due to the resistance of the trip wire or, at any rate, consti-tutes an upper limit of this resistanäe. In this case, the resistance of

the trip wire calculated by means of the method outlined in the above varies from 50 to 100 per cent of the observed change in. resistance.

Appendix 2

Mean Length of Laminar Surfaces

Assume that the mean coefficient of friction for a rectangular

surface can be represented by the formula

C1 = 1,328 Re"2

if the flow is laminar, and by the formula

C1 = 0,074 Re115 if the flow is turbulent. '

What length X must be inserted in Re in order that a correct

value of the force of friction shall be obtained for the triangular,

- bx

hatched area bounded by the straight line y = - --- + b and the

co-ordinate axes in Fig. 19, if the above formulae are to be used without any changes?

1. Laminar Flow.

For each element dy, the resistance is

U2

IUx\'12

dR=1,328

(45)

Fig. 19.

The resistance can also be written bc

B = A (XT)"2

9c

Therefore XT

= 16

's 13 fer cent greater than the mean length

of the area.

2. Turbulent Flow.

For each elethent dy, the resistance is

-

U2 IU \_/5

dR=O,O74

2

k-)

xdy

or

B

= TBb

The resistance can also be written

bc

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