A preliminary investigation of the discrepances between the calculated and measured wavemaking of hull forms

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A

PRELIMINARY

INVESTIGATION

OF THE

DISCREPANCIES BETWEEN THE CALCULATED

AND MEASURED WAVEMAKING OF HULL FORMS

By JOHN R. SHEARER, B.Sc.

(Communication from the National Physical Laboratory) 17th November, 1950

SYNOPS1S.The available methods of calculating wave phenomena far hull forms

have been considered from the point of view of their application_{as a means of}

analysing practical ship hulls in terms of resistance. For this purpose the

approxi-mate solution due to Havelock has been found to be most suitable.

A series of calculations and experiments has been made to assess the limits of application of the Havelock method. _{So far this work has been limited to simple}

narrow hullforEs. Comparisons of measured and calculated profiles at-id wave resistance have been made. The effects of preventing sinkage and trimon the

models. and of varying the vertical and horizontal subdivision of the hull in the calculations have been investigated.

The approximate calculations have been found to agree well with exact calculations made by Wigley, and to reproduce the general features of the

measured results. _{Agreement between measured and calculated results is reduced} if the subdivision of the hull is reduced. _{The main discrepancies for the models} used appear to be due to neglect of viscosity and to interference_{between the wave} system and the hull form.

Consideration has been given to the possibility of a quantitative analysis of

the discrepancies, and a tentative programme for further investigationis suggested.

1. Introduction

RESEARCH

on ship resistance has been largely empirical, and

although much of the basic mathematical theory

_{has been}

developed, little systematic analysis has been attempted. There

are two main reasons for this:

_{first, that practical ship forms do not}

readily lend themselves to mathematical representation, and secondly, that the theory involves expressions which have proved to be calculable

for only a limited range of conditions. _{There is, however, a considerable}

need for an analytical technique which might be applied to the control

and interpretation of experimental work, and to the analysis of hull

forms in terms of resistance. _{Some of the theoretical solutions appear}

to offer possibilities in this direction.

_{The limits of application of}

these solutions, and the order of the

errors involved cannot easily be estimated, and before practical uses can be considered we must study these points in more detail.

Preliminary work has been done on this at the National Physical Laboratory,

and although the investigation has been limited by the continuing pressure of

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A PRE INVESTIGATION OF THE DISCREPANCIES BETWEEN THE CA AND MEASURED WAVEMAKING OF HULL FORMS

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rmar 2 to MIR LAVIE409 oPlesistance

A hull Moving on the free surface of a liquid presents ahydro-dynamical

problem with clearly defined boundary conditions. A complete theoretical solution would require to satisfy the conditions at the boundaries of the fluid, at the surface of the hull, and at the free surface of the liquid. It would also have to satisfy the equations of wave motion, and of viscous flow. In order

to allow practical study, we usually assume that the resistance can be separated

into components which are mutually independent. These have been placed

under various headings, i.e. skin friction, wavemaking, eddyresistance, and so on. For the purpose of this investigation it will be more convenient to consider

the factors under the headings given below:

Skin friction. This is assumed to be the resistance of a flat plane of the same roughness, length, and surface area as the hull_ We therefore neglect any effect due to wave motion.

Wavemaking% This is assumed to be the resistance of a similar hull

in a non-viscous fluid. We therefore neglect both the direct effects of viscosity, and the effect of changes in relative velocity due to skin friction.

Effect on skin friction of wave motion, e. The interference of (b) on (a).

-(d) Effect on wavemaking ofboundary layer flow, i.e. the interference of (a) on (b).

(a) and (b) are first order effects, (c) and (d) are secondary. Obviously these

secondary effects will react on each other, and the process might be continued by successive approximations until a complete solution was reached.

Although skin friction is the major part ofthe resistance of most ships; there is little evidence to suggest that it can be much influenced by the design

of the hull form, so long as turbulent flow conditions are assumed. Any

analysis of the relation between hull form and resistance will therefore be mainly concerned with wavemaking, although the interference effects with skin friction may also be significant.

Methods of Calculating Wavemaking

Practical S-Olutions of the equations of wave motion for a hull moving on the free surface of a liquid have been obtained only by making certain simplifying

assumptions:

That the velocity changes round the hull are small relative to the

main-stream velocity.

That the height of the waves generated is small relative to their length. That sinkage and trim do not affect the wavemaking.

That the angles made by the hull surface to the centre-line plane= ate

small.

That the fluid is non-viscous.

For these assumed conditions, calculations of wave resistance have been

made by Mr. Wigley. In these calculations the hull is replaced by a continuous

distribution of infinitesimal sources over the centre plane of the hull, the complete solution being obtained by integration over the distribution. This process is possible only for hulls defined by simple algebraic expressions.

The analysis is extremely complex, and this factor, and the limited range of application, make the method unsuitable for our purpose.

Within recent years two methods for the approximate calculation of wave, 'making have been developed. Themethods are somewhat similar. The hull is represented by a distribution of simple elements for which the wavemaking

le

1. ivo,I4

44

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A PRELIMINARY INVESTIGATION OF THE DISCREPANCIES 45

BETWEEN THE CALCULATED AND MEASURED WAVEMAKING OF HULL FORMS

characteristics can be calculated. The number and distribution of these elements are arbitrary and the elements themselves are defined by coefficients which can

be calculated from normal design data for the hull. The wavemaking characteristics of the hull are obtained by summation of the corresponding characteristics of its component elements. The methods are, in general,

subject to the same limiting assumptions as the complete solution.

The first of these approximate solutions, historically, is due to Dr. Guilloton,1

The hull form is represented by the summation of a system of simple wedge

shaped elements. These are defined in terms of the second differences of hull

offsets, and can therefore be computed for any normal form. Dr. Guilloton

has developed and tabulated functions representing the wave profiles for these

wedges, and demonstrated that to a first order of accuracy the wave profile of the hull can be obtained by summation of the appropriate profile ordinates for the elementary wedges. A method of calculating wave resistance by a

graphical integration of the pressures corresponding to these profiles is suggested. The wedges first used were defined by straight sections and

water-lines. It is understood that Dr. Guilloton is now using elements defined by parabolic lines, which allow the hull to be represented with greater accuracy. The second approximate solution was suggested by Professor Havelock'.

In this case the hull is replaced by a small number of sources of finite strength distributed over its centre-line plane. The wave resistance is given by expressions

based on the forces acting between the sources in the distribution.

Expressions for the wave profile due to a single source of unit strength have been given by Mr. Wigley'.

In the initial stages of this investigation both methods were tried. The

profiles obtained were very similar, and each method appears to have certain

advantages and certain disadvantages. The second method was finally adopted

for the following reasons:

The wave resistance and wave profiles can be calculated independently, but for the same distribution of elements. Calculation of the resistance

from the profile itself, as suggested by Dr. Guilloton, involves the calculation of small differences between relatively large quantities.

These quantities are obtained by graphical integration, and high

accuracy does not appear to be practicable.

The source distribution is obtained from first differences of section

area curve ordinates. The Guilloton method involves the use of

second differences of hull offsets, and it was found in practice that in order to obtain reasonable values for these, considerable fairing of

first differences was required.

The profile functions for unit sources had been computed and tabulated to a high degree of accuracy and for a wide range of

conditions, and tables of these were available.

The possibility of analysis of discrepancies by modifying the distribution

was envisaged. While this process might be possible with either

method it was considered that the method of sources had advantages.

4. Calculations of Source Distribution and Equivalent Wave Resistance and Wave Profiles

The method of obtaining the source distribution representing a given hull,

and of calculating the corresponding wave resistance is that given by Professor Havelock'. The principal expressions used are quoted below. Cartesian

co-ordinates are used throughout, x being positive in the direction of motion,

and the plane of xy being the undisturbed water surface. z is positive vertically upwards.

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3

46 A PRELIMINARY INVESTIGATION OF THE DISCREPANCIES

BETWEEN THE CALCULATED AND MEASURED WAVEMAICING OF HULL FORMS

Calculations of Source Distribution

The origin of co-ordinates is assumed to be at the intersection of the midship

section, centre, and water planes. The hull is sub-divided by transverse

sections at x xa, xa .etc., the corresponding section areas being S S2, S3. .

The portion of the hull between two adjacent sections S, and Sa might be

exactly represented by a distribution of infinitesimal sources over the part of the

surface between these planes. As an approximation this distribution is

concentrated as a single source of strength M given by

Al= Li, (S2Si)

... ... _ ... (1)

Where v is the forward velocity of the hull. This source lies in the centre line

plane xz at a depth below the surface equal to the depth of the centroid of

the area difference (S2S1). Its longitudinal position is given by:

x0.9 x2 [ K12) Si (x1 xa)] Si) . .. . ..

where V(10 is the volume of the hull between the planes of Si and S2. If the hull is subdivided in depth as well as in length, Si and S2 are the areas of the sections between adjacent waterplanes. Sources are usually positive in the forebody and negative in the afterbody.

Expressions for Wave Resthance

For a source distribution calculated in this way the wave resistance RH, isr

given by the equation:

RH,=167T/c2pf(12-1- J2) cosh2u du . (3)

Pte.

. ktiCoShz

where

/

EM,

e sin (kxrldshu),

--kzr cgsh2 u

and

J

EAT,

e cds (k.ec Cosh u)

In these expressions k=g/v2 and u is a parameter defining the range of

integration. The typical source of strength Mr lies in the centre plane of the

hull at a depth z, below the surface and at a distance x, from themidship

section. The summation is taken over the source distribution.

Wave Profile due to a Source Distribution

The following expressions for the wave profile due to a single source, are

quoted from Reference (3). The origin is in the free surface and vertically over the source. We therefore consider a typical source of strength Mr at (0, 0, zr).

The wave profile ordinate at any distance x from the origin is expressed in

the above reference as the sum of two quantities w and so that

t.=

+ igc Atil; k,-...-±f/2k .- 4.

7711

8k111, f kzrsec20

where

=

- Sec3 4' ,cos(kx sec 0) deo . ...,...(5)

8kM,.

F

v 10001

(2)

(5)

and

7r/2 _mx cos 0

4M,/sec

df

(k sec2 0 sin mz,--In cos Inz,.)m dm k2sec40 m2

0

8 Mr

=- ₍₆₎

VZr

The integral F in (5) has been computed by the Mathematics Division of the

National Physical Laboratory in terms of two variables cc and fi, (a2= kzr, ft = kx) for a range of a from 0.2 to l0 and offt from 0 to 60*. The integral I, in (6) has been computed by Mr. Wigley and a table of values is published in Reference (3). A complete computation of this integralover the same range as is now being carried out by Mathematics Division and it is hoped that this work will shortly be completed.

is zero for x positive, i.e. ahead of the source. is positive for x positive and negative for x negative. Typical curves of and are shown in Fig. 1.

represents the wave disturbance due to the source, and Tc the non-wave or local disturbance. the total disturbance, is continuous.

5. Equivalence of Calculated and Measured Wave Phenomena

The expressions given in the last section have been obtained by making certain simplifying assumptions. These are, in the main, those given in

Section (3). _{In addition we have assumed that the depth of water is infinite}

and that the waterplane is unbounded. In obtaining the source distribution

we have neglected the inertia coefficients for longitudinal motion, and assumed that the hull is narrow so that it can be represented by a distribution on the centre plane. _{In calculating the wave disturbance we have taken the} boundary condition at the free surface, i.e. that the pressure there is constant =atmospheric pressure, as being satisfied at the plane

z = 0 instead of at

the disturbed surface z = and we have neglected the effects of interference

between the hull and the wave system itself. We have, of course, also

neglected viscosity.

The exact physical picture of the system is not thereforeeasy to visualize.

In effect we have calculated the fluid motion due to a distribution of sources advancing through a non-viscous fluid, on the assumption that the surface

remains plane, and thence calculated the wave system resulting from this fluid

motion as if the distribution itself were no longer present. All interference

effects between the wave system and the source distribution are therefore

neglected.

The distribution of sources does, in fact, represent a stream form of some sort and it would be possible, though somewhat laborious, to calculate its

shape. _{In a contribution to the discussion on Mr. Wigley's paper', Dr.}

Guilloton has suggested that since this stream form is made up of sources of

finite strength its leading edge must be of finite radius, as must be the case with

all its component elements. He has suggested that in the limit when the vertical distribution of sources is carried to the free surface this will lead to a

local infinity in the wave profile. There appears to be some theoretical justifica-tion for this criticism.

Since it is clear that the physical conditions implied by these assumptions

cannot be exactly reproduced in practice, it will be more useful from the point of view of the present investigation to consider the hull as being replaced by a wavemaking system whose equivalence as such requires to be demonstrated. The

Photostat copies of these tables can be obtained on application to Mathematics Division (Ref.

Ma/16/1502). A copy has been deposited by the Author in the Library of this Institution.

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BETWEEN THE CALCULATED AND MEASURED WAVEMAIUNG OF HULL FORMS

final test of this must in all cases be the accurate reproduction of measured phenomena. The equivalence of the system in other respects is of interest, but is not relevant to this study. The point raised by Dr. Guilloton therefore becomes a question of determining the vertical subdivision of the hull which

will give sufficient accuracy in the resistance calculations without introducing errors in the profile. It is nevertheless of considerable importance and will be discussed further at a later stage.

6. Questions to be Investigated

In any comparison between the measured wave characteristics for a given hull, and the calculated characteristics for the equivalent source distribution we shall expect to find discrepancies depending on the orderand significance

of the factors assumed or neglected. We can make a rough classification of these. First there are factors which are due to assumptions regarding the fluid

itself, i.e. neglect of surface tension and of viscosity. These will apply to any

hull form but may be expected to become less significant as speed and beam

are increased. Secondly, there are those factors which depend on the hull

form, i.e. assumptions regarding the slope of the hull surface to the centre

plane. Thirdly, there is a group of factors which may be expected to produce

an increasing order of discrepancy as the breadth of the model is increased. In this group are the assumptions regarding wave height, wave velocities, the

influence of the inertia coefficient, and sinkage and trim. And fourthly, there

are factors which will depend on the subdivision of the hull, e.g. the question raised by Dr. Guilloton.

We have seen that the source distribution given by equation (1) can be

calculated for any hull form. This does not, of course, mean that the wave

resistance and wave profile can be correctly calculated for any form. In

practice the accuracy with which these will be reproduced will depend on the

degree to which the actual hull violates the assumed conditions. The grouping

of the various factors, as mentioned above, allows a systematic investigation of the effects of these initial assumptions. Certain of the factors can be

eliminated experimentally, e.g. the effects of sinkage and trim. That of viscosity will be the major cause of discrepancies for narrow hulls, while increasing beam

will increase the effects of wave height, and so on. These considerations suggested the following tentative programme:

Tests with hull forms which do not seriously violate the main

assumptions. These should cover a reasonable variation in form and

should allow us to check the comparison between measured and

calculated results, the comparison between results calculated by

exact and approximate methods, the effect of preventing sinkage and change of trim and the effect of varying the vertical and longitudinal

subdivision of the hull.

On the basis of the results of these tests we should consider the

possibility of approximating to the effect of the main discrepancies

by modifications to the source distribution.

The effects due to variations in the beam should then be investigated

using a series of similar models of increasing beam.

On the basis of these results the possibility of extending the analysis

to practical ship forms should be investigated.

So far, work has been done on (a) and some consideration given to (b).

The programme is, of course, tentative and we shall expect to modify it in the light of experience.

0)

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-A PRELIMIN-ARY INVESTIG-ATION OF THE DISCREP-ANCIES 49 BETWEEN THE CALCULATED AND MEASURED WAVEMAKING OF HULL FORMS

7. Choice of Hull Forms for Study

The requirements laid down under (a) in the previous section are met by the family of parabolic forms developed by Mr. Wigley4. These forms are defined by the

equation:--

(1 d",) (1+ [a i]

a (7)

-2 r4

Three members of this family were selected, with values of a -= 1.0, 0.6

and -- 0.2.

These are respectively Models Nos. 1805B, 1805A and 1846B

described in the above reference. They are now renumbered 3012, 2891 and

2892. The models are symmetrical fore and aft, and have similar transverse

sections throughout the length. The stem and stern are vertical. Details of

these forms are given in Fig. 3. It will be seen that they cover a considerable range of fullness. Model 3012 has hollow waterlines and an angle of entrance of zero. Model 2891 has hollow lines and an angle of entrance of 4.5', and 2892 has convex lines and an angle of entrance of 12'. Calculations of wave

resistance for these models using the complete integrals were made by Mr.

Wigley and the results have been published4.

8. Practical Calculations

In view of the objects of this investigation it was essential to ensure at this

stage that no serious discrepancies were likely to occur through lack of

refine-ment in the computation. A subdivision of the hull by forty equidistant

transverse sections and three equidistant waterplanes was therefore generally

adopted. At low speeds a subdivision of this order is necessary since the

distance between sections must not be greater than the length of the first

elementary wave crest. At higher speeds a wider spacing might have been adopted hut it was found to be generally more convenient to keep to the same distribution throughout. In the profile calculations, however, a reduced subdivision was used on the lowest layer at high speeds. A number of

additional profile calculations were made for twenty stations and one layer, for twenty stations and three layers, and for forty stations and four layers.

First differences of section areas along the three layers were calculated directly from the equations to the hulls. For a non-mathematical form these would, of course, be obtained by means of a planimeter. Equivalent source

strengths for each speed were then obtained from Equation (1). The sources

are positive in the fore-body and negative in the after-body. For the close

subdivision used here, the longitudinal position of the sources was assumed to be midway between the sections.

The wave resistance was calculated from Equation (2). This work was considerably simplified by taking advantage of certain features or the hull

forms. Since the models are symmetrical about amidships the source strengths

in the fore-body are equal to the source strengths in the after-body, but, of course, of opposite sign. The cosine terms in the expression for J therefore

sum to zero and the wave resistance is then given by:

= 167r k2 pf 12cosh2 u du

For these hulls the sections are of the same form throughout. The ratios

of area differences for the three layers, and the depths of the centroids of these

area differences, are therefore the same along each model, and for all models

in the series, and are equal to the ratios and depths of centroids of the sections

themselves. Let S be any section area and the corresponding section areas

for the three layers be Sa, Sb and Sc with centroids at depths Za, Zb, ze. Then

the exponential term in I becomes

sa kza cosh2 u _Sb _kzbcosh' u Sc _kz,cosh2 u

--s e

-I--

e

+

e

S S

)

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50 A PRELIMINARY INVESTIGATION OF THE DISCREPANCIES BETWEEN THE CALCULATED AND MEASURED WAVEMAICING OF HULL FORMS

Since this term is independent of x it will apply to all sections for any given

value of k and u. If calculations are made for a standard length and

sub-division of the hull then the values of xr in the sine terms are fixed and we can

compute these terms for each k value. These can then be used for any other hull of this length provided the same subdivision is adopted. Sine terms had already been calculated for a number of values of k by Mr. Emerson for a length of 400 feet, and for forty stations. This arrangement was therefore adopted, additional k values being introduced as required. The computation then reduced to a summation of the products Mr sin (kz, cosh u) over the distribution of sources, for each value of u. The totals were then multiplied

by the corresponding exponential terms. The products P cosh' u were plotted to a base of u and integrated by planimeter. This product converges fairly rapidly and it was found that a range of u at intervals of 0.1 or 0.2 up to about

3.0 was generally sufficient. The resistance results were converted into the

usual non-dimensional (c) coefficients.

Profile calculations were made for the same longitudinal and vertical

distribution of sources, and for a range of six speeds for each model. These

speeds were selected at an early stage in the study in order to allow a direct correlation with the previous experiments with these hull forms. A certain amount of interpolation of profile functions would have been saved by using the same speeds for all models. The profile was obtained by interpolation for appropriate values of a and for a longitudinal spacing equal to the section spacing on the model. The two functions were multiplied by the appropriate factors and added giving which is equivalent to the wave profile for a single source of unit strength at speed v and depth Zr. The source strengths

were multiplied by the equivalent values of iiMr and summed over the

distribution at each station. The three layers were treated independently and

the ordinates added to give the actual wave profile ordinates for the total

distribution.

The calculations were carried out with the aid of hand and electric calculating machines.

9. Experimental Work

Models 2891 and 2892 were made in paraffin wax in the usual way. Model 3012 was made partly in wax, but the very fine ends were finished in wood and sheet brass. Vertical stations were marked off on the hull at 0, 1,

4, 2, etc. A vertical scale with a spacing of 0-2 in. was marked off on each

station.

The experiments were carried out for two towing conditions: with the model towed normally, i.e. free to change trim.

with the model rigidly attached to the carriage at the correct diaught relative to the undisturbed water surface.

Wave profiles, wave resistance, and sinkage and trim were measured

in condition (a), and wave profiles and wave resistance in condition (b).

Profiles and sinkage and trim were measured by direct observation. In

addition, photographs were taken of the bow crest as a guide in plotting.

In condition (a) the finer models became rather unstable at the higher speeds. In condition (b) the long period oscillations of the water surface in the tank

became significant and long intervals were required between runs. The upper

speed limit for measurement of the profiles was determined on practical grounds.

At low speeds the profile along the side of a model is usually clearly defined,

but it is found that as speed increases a thin film of water rises up the stem for

a short distance and spills back along the hull surface effectively masking the

true profile. At speeds above about 12 feet per second for the fullest model (2892) it was found to be almost impossible to define the correct profile line.

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This effect is probably due to viscosity and surface tension and to the fact that

on an actual model the stem always has finite radius. The actual volume of

water involved is very small and it can be safely assumed that it will not

influence the resistance. An attempt was made to measure the water level ahead of the model. No significant change was found except very close to

the stem.

The resistance for condition (a) was measured on the normal resistance

dynamometer and presented no difficulty. For condition (b) it was necessary to prevent vertical and lateral movement of the model while allowing longi-tudinal forces to be measured. At higher speeds the effect of restraining

sinkage is to introduce fairly large vertical forces. In order to ensure that the

longitudinal components of these vertical forces are negligible, the fore-and-aft

measuring movement must be small. The arrangement adopted is shown

diagrammatically in Fig. 1. The model was suspended directly on two stiff vertical cantilever springs, which provided both the vertical restraint and the

towing force. The main part of the resistance was measured by a weight and pulley system, the balance, which was about 2 lb., being taken by the springs. With this arrangement the measuring movement at the model was less than 0.01 in. This was measured by an electric capacity gauge, the signal being

converted to a current variation which was amplified and recorded on a

recording milliameter. It was found that with this small measuring movement

great care was required in setting up the apparatus. The amplifier used had

sufficient gain to allow the use of a considerably stiffer spring system. It was

found, however, that if the measuring movement was made too small the carriage structure itself did not provide a sufficiently rigid reference standard

of length. The vertical load due to restraining sinkage at the upper end of the speed range was of the order of 60 lb. The measuring arrangement was checked under this load, and it was found that this did not alter the calibration curve more than about 1 per cent. at the extreme deflection of the spring. In practice the weights on the scale pan were adjusted to keep the deflections as near zero as possible. The arrangement as used was in the nature of a " mock-up " but it was found to be reasonably satisfactory in use and enabled information to be obtained which could not readily be obtained by any other

method.

Since an absolute, rather than a comparative, estimate of model resistance was required, no turbulence-stimulating devices were fitted because of the

difficulty of assessing the resistance of the devices themselves. Observations of flow by ink streams were made for Models 2891 and 2892, and during actual

tests various struts were tried in front of the models. These all indicated that the flow was substantially turbulent.

The experiments were carried out in No. 2 Tank which has a cross section of 20 ft. x 9 ft. depth.

10. Accuracy of Experimental Results

The accuracy of the measured profiles is about ± 0.05 in. at low speeds and probably about ± 0.10 in. at high speeds, when it was difficult to assess the true profile. Sinkage and trim readings are accurate to about the nearest

0.05 in. This is believed to be about the greatest practicable accuracy for readings of this kind in view of the nature of the profiles themselves. Runs were repeated in cases of doubt. The profile ordinates at the extreme ends of

Model 3012 up to say station and aft of 9/ station are slightly suspect since it was found impossible to prevent the fine metal ends from bending slightly under pressure during runs. At the highest speed with this model the profile ordinates at the extreme aft end did not repeat satisfactorily on successive runs and were therefore not plotted.

Resistance results for the free-running condition showed a scatter of not more

than ± 1 per cent.

For the fixed condition the natural frequency of the

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A PRELIMINARY INVESTIGATION OF THE DISCREPANCIES

to minor speed fluctuations of the carriage. The scatter for this condition

was therefore considerably greater. Runs were, however, repeated as required,

and the anal checked spots showed a scatter of about ± 5 per cent. As has already been stated, the apparatus was at an early stage of development and there is no doubt that better performance could be obtained. The general shape and position of the @ curve was, however, in no doubt, but it was felt that the scatter of the spots did not justify any attempt to plot the minor inter-ference humps at low speeds.

The resistance measurements might be expected to be 2 per cent. to 3 per cent.

too high at the upper end of the speed range due to tank wall and bottom

effects in No. 2 Tank.

11. Presentation of Results

Profiles

Profiles for the three models are shown in Figs. 4, 5 and 6 and in Tables A, B and C. All profiles are referred to the undisturbed water-surface level, sinkage and trim corrections being made as appropriate. Calculated profiles showing the effect of varying the subdivision of the hull are shown in Fig. 7.

Sinkage and trim measurements are plotted and tabulated in Fig. 8 for the

three models. These have been plotted to a base of Froude No. f=v1VgL

A scale of VI VT, has been added.

Resistance

The resistance results are plotted as curves of c to base Froude No. for the three models in Figs. 9, 10 and 11. The values are also tabulated. A

scale of V/ VL has been added to these diagrams. In order to bring the results into a position which would allow direct comparison, the skin friction resistance

has in all cases been subtracted from the measured results. The measured results are therefore residuary @ while the calculated results are wavemaking

Schoenherr friction coefficients were used, the results being corrected to

59° F. (15' C). For the free condition the values were calculated in the

usual way. When vertical movement is restrained, i.e. the hull is no longer floating, the effective displacement at any given speed must be estimated. In this condition the CC) coefficient tends to lose its significance. As a reasonable

approximation it has been assumed that the change of displacement at any speed is that equivalent to the sinkage of the free model at that speed. The displacements used in calculations of@ values for this condition were therefore adjusted in this way. Slightly greater precision might be obtained by computing the displacement to the measured profile line. The skin-friction resistance for this condition was also modified to take account of the reduced wetted surface. The skin friction values are shown.

Values of wave Cc) for the exact calculations are also given in these diagrams. These were plotted from figures published by Mr. Wigley4.

12. Discussion of Results

(a) General Comparison of Measured and Calculated Results

It is clear from Figs. 4, 5 and 6, and 9, 10 and 11, that the calculations do,

in fact, reproduce the main features of the wavemaking with reasonable

accuracy. Agreement is, in general, better for the finer models. The profiles agree fairly well at the bow, and the discrepancies tend to increase towards the

aft end. Agreement breaks down to some extent for the fullest model at high

speeds. The interference humps in the curves of Cc) are much more marked

for the calculated values. 52

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--amiwuram.F"-zi

-,51.1Mitiffinzmr-A PRELIMIN-,51.1Mitiffinzmr-ARY INVESTIG-,51.1Mitiffinzmr-ATION OF THE DISCREP-,51.1Mitiffinzmr-ANCIES 53 BETWEEN THE CALCULATED AND MEASURED WAVEMAKING OF HULL FORMS

The form of the curves for the fullest model, No. 2892, suggeststhat, in

spite of the checks made, the flow at low speeds was transitional.

Comparison of Wave Resistance calculated by Approximate and Exact

Methods

Figs. 9, 10 and 11, show good agreement between the exact and the

approxi-mate calculations over the whole speed range.

Effect of Sinkage and Trim

In general the calculated results agree better with the results measured with the model free to trim, than with those for the fixed model. This is true both of resistance and wave profiles. The wave profiles indicate considerably

reduced wavemaking in the latter condition, and this is confirmed by the

resistance curves.

Effect of varying the Vertical and Longitudinal Sub-Division of the Hull

As already stated, the profile calculations were made for a distribution of sources in three horizontal lines of forty sources each. The effect of varying

this distribution was tried for the fullest model No. 2892, as this is the model giving the worst general agreement, and might therefore be expected to be most sensitive to changes of this kind. Calculations using a single line of twenty sources were made for speeds of 5- 9, 7- 08 and 8- 35 feet per second, and an additional calculation using four lines of forty sources was made for

8-35 feet per second. This last distribution was obtained by halving the upper

layer. This gives the smallest depth of source that can be used at this speed, with the existing ranee of values of the wave profile functions. The profiles calculated for these distributions are compared in Fig. 7 with the corresponding

profiles for the standard distribution. The reduced subdivision has the effect of reducing the bow wave, the effect being slightly greater at low speeds. All the main features of the profile are, however, reproduced. The effect of

increasing the vertical subdivision is only evident at the bow and stern in the

case tried. The most significant effect is an increase in the height of the bow

wave. This leads to better agreement with the measured value. It must, however, be recognized that this may also tend to confirm Dr. Guilloton's suggestion, i.e. that increasing the vertical subdivision would lead in the limit

to an infinity in the profile. The existing range of values of the profile functions

does not, however, allow this to be checked. The possibility of computing

these functions for smaller values of a is under consideration.

13. Analysis of Discrepancies

The measured results for the models in the free condition represent what is,

in effect, the complete solution to the fluid motion, while the calculated results represent a first-order solution limited by the initial assumptions. The difference

between the two represents the effect of these various assumptions, and the effect of second-order interference between wavemaking and skin friction. Of the initial assumptions, we have obtained results which suggest that the effect of sinkage and trim is not serious, and we have checked that no serious

new errors are introduced by using a finite distribution of sources. The remaining assumptions which may influence the results are those regarding the

form of the hull and the free surface. The first of these has to some extent

been investigated by Wigley and Lunde' who have checked the effect of

increasing the fullness of the midship section. Lunde6 has also tried the effect

of using a distribution of sources in breadth, as well as in length and depth, as

a means of representing a wider hull. In neither case do the results published suggest that these will be major causes of error on the models here considered. The most obvious effect of the wave disturbance on the hull itself is that the effective wavemaking form changes. This might be represented by calculating

(12)

the source distribution equivalent to the immersed section areas below the profile line, i.e. we might calculate the wave profile to a first order and then use this to work out a second-order correction. The Author has tried this in

a number of cases. The results obtained in this way are interesting but inconclusive. _{The agreement between the measured and calculated profile}

was improved at the first crest, the slight lag of the calculated wave being

largely eliminated. _{Further aft the corrections were in the right direction,} but, in general, greatly exaggerated. _{It is, therefore, concluded that while this} is evidently an important factor, it is not the only one.

The other factors which might be expected to influence the wavemakingare surface tension, damping of the wave system due to viscosity, and boundary

layer flow. For a model of this size the effect of surface tension must be small

except ahead of the model where the pressure disturbance itself is very small.

This factor is not important in the present study but it is noted that while

theory would suggest that there should be a rise in surface level forward of the

stem, of the general form of the function very little disturbance could actually be measured. _{Viscosity can also only have a small direct influence} on the results. That is to say, within the length of the hull, no appreciable

damping of the wave system will occur. The major factor is therefore boundary

layer flow.

The influence of the boundary layer flow on the source distribution representing

the hull may be considered in two ways. In obtaining the initial distribution

the strength of each source was calculated for the same velocity, and for section

area differences appropriate to the source positions. Skin friction may be considered either to introduce a progressive reduction in this velocity towards the aft end of the distribution, or it may be considered that the effective

wave-making form is the envelope of the boundary layer. On the usual assumption that the flow outside a boundary layer is effectively non-viscous, the latter

approach is more convenient. This again suggests the possibility of

representa-tion by a modificarepresenta-tion to the main source distriburepresenta-tion. Tentative experiments

in this direction have been made by Havelock7 and the empirical viscosity

corrections used by Wigley8 have a similar effect.

Reviewing the position so far, it has been seen that a given hull can be

represented by a source distribution which is, within certain limiting conditions,

equivalent to it as a wavemaking system. We must now consider the

possibility of modifying this source distribution in such a way as to make the

system equivalent to the actual conditions, i.e. without these assumptions. Two

factors have been tentatively studied in this way, as already described. The

order and nature of the corrections to be applied are difficult to assess and the Author has therefore given consideration to the possibility of a more direct

approach, that is to find out whether it is possible to calculate the actual source

distribution required to bring the measured and calculated results into

agreement.

For this purpose the wave profile is more convenient than the wave resistance,

because it is more closely related to the hull form itself. If we suppose that any second order effects can be represented by an additional single line of sources at the mean depth of the distribution, then for any given speed, the

corresponding wave profile functions

are known. We also know the

corrections to the calculated profile which are required, and it is possible from

this information to construct a set of linear simultaneous equations equal in number to the number of stations considered. The Author has used twenty stations for this purpose and found that the twenty equations obtained can be solved to any required degree of accuracy by a simple relaxation method. Solutions obtained in this way are not necessarily unique but since no source

in these distributions can seriously influence the profiles ahead of it alternative solutions are unlikely. Failure to obtain a solution would, of course, mean that the corrections required could not, in fact, be represented by a single line

(13)

A PRELIMINARY INVESTIGATION OF THE DISCREPANCIES 55 BETWEEN THE CALCULATED AND MEASURED WAVEMAKING OF HULL FORMS

of sources on the centre-line. Except at the lowest speed, when twenty sources are scarcely sufficient to represent the system, no great difficulty was found in

obtaining solutions. Since the data used involved the differences between measured and calculated profiles no great accuracy was sought. The source

distribution obtained showed unmistakable relations to the profile form, and in some cases to the main distribution. A general increase in source strength towards the aft end was also apparent. Since the sources in the after-body are negative, this is equivalent to a reduction in the wavemaking of the after-body and is consistent with the expected influence of the boundary layer. However, the results obtained were not very consistent and it seems probable that certain

of the factors involved cannot be represented by a simple distribution such

as this.

When we consider the effect which these modifications to the distribution

would have on the resistance calculations we are faced with a difficulty. One

of the conditions governing the use of sources to represent an actual body is

that the algebraic sum of the source strengths for the distribution must be zero.

If it is not, then the distribution does not represent a closed body. Empirical

alterations to the distribution do not necessarily fulfil this condition. Up to

the present it has not been possible to try the effect of this on the resistance

calculations except in a few simple cases. The evidence of these, however, is that the fact that the streamlines do not close behind the body is not necessarily

serious. It may in fact be taken to represent the effect of the frictional wake. We must, however, approach this question with considerable caution, and the

point is receiving further study. However, it is evident that the assumed effect

of viscosityto reduce the wavemaking effect of the after-bodywill influence the curve of wave resistance in the right direction, i.e. it will reduce the inter-ference humps as has already been demonstrated by Professor Havelock and

Mr. Wigley.

Further factors which must be taken into account in considering the

differences between measured and calculated results, are virtual mass, and the

effect of the wave motion on the skin friction. In neither case can exact estimates be made. Virtual mass would appear as an effective increase in source strength, but its distribution along the hull is difficult to assess. For the present we can assume that it will be included in the secondary source

distributions already discussed. The effect of the wave motion on the skin friction is due partly to increased particle velocities round the curved surface

of the hull, and partly to changes in the wetted surface. Increases of 5 per cent.

to 7 per cent. in the skin friction resistance due to this, have been suggested. These increases would account for an appreciable part of the discrepancies

between the calculated and measured resistance curves.

14. Conclusions

This paper is a record of the first results of an attempt to bring our present theoretical knowledge of ship resistance into relation to the practical needs of ship design and research work. We are concerned therefore with the problem

of finding the simplest practicable solution which will give sufficient refinement

for our purpose. We must of course recognize that it is desirable that all steps in this process should be brought into line with a complete theory, but

we may, for the present, be justified in accepting purely empirical tests, provided that we do not attempt to predict what will occur beyond their limits. Bearing

this in mind we can draw the following general conclusions:

The approximate calculations, using three lines of forty sources each,

reproduces the main features of the wavemaking.

Calculations for this distribution agree well at all speeds with those for an infinite distribution.

Reduction in the number of sources to twenty still allows the main features of the wavemaking to be reproduced.

(14)

The physical equivalent of the assumptions regarding sinkage and trim is most accurately obtained when the model is allowed to take up its natural trimmed attitude.

The major discrepancies in the theoretical results may reasonably be attributed to the initial assumptions, rather than to the theoretical

method itself.

For these models the main discrepancies are due to the effect of the profile on the distribution, and to the effect of the boundary layer. It is possible to calculate simple source distributions which will give

the correct wave profile and it may be possible to calculate the

corresponding wave resistance.

Caution is required in applying (b) to the profile calculation and it is obvious that further investigation is required in connexion with (hand (g). It is evident

that even these models are too wide to fulfil the initial assumptions without

serious error and it would appear that the effect of the boundary layer can only be isolated on a considerably narrower model. Variation of the beam in both

directions should provide useful data on several points of difficulty. When we come to consider models of greater beam it will almost certainly be necessary to

consider a distribution of sources in breadth as proposed by Professor Lunde. The computing processes involved in this work are not difficult and it is quite possible to programme the calculations in such a way that they can be handled by unskilled computers. The major difficulty encountered is simply the physical volume of this computing work, and the Author is of the opinion that with existing computing techniques we cannot consider any solution of appreciably greater complexity to be practicable for our purpose.

It is, of

course, probable that with the development of electronic digital computing machines much more complex solutions will be possible, but obviously this method must be limited to large research stations. The application to this work of the A.C.E. machine now being developed at the National Physical

Laboratory is a future possibility which has received some consideration.

In spite of the difficulties encountered, the Author is of the opinion that the

results, so far, are, on the whole, encouraging, and no major snag has appeared which would prevent this method being used as a practical method of analysis.

In this connexion, it is interesting to note that in two recent papers,' Emerson has shown that the resistance characteristics of practical merchant ship forms can be related to the curve of slopes of the section area curve. This curve is, of course, equivalent to the source distribution for a single line of sources, and

it is not difficult to visualize that with further knowledge of the implications and limitations of this theory we may be able by a combination of theoretical and empirical work to approach the question of the optimum distribution for

any given conditions.

The work described above has been carried out as part of the research

programme of the National Physical Laboratory, and this paper is published

by permission of the Director of the Laboratory.

Acknowledgements

An investigation of the kind described in this paper involves a very

considerable amount of computation, analysis, and experimental work. The Author would like to acknowledge the valuable work which has been done by members of the staff of Ship Division.

The Author would also like to acknowledge the invaluable advice given by

Mr. Wigley in connexion with the interpretation of the mathematical solutions

and many other points.

N(d)

I(e)

(15)

REFERENCES

R. Gum OTON (a)" Contributional'Etude Ides Carenes Minces." Editions Science et Industrie 1939.

(b)" A New Method of Calculating Wave Profiles and Wave Resistance of Ships." I.N.A. Vol. 82, 1940.

T. H. HAVELOCK, "The Approximate Calculation of Wave Resistance at High Speed." N.E.C.Inst. Vol. 60. 1943-44.

W. C. S. WIGLEY, " L'Etat Actuel des Calculs de Resistance de Vagues." Bulletin de l'Association Maritime et Aeronautique, No. 48, 1949.

W. C. S. WIGLEY, "The Wave Resistance of Ships." Congres International des Ingenieurs Navals. Liege. 1939.

W. C. S. WIGLEY and J. K. LUNDE, " Calculated and Observed Wave Resistance

for a Series of Forms of Fuller Midsections." I.N.A. Vol. 90. 1948.

J. K. LUNDE, "Wave Resistance Calculations at Higher Speeds." I.N.A.

Vol. 91. 1949.

T. H. HAVELOCK, "Calculations Illustrating the Effect of Boundary Layer on Wave Resistance." I.N.A. Vol. 90. 1948.

W. C. S. WIGLEY, "Effects of Viscosity on the Wavemaking of Ships."

Inst. of E. & S. in. Scot. Vol. 81. 1937-38.

A. EMERSON and N. A. WITNEY, (a)" Experiment Work on Merchant Ship

Models during the War." N.E.C.Instn. Vol. 64. 1948.

(b)" A Review of Ship Model Data." N.E.C.Instn. Vol. 66. 1950. (1)

(16)

NOMENCLATURE

The nomenclature used in this paper is generally that of Reference I. The source strength has, however, been represented by M instead of m in order to avoid confusion

with the parameter used by Mr. Wigley in Equations (5) and (6).

M Source strength

Source strength at (x,., 0, Zr)

Half length of model = Half breadth of model Draft of model

RH, Wave resistance

p = Density of fluid in gravitational units = = 32.2 ft./sec.2 =-- 981 cms./ sec.'

Velocity of hull V = Speed in knots k =g/v2

S, S,= Areas of transverse sections of hull. I)

See equation (3)

= Ordinate of wave profile. Its components and are defined in

equations (5) and (6)

mx cos 0

, TI2

= Zrsec

0 dO (ksec2 0 sinmzr cos mzr) 01dm k2 sec4 0 + ,n2 0 2r z _kzr sec2 0 = 1000f sec' 0 e Al(kzr) = kx gL .cos (kx sec0) dO

=

= d = g = e sec 0.

(17)

TN 0

I 0

Fig. 1-Form of the w, land Functions. For a Single Source Mr

Fig. 2-Diagrammatic Outline of Method of Measuring Resistance at Fixed Trim

9

LOAD WATER LINES

MOD EL N0 2821 MODEL N5 2892 MODEL N5 3012 -8-OFT ---3 a 7

---Fig. 3 CARRIAGE FRAME SCALE PAN

CAPACITY GAUGE,-..

I5

2" X SPRING

SCRFFN FREE LENGTH 12'

4

6

PRINCIPAL DIMENSIONS OF MODEL hic, 2891, 2892 &

-3012

L.W.L

1-0,11

LARGEST SECTION ORDINATES FOR ALL MODELS (MAXIMUM

-LARGEST SECTION

FOR ALL MODELS

h_t

SYSTEM OF CO-ORDINATES USED WATER-LINE BEAM ci 10 0.993 5/6 d0.972 3/4 d

0 9375 373 d0.859 0.826Viz d 0-750I/2 d 5/te0.6600 5551/3 d V40-437d 0-30d 0160d KEEL0

"9.T.P- -,51',It .TP.e I,7Fer'.."b IN FT. tNat°, ai'FF'`, tg'6.." 'ciVENNL.E HULL FORM EQUATION 2891 16.0' 1.5' 1-0' 586 36-6n0-392 0-587 4+5° -D-1.6%)=40.6C-icriD-11 2892 16-0' 1.5' 1-0' 692 37-6n 0-462 0-693 12-0° + -D-0.80+0'3(f)43D-V7

3012 16.0' 1-5' 1-0' 533 36-In 0-355 0+533 00 -t- 1--(f)92[1-+ ()2]

STATIONNQ 5-SC 4:1&51/24 &6 31/2 &61/2 3&7 21/2 &71/22 &B 11/2 &81/2 1 B, 9 3/4 &91/4r/z &91/2V4&93/40 0 & 10 MODELIV2891 I 0 0 954 0 .937 0.861 0.759 0.6375 0.502 0.360 0.222 0.157 0.098 0.045 0 ..2892 10 0.992 0.968 0.926 0.869 0-7875 0.656 0.560 0406 0-316 0-221 0.115 0 3012 I. 0 0.980 0.922 0.828 0-706 056250410 0-260 0.1295 0-077 0.036 0.009 0 CARRIAGE FRAME 1.4 FORWARD 2" X I/4. SPRING FREE LENGTH 12" TOP OF MODEL o)

LOAD WATER LINE ORDINATES (MAXIMUM BEAM

-+C

1 I 1

12 0 0 6.0 +ao 0

SCALE OF X NEGATIVE MY SCALE OF POSITIVE

5

-5 FT

IN LB5

(18)

ZO

V 20'

10

STERN

OBSERVED WITH MODEL FREE to TRIM OBSERVED WITH MODEL FIXED CALCULATED

'S 5 4 _O

STATIONS 1.6 " APART BOW

Fig..4--M,Oel No. 2891: Comparison of Calculated and

Observed Wave-Profiles 2 ao o 10 20 ,s. \.---\ , (i - &772) f - 0.2295

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66 A PRELIMINARY INVESTIGATION OF THE DISCREPANCIES BETWEEN THE CALCULATED AND MEASURED WAVEMAKINp OF HULL FORMS

0

30

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(19)

co

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A PRELIMINARY INVESTIGATION OF THE DISCREPANCIES 61 BETWEEN THE CALCULATED AND MEASURED WAVEMAKING OF HULL FORMS

0 9 7 6 5 4 3 2

STERN STATIONS I 6, APART BOW

Fig. 5(a)Model No.2892: Comparison of Calculated and

Observed Wave-Profiles

(20)

OBSERVED WITH MODEL FREE TO TRIM OBSERVED WITH MODEL FIXED

CALCULATED 70 6. 5.0 o 11 4-0 0 30 Li l0 2-0 -3.0 io 8 STERN 7 6 5 4 3 STATIONS IS Fr

Fig. 5(b)Model No. 2892: Comparison of Calculated and

Observed Wave-Profiles 40 10 0 BOW 3 0 ", 0 30 CC , \ \ , --- --- --- .. ( - -235)

f

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(21)

2

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A PRELIMINARY INVESTIGATION OF THE DISCREPANCIES 63,

OBSERVEDWITH MODELFREETO TRIM

OBSERVED WITHMODEL FIXED CALCULATED. 01

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Fig. 6Model No. 3012: Comparison of Calculated and

Obfierved Wave-Profiles

4.0

₅ ₄ ₃ ₂ ₀ 10 30 0 0

(22)

8.35 /-0368 20 40 STATIONS 1 FT/SEC . _ 1 I LINE 4 LINES 1 1

---A

nal

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Fig. 7Model No. 2892: Effect of Varying the Suh-Divison

0+3 0

2 c,

ua Li I 0 0 LT) Lu LT_ -1.0 0 0 2 +1 Li Li 0 -2.0 (5 -I. -2.0

(23)

CURVES OF SINKAGE OF CENTRE OF GRAVITY &

CHANGE OF TRIM FOR MODELS a89, 2892.A 3013

-0,10 0.15 0.20- 025 030 045 04O045 050 055

II I I 1

Iv'

111 I

0.4 04 0.6

07 04 05

poz .tI 2 4..3 11.4 1.5 1,6

Fig. 9-Comparison of Calculated and Med.i.ured Wa ye Resistance for Model No. 2891

Iiiiirm"

um

Se, -- --- 1 $3 1 T -Si - SINKAGE Se - SINKAGE OF C.G OF C G FOR MODEL FOR MODEL I' N' N, 4 28911 2892

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WAVE RESISTANCE FOR MODEL N9 2802 131ECIIIIMINIMIN11+11111r1l 60111213111CFIENEUEMPICEIKev.

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nagam 0.19 0-20 0.21 -0-20 -0-20 -0.2C tiFOINEEM5MIEIMIE0131 *22-I. D 1221Ing-Mle+0.12 -1-23 0-23 -0-20 ONDIMPIR.F.E+0.05. 025-0-25 -020 ON11111:/ 0 0-26-026-0-209 0-0.pK--05a.- 5-29-0-29 -0215 CIRENCE+o101 06 0-30-030-024 D.),EgIDIAMINI 42 .32 -0-32 -0-28 [i]Fariniff5V1 034-0-34 -0-31 ETZ111EIGMF4 -25 0-39-0-39 -037 0.280+0-12 +022 39 0-41-0-41-043 0-295 +012+021 -0-46-045-049 0300 +0.02 +0.21 +.28-051 -049-0-56 0-310 0094021 +.09 0-55 -a5o 1-063 0-320 005+023-.18 0.65-0.55-049 0-328-0-08 +025-.43-0-70-0-59 -076 0-335-015+023 -075 -0.61-040 0345-041+0.12 0 -0-52 -0-65 -086 0360-100 -0.13 -0.93-075-095 0-390-232-1-00 -1-16-0.93-1-10 0410-320-142 -1.25 -1.08 -120 0430-4.0 -275 -4-65 -1.30 -1.20 -1-25 0-450-4-85-3.74 -5.33 -1-30-1-28-126 0.475-573 499 -1-30 -1-30-1 21 0-490-6-14 -1.31 -1-14 0510-671 - -7.30-1.36 -0.99 0-530-7-24 -1-38 0,550 -7-69 -1-35

'BETWEEN THE CALCULATED AND MEASURED WAVEMAICING OF HULL FORMS

Fig. 8-Curves of Sinkage of Centre of Gravity and Change of Trim

for Models 2891, 2892 and 3012

1.2

0-4

-T,

(24)

-Fig. 10-Comparison of Calculated and Measured Wave Resistance for Model No.2892

Fig. 11-Comparison of Calculated and Measured Wave Resistance for Model No. 3012

1 -0 08

0

0-6 0 4 0-2 0

COMPARISON OF CALCULATED 8. MEASURED

a

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WAVE RESISTANCE FOR MODELN.2892

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COMPARISONOFCALCULATED & MEASURED {-W.,-_0.12 41

42 0

02 C 3

WAVE RESISTANCE FOR MODEL N9 3012 0.14 0.002

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RESIDUARYC.)FOR MODEL FREETO

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(25)

BETWEEN THE CALCULATED AND MEASURED WAVEMAKING OF HULL 'FORMS

TABLE I-Profile ,Ordinates: Moslel No 2891

41.

+-t

T7,7.

TABLE 2-Profile Ordinates: Model No. 2892

COND MODEL FREE MODEL FIXED CALCULATED ORDINATES

IT= 5.21 6.08 6.85 7.26 8-74 11.42 5-21 6.08 6.85 7-26 8-74 11.42 5-21 6.08 6.85 7.26 8-74 11.42 $r0 I/4 +.43 +.43 4-54 +.55 +55 +.60 +.64 +57 +328 4-289 +339 4-39 425+26 +-97 +1.23 +1-32 +1,40+1-48 +1.60 +.97 +1.08 +1.19 +1.40 +1.51 +1.45 +.856 +.841 +908 +1.00 +.981 +98 Va +93 +i.34 +1-48 +1.60+1.86 +2.18 +1.02 1.24 +1.51 +1.70 +1-87 +2.11 4.060 +1.206 +1.358 41-40 +1.541 +1-60 1 I 4-61 +.97 +1.37 +1.62+2-00+2-82 +.54 +1.01 +1.32 +1.58 +2.08 +2-85 +647 4064+1443 +1.60 +2073+2.45 II/2 - - 11 +.27 +-75 +1.09 +1.73 +2.89 - I8 +0-25 +.70 +.99 +1.139 +2.97 -.019 +.536 1.078 +1.22 +1-991 +2-64 2 --56 --42 0 +-32 +1.21 +2.72 -.55 - .45 -0.10 +.16 +1.36 +2.65 --48! -100 +.406 4- .63 +1.593 +2.35 24 --47 -.87 -.70 -56 +-46 +2.15 --52 -.88 - -88 --64 4- -39 +1.85 -569 -609 -.241 -.07 +.993 +1.84 3 --31 -1-00 -1.23 -1.145 --54 +1-35 -.32 -.92 -1.3 -1-18 --62 +66 -.465 -888 -.759 -.73 +.231 +116 31/2 --18 -.78 -1.32 -1.49 -1-30+42 -.27 --77 -1-3 -1.39 -1.31 -.29 -.220 -.908 -1.234 -1-28 -496+35 4 -.30 -.65 -1-32 -1.55 - 1-90 - -52 -.27 -54 -1-06 -1.37 -1.71 =1-49 -.190 -.712 -1.423 -1.65 -1.240 --52 41/i --47 --42 -1-09 -1.43 -2.13 -134 -.43 -.42 --89 -1.23 -1.96 -2.17 -.377 --443-1-369 -1.80 -1.899 -1-30 5 -.66 -36 --53 -1-20 -2-26-2-09 --54 -.41 -.66 -1-00 -1.99 -2-41 --679-247 -1.163 -1.73 -2.411 -1.98 5%i -.55 -.46 --71 -.99 -2-19 -2-39 -.42 -39 -52 -79 -1.84 -2-56 -.749 -.204 -.836 -I.50 -2-745-2-66 - -44 58 -.52 -.77 =2.07 -2.71 -.29 -.39 --42 -.59 -1-64-2-56-642-285-522-1-13 -2.850 -3.18 ! 6V2--32 -.64 -47 -59 -1-95-3-07 -18 -.38 -32 --44 -1-42 -257 -.378 -.432 -296 - .71 -2-735-3-56 7 -.35 -.65 --45-43-1-77-3-22 --04 -41 -.25 -.28 -1.19 - 2.58 -.139 - .667 -.142 - -30 -2.423-3-78 74--23 -.43 --50-31 -1.37 -3.12 -.07 -.36 -23 --16--97-2-52--046--548--114 +-01 -1-926-3-78 , 8 --23-39 --46 -20 -97 -2-73 -.08 --12 -.21 --08 --60-2-40-079-368-114 +.23 -1-275-3-59 els -.07 +.05 -.25 --07 --50 -2.82 -.06 +14 -18 +05 -.23 -2-1S --104 -070 --064 +.40 --496-3-16 9 4-10 +.23 --03 +08 +.15 -2-34 +.10 4-24 -.02 +135 +17 -1.82 +.008 +361 #099 +.50 +.353 -2-50 +.16 +42 16 +.32 +.79 -1.81 4-39 +.53 +.37 +.33 +.77 -1-22 +278 +729 +367 +.67 +1.227 -1-66 STh i0 4-70 + 89 +.93 +85 +1-37 - -23 _+ .70 _+.92 +94 +.85 +1-38 --64 +751 +1.038 +.858 +.98 +?.?3c --85 rC084D4 - MODEL _

FREE MODEL FIXED I CALCULATED - ORDINATES - f v . 5.21 5.90 6-85 7-08 8-35 11-24 5 . El 5-90 6-85 7.08 8-35 -11.24 5.2i 5.90 6-85 7.06 8-35 11-24 S0 +90 +1.02 +64 +.74 4-70 +91 4 .81 +84 +80 +.60 4. 91 ±90 4-515 4-694+631 +-60 +.561 +.668 +2.25 +2.62 +2.77 +2-88 4325 +3.25 +204 +2-34 +2.60 +2-50+3.00 +302 +1.785 +2-002+1-379 +2.02 +2.061+2.384 I/2 +1.65 +2.45 +2.99 +3.18 +398 +4-83 +1,65 +2-22 +2.82 +2.92 +3.89 +4.51 4(896+2-429+2- +2-78 +2-950-1-3.325 1 -.25 +-61 +1.45 +1.82 +3.00 +5.44+.08 +-61 +1.67 +1.69+2-88 +6.11 +.250 +1.059 +1.760 +1.98 +2-6204541 114 -1-13 --87 --53 - .20 +1.26 +385 -1-08 -.95 --19 + -02 +1-IS +4-02 -1-098-433 +319+0-52 +1.290+1.289! 2 -1.17 .-1-55 -1-44 -1.27 - .30 +1.62 -1.02 -1.40 -1.28 - 1.19 - -33 4- 1.79 -1-366-1-309-762 -0.57 +145+ 184

2ji --43 -1.30 -1.85 -1.91 -1.69 -65 --35 -HO -1.55 -1-61 -1.31 +12 - -so! -1-546-1-470-1-36 --761 -.826

3 --03 --85 -1-84 -1.97 -2.21 - -40 +-21 -.55 -1-36 - 1.52 -1.81 --99 --006-1-177 -1.761 -1-79 -1-430-1-431 3/4 +.03 -.27 -1.35 -1.70 -2-21 -1.25 +.14 -.02 -1-04 -1.22 -1.83 -1.75 +-504- .475 -1.679 -1-83 -1-890 -1-858 4 -.12 -+11 - -90 -1.I7 -2-13 -1-71 --05 -.22 --57 --82 -1-69-1-53 +275+132-1261 -1.52 -2-116-2-079, 02 --50 +09 --50 - -75 -1.98 -1-50-42 -.09 -.26 --46 -1-48 -1.87 --366+355 -.721 -1.00 -2-150-2-145 5 - .61 - -18 -.30 - .47 -1-66 -2.06 - 43 -.10 +05 --16 -1.30 -1.87 --836+143 -.139 -0.45 -2.070-2.033 31/2 --50 --55 -.22 - .22 -1.41 -2.21 --26 -.32 +.06 +-06 - .95 -1.84 -851 --300+25! +0.05 -1.741-1-686 6 -.33 -.76 --27 --24 -1.20 -2-12 --12 -.54 +.08 +-09 --73 +I.92, - -437 -.884 +.427+0-33 -1.348-I-272 61/2 -17 --68 --45 --37 -1.09 -2-44-21 -.62 -.28 --09 --56-2-13 --032 -1.205 +307 +0.41 -.930-859. 7 --20 -.59 -.72 --60-65 -2.62 -43 -42 -.52 -.22 - -50 -2.12 + -108 -1.171 --114-0-75--4995--469 7Y2--42 --34 -.89 --81 --70-2-85-43-40-69-47 - .48 -1.97 -.098 -.762 -.622 -0.16 - .147 -.167 5

-50 -

-95 --94 --65-3-02-38-33-80-76 - -44 -2-06 -.511 --300--050-0-58+-085+-013 ea-.44 --30 -91 --94-62 -3-00 -36 -27 --85-83 - -48 -2.12 - .802 +.028 -1.254-aos +-199 +.067 9 -32 --27 -.80 --91 --GO -3-20-18 -.25 -.95 -.97 - .5 -2.32 -.543 +.103 -1-271 -1-31 +233+053 9Ya 0 -.20 -57 --60 --57 -3.12 -.07 -.16 --17 - -49 - -48 -2.24 - -091 --092-1.003-1.42 t .103-.120 6,"10 +.85 +1.13 429 +85 +89 -1-67 4- 94 +1-12 +1.22 +90 4- -66 -i-12 + 6Il +.021 --220-0-66 +198 -.052 67 +44 +.40 +-40+ -74 -6

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