Experimental determination of wave resistance of a ship model from lateral wave-slope measuerements

Webb Institute of Naval Architecture Glen Cove, New York

EXPERIMENTAL DETERMINATION OF WAVE RESISTANCE OF A SHIP MODEL FROM LATERAL WAVE-SLOPE MEASUREMENTS

by

Lawrence W. Ward

June 1968

This research was carried out under the Naval Ship Systems Coijuijand, General Hydromechanics Research Program administered by the Naval Ship Research and Development Center. Prepared under the Office of Naval Research Contract Nonr-4152 (QQ).

The section of this report titled "Theory" was

prepared as Lecture Notes under sponsorship of the Office of Naval Research Contract Nonr (G)-0001I-67, Project NR-062-305.

This document has been approved for public release and sale; its distribution is unlimited.

ABS TRACT

A new method of determining ship model wave resistance from the wave pattern generated during a test in a model tank is outlined. This method is based on a Fourier analysis of lateral wave-slope _data

taken on a longitudinal cut parallel to the model path. Results of an exploratory test series carried out at Webb Institute _{are given} and compared with previous results using a different method. The comparison is very encouraging and indicates that a new technique has been established which has many beneficial features, _including potential application in full-scale research. _{Details of an electrical} wave-slope measuring probe for use in the model tank _{are included in} an appendix.

TABLE OF CONTENTS

Page No.

ABSTRACT 1

TABLE OF CONTENTS ii

LIST OF FIGURES AND TABLES iii

NOMENCLATURE iv

INTRODUCTION i

THEORY 3

Momentum Analysis

Fourier Wave-Spectrum Model

Stationary Wave-Pattern Requirement Eggers w u Parameters

Wave Resistance Formula Choice of Path

EXPERIMENTS 14

Tests Analysis Results

CONCLUSIONS AND RECOMMENDATIONS 30

ACKNM1EDGMENTS 32

APPENDIX "A': DEVELOPMENT OF THE WAVE-SLOPE MEASURING PROBE 33

APPENDIX "B": COMPUTER PROGRAM 37

REFERENCES 40

INITIAL DISTRIBUTION LIST

DD FORM 1473 - DOCUMENT CONTROL DATA FILE INDEX CARDS (LOOSE)

-11-LIST OF FIGURES AND TABLES

Page No.

Figure 1: Energy Balance Control Volume 4

Figure 2: Test Geometry 15

Figure 3: Sample Record (Run G-2) 19

Figure 4: Results of Lateral Wave-Slope Tests on Series 60, 22

0.60 Block Model

Figure 5 (a,b,c,d,e): Lateral Wave-Slope Spectrum Results: 24

Runs D-1, G-2, J-1, J-2, anzi J-3.

Figure A-l: Wave-Slope Probe and Calibrating Plate 34

Figure A-2: Wave-Slope Probe Circuit ₃₅

Figure B-1: Computer Program 38

Table I: Characteristics of the Series 60 Model 14

NUMEN CLATIJRE

Letters and Symbols

A, B, C, D constants, functions, or points

A wave amplitude, ft.

B model beam, ft.

b breadth of model tank, ft.

C Fourier Cosine-transform; also 2

resistance coefficient

R/f/SV

Cb, C block and prismatic coefficients

e distance of cylinder and model off

tank centerline, ft.

E energy in wave system,

ft. lbs.

F Froude Number = V

/.Ji

gr

acceleration of gravity, ft./sec.2

H model draft, ft.

h depth of water in model tank,

ft. k wave number, f t. K constant L model length, ft. pressure, lbs./ft.2 R resistance, lb. s

element of length along path AB, ft.

-iv-S _{Fourier Sinetransform; also wetted} surface, ft.

u, y, w fluid velocity components in the

x, y, z directions, ft./sec.

uV , Eggers' parameters, see Equation (15)

V _{model speed, ft./sec.}

V _{fluid velocity normal to AB, ft./sec.}

w _{model weight, lb.}

w

X, y, z

energy transfer into wave system from rear plane, ft. lbs./sec.

Cartesian co-ordinates, ft. (see Fígure 2) is vertically upwards.

angle of path With the x-axis, radians

speed constant gL/V2 = 1/F2 displacement, lbs.

wave elevation, ft.

lateral wave slope radians wave direction angle

wave length, ft. index = 1, 2,

3 density of fluid, slugs/ft. ve1ocit potential, suh that

u =

Y)(,

etc., ft. ¡sec.

Wave coordinate = x cosQ + y sin9, ft. ohms electrical resistance

Subscripts

o centerline wave properties

y lateral slope

I.T.T.C. International Towing Tank Conference t, f, r, w total, frictional, residual, and

wave parts Superscripts, etc. Q Q, Q (Illustrated for "Q") time rate of change of Q

Q per unit distance in the x direction. Primes are also used to denote variations in a quantity.

average value of Q (see footnote page 6 )

-INTRODUCT ION

In recent years several methods have been developed to determine ship model wave resistance experimentally in the model tank

from measurements of the wave system generated by the model during a

test. Interest in this subject was stimulated by a conference at Ann

l'

Arbor sponsored by the Office of Naval Research. The methods, several of which have been found to be successful, stem from essentially the same assumptions of linearized potential wave theory, but differ in

the type of wave characteristic measured (e.g. _{wave elevation, slope,} or force on a fixed object) and the type of path along which data are taken (e.g. a longitudinal or transverse cut, relative to the model

path). Most depend upon a Fourier analysis of the data and yield the

spectrum of the wave pattern as well as the wave resistance _{as the} result. It should be noted that it is not necessary to _{assume the} validity of any ship-wave generation theory (e.g. thin ship theory) for the derivation of these methods. An up-to-date picture of this work including a complete bibliography of _{papers related to this} subject is contained in Reference (2).

The work done by the author at Webb Institute _{over the past} few years3'4 has primarily concerned his so-called "X-Y Method" which uses the x and y components of the lateral force exerted by the _wave system on a long fixed vertical circular cylinder, and which _{does not} require a Fourier analysis but yields the _{wave resistance directly.} A major advantage of this scheme, as compared _{to those using wave} elevation data, was found to be the finite length or "self truncation"

advantage has been tried utilizing lateral wave-slope data taken along a longitudinal cut. This method was first suggested along with a number of others by Sharma5 in 1963 and yields the wave-slope spectrum as well as the wave resistance. Tests run at Webb Institute in June 1965, just

before the author left for a one year National Science Foundation

Fellowship in Hamburg, Germany, have now been analyzed, and this method has been found to be quite convenient and successful in the smaller facility at Webb. An attempt to include this method among others during a test series in the Hamburg model tank2 did not meet with similar success, mainly because of the difficulties encountered there in connection with

the slope probe instrumentation problem. However, in connection with the latter effort computational and theoretical evidence was found2 which indicated this method to be consistent with some of the others which were used in the actual tests. This paper will outline the lateral

wave-slope method and underlying theory briefly and present the test results and experience gained to date in its use.

-2-THEORY

The basic underlying theory of wave survey methods in general can be found in several references, the latest being that by Eggers, Sharma, and Ward2. In this section a brief outline will be given of the theory needed for the lateral wave-slope method in particular, with emphasis on understanding the significance of the various steps involved in carrying

it out.

Momentum Analysis

We consider the situation shown in Figure 1* of a model traveling at constant speed V along the centerline of a long tank of breadth b and depth h, and a set of axes x, y, z moving with the model. We consider potential flow and small surface disturbances, . Consider a

volumetric region I, fixed in space, whose side boundaries are formed by the lines ABCDC'B' projected down from the free surface except in the region of the model sides (where it is formed by them) and whose bottom surface is the tank bottom, both of the latter within the lines ABCDC'B'. We consider path with symmetry about the tank centerline, and therefore only the region ABCD on one side of the tank need be considered and the

results doubled. The line CD is drawn sufficiently far ahead of the model so that there is no disturbance there. _{The line AB in general forms a curve} as shown. _{The model moves ahead at the speed V and creates waves within}

m

region I. _{This wave system is often idealized to the concept of a transverse}

7AJK S/&

'//// / /y/ // / / / / /////////// ////// ////i//////////////J/

_1c

î

/8

bi

N

/2

N

0

I

_14

Cís L ì,v

'5

-5,

//////// ////////7

/////////////////////////////

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f

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ZIÁ/' 507 7W

FLL-v4? T/AJ

-

4 L NCE

C,V77OL L'Z

/1ODLL

C,

shown in Figure 1 and is often referred to as the Kelvin _{wave pattern.} The wave system grows within this region and therefore the energy in the wave system increases at the rate:

E

=

E'V

(1)

where E' _{is twice the average total wave energy in a strip of unit} width in the x direction along the line AB extended down through the depth h. _{This growth of energy must be supplied by the work done at} the boundaries of the region, this rate of doing work being given _by integrals over the boundaries of the fluid of the pressure p times

the velocity, V, normal to the boundary. There are no contributions to W from the tank sides BC and B'C' or the front plane CDC' or the tank bottom due to the normal velocity being _{zero in each case, nor} from the free surface, due to the pressure there being zero ambient. Along the model sides there is work being put in, due to the action of the pressures there in conjunction with the forward motion of the model, at the rate R V , where R _{is the wave resistance, which is}

w m

w

the only resistance possible in the present _{case in which an ideal} fluid and small disturbances are assumed. _{Along the prismatic}

surface AB we have the work done by the waves in region II on region I, i.e. the "work done on the rear plane":

have -

W

. For the general case shown where the curve AB makes an angle ( ¶'/ -

o<

₎ locally with the x axis we have:

Vb -

U- -

(3)

where:

dx

c and,

d

u and w are the fluid velocities in the x and y directions. The pressure p is given by the linearized Bernoulli relationship:

-

ru

-

a

(4)

where terms of the order of the fluid velocities squared have been neglected in relationship to the unsteady pressure term

f

U

Since the value of for use in the energy relationship must be an average value with respect to time' at a fixed plane, the pressute times velocity terms represented by the first and third terms in Equation (6) will average to zero. Thus we have:

* The time average Q of a quantity Q is defined as:

=

wheret is the period of fluctuation of Q or the limit can be taken

as

-6-WAß =

ZfV

d

u(u-vr<) d

(5)

4ß h

Since this is the only work term other than that due to the towing force we have, equating the total work put into region I to the average rate of growth of energy in the wave system in this region:

\ty3

Vrn

(6)

The total energy E' is equal to the sum of the potential and kinetic energies per unit distance in the x direction:

=

fJdj(U2+v2W2)dz

(7)

AB

where ¿ t()()is the wave elevation and w is the fluid velocity in the z direction. Combining Equations (5),(6), and (7) gives the

"1st 1-lavelock Formula":

-if

dj(Cu2vz+w2)

(8a)

_zpJd

_{fucu-vi<) d}

for the general path AB. Two special cases are of interest: For

plane perpendicular to the direction of motion we set c,< _Q giving

b//L

Tad

This is the form used for a transverse cut method. _{For a longitudinal} cut method we set giving:

r

=

p/dx

Uyc/

Fourier Wave-Spectrum Model

A study of Equation (8a) will convince the reader that this relationship does not form a convenient basis for an experimental method, as it requires determination not only of the wave elevation along the path AB but of the fluid velocities u, y, w throughout the

depth of the fluid along the vertical plane formed by the line AB extended in depth into the fluid. We wish to use potential wave

theory, assuming free gravity waves of small amplitude, in effect to predict these fluid velocities from knowledge of the wave elevation

(or in the present case from knowledge of the wave slope in the y

direction y ). This will allow us to "trade-off" more detailed knowledge of the function on the surface for the next-to-impossible

integration in depth. This would be quite simple for a single component wave of amplitude A and length traveling in a direction e to

the x axis: and c =

t

Accsc.

where: (-

XCQSO +ScS

k=

iî/->

Potential gravity wave theory would then predict:

-8-(9)

is the wave speed in deep water which is assumed. The

u=

(10)

V =

w

-where:

-kc

e

COS)

wave system which is likely to be generated by even a simple traveling disturbance is quite complex, however6. For this reason we must use a spectrum representation:

t

=

j[Acoskc

4-e(e) SLr

_de

(11)

This can be seen to be the superposition of _{a large number of waves of} amplitude A(Q)d _{traveling in different directions with a varying phase} relationship allowed by B(@). _{A similar spectrum form exists for}

and therefore the velocities u, y, w are known in terms of the spectrum functions A and B. _{Moreover simple differentiation would result in}

the _{or "lateral wave-slope" spectrum:}

j

[Ae)

5

IS

SH

kc]

de

(12)

where

Ae)

₃₍

, etc.

It can therefore be seen that knowledge of the _{spectrum of one characteristic} of the wave system (say _{) will in theory allow any other}

to be derived analytically. _{This is an important point which will}

allow one to pick the most convenient data to _{measure experimentally, regardless of which} type of spectrum is needed. _{Some typical wave spectra can be seen in} Figures 5(a)-(e), and further discussion _{on the matter of wave patterns} can be found in Reference 6.

Stationary Wave-Pattern Requirement

Another very important fact about the ship wave system is that it must be stationary with relation to the ship. The corresponding relationship between

k

and Q, which can be derived either geometrically or from mathematical reasoning , is:

\ç sec.2e

(13)

where:

_{/ v}

Other useful relationships are:

cose

c

0o&

(14)

Using the foregoing relationships the argument

(&) of

where c is the wave speed, and the zero subscript denotes the wave component for e = O, or the "centerline" wave. It should be noted that all other waves are shorter and slower than the centerline wave, and that c_O = V

ir.

Eggers w, u Parameters

-JO-the wave functions can be written:

where: _W1,

u3,

WN+- U-v(:i

k0

SecG

The factors w.,, and _{uVare the convenient "wave numbers in the}

8

x and y directions" introduced by Eggers , and are used as parameters

in the analysis to follow. For this purpose however they are non-dimensionalized as follows:

WV'

W/

U/k0

(16)

This form will be used with the primes not written in this text. It

is seen that now there is only one parameter in the spectrum representation; either 9, w or uvwith the other two derived from it. Many different

forms of the previous equations and those to follow _{are therefore possible.}

Wave Resistance Formula

With the wave spectrum model now established, the ist 1-lavelock

Forumla (8) can now be converted to integrations involving only _{the function}

t

that is the surface wave elevation. _{The resulting equations will appear}

either (a) in terms of the wave spectrum functions A and B _{or (b) in terms} of the related Fourier transform coefficients (denoted _{S, C, etc.) of some} type of wave data ( , tetc.) along some path (longitudinal or transverse cut, etc.), or (c),in the special case of the X-Y Method, _{in terms of a}

I

direct integral of a-function along a path. The latter is fully discussed in Reference (3). The first mentioned is the classical "2nd Havelock Formula"9:

1r/

Zia

2.

2-pV

I(A +B )

COO d

(17)

Jo

Which is extremely simple in form and requires only the magnitude of the wave spectrum A(Q), B(Q) to be known. The second-mentioned is very closely related to it and gives, for lateral-slope data taken on an infinite longitudinal cut5:

r

where:

1SLr

i

t

_Co

Lwx dx

are the sine and cosine transforms of the lateral slope data along a line y = constant. The infinite limits are in practice replaced by a finite integration over the extent of the signal which for the

lateral slope data is finite due to its self-truncation characteristics noted previously. It is interesting to note that the Q form of (19):

z

R

Z'rr

(c.2)

C0s30

1e

(20)

R

_WZ1T

fv

'IÖ

1s

t' 2.

dU-i,

çsj

w4(w-i)

12 -(18) (19)

is very similar in form to (17). The Eggers' parameters u,and w, are more convenient to use however in the computations.

Choice of Path

It should be noted that the results R and S , C should not

w y y

depend on the path chosen; however both theoretical reasoning and computational experience show that the line y= constant should be neither too near the model or the tank wall for accurate results. The

latter is due to tank wall reflections, not considered in the foregoing, and the former due to the presence near the model of a "local" wave

system for which the wave spectrum model as given in Equations (11) and (12) is not a valid representation. Further guidance on this question can be obtained in Reference (2). En the case of lateral slope data, for reasonable distances of the probe from the model, tank wall

reflection in a tank of normal breadth does not occur until the record has truncated itself; an advantage which this method shares with the X-Y Method3, and which is not in general true of other longitudinal

cut methods2. If the tank breadth is a critical factor, some advantage can be gained by employing "off-centerline towing" as described in

Reference (3) and shown in Figure 2. A longitudinal cut is accomplished mOSt easily by employing a fixed measuring location over the water surface rear the tank centerline and letting the model pass by.

In the foregoing, port-starboard syrrirnetry of the wave system and therefore of the spectrum has been assumed and therefore only one path on one side of the model is necessary. _{If such symmetry cannot}

EXPERIMENTS

Tests

Experiments were carried out on June 4, 1965 at Webb Institute

on a 5'-O" model of the Series 60, 0.60 Block Coefficient form. Particulars of this hull form are given in Table I.

Table I: Characteristics of the Series 60 Model (See Reference 10)

Length Between Perpendiculars L 5.000 ft.

Waterline Length 5.084 ft. Beam B 0.667 ft. Draft H 0.267 ft. Displacement 31.170 lbs. FW Wetted Surface S 4.262 ft.2 Block Coefficient Cb 0.600 Prismatic Coefficient C 0.614 p Displacement-Length Ratio 122

Stimulators: 1/8" Diameter x .050" height pins spaced 0.275" placed 4" aft of bow

The model basin at Webb Institute is 93 ft. long with a 10 ft. wide by 5 ft. deep rectangular section and a carriage on an overhead rail. The model was towed at a distance e

= 6m,

off the tank centerline with a slope measuring probe mounted at a distance 6-7/8 in. off the tank centerline on a fixed arm at about the mid-length of the tank but on the opposite side, as show-n in Figure 2. This maximizes the length

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WAVE

of the resulting record along the line AB before wave reflection from the sides of the tank take place.

Records were taken of the Lateral slope of the waves caused by the model measured by a three-wire resistance type slope gage developed in conjunction with the work and described in detail in Appendix A. This instrument was designed to produce a linear reading, on a two channel Brush RD-4622-OO amplifier and Mark II recorder, proportional to the wave-slope in the y direction over a range of ± 0.20 radians and to ignore changes in the wave elevation over the maximum range encountered. Three repeat runs each were made at three different speeds over the range covered previously for this model in the "X-Y'Method" tests, in order

to provide an exploratory series to examine the method itself. These

conditions as well as the resulting resistance values are shown in Table iiJ

-Table II: Test Results from the Lateral Slope Tests Run on 6/4/65 on the Series 60, 0.60 Block Model

Froude Effective No. Caic. Resistance Run No. Model Speed Number Tank Width Steps Coeff t.

Vm, ft./sec. _Fr

_beffft

,' max

c +27. D-1 3.54 .279 12.2 51 .655 D-1 3.54 .279 12.2 76 .666 = D-2 3.51 .277 12.0 51 .549 D-3 3.50 .276 12.0 51 .494 G-1 4.29 .338 18.0 51 1.576 G-2 4.30 .339 18.1 51 1.480 G-3 4.30 .339 18.1 51 1.533 J-1 5.09 .401 25.3 5 5.759 J-2 5.07 .400 25.1 51 5.772 J-3 5.09 .401 25.3 51 5.994

A sample record, Run G-2, is shown in Figure 3. It can be seen that this record is free from electrical or mechanical "noise". The event mark indicates passage of the model bow past the measuring station. The recorder was run at its maximum paper speed of l25mm/sec. to obtain

an easily readable record. The calibration factor for the tests, converted to the sensitivity setting of this tape, was 85.0 mm radian, corresponding to a full scale tape reading of ± 0.17 radians. Less than 1.0 ILIL deflection was noted over a change of elevation of ± 1.0". As can be seen in Figure 3,

the record was practically self-truncating before wave reflection from the tank walls took place. The tests were all accomplished, including set-up of equipment and calibrations before and after, in one working day.

Although turbulence stimulation is not as important in this type of testing as in the case of ordinary resistance tests, a turbulance stimulation pin configuration found suitable for this model in the high speed range was fitted as shown in Table I, however the tank water was not heated but was at the ambient temperature of approximately 72°F. The regular side cylindrical wave dampers and end corner beaches were in place during the tests. Speed measurement was accomplished by means of a timer acting between switches on the raíl 35.0 ft. apart, and the model was

towed by an arm on a fixed strut on the carriage.

Analysis

Prior to actual analysis, each tape (e.g.Fig. 3) was divided into 2.5 mm spacing ( the divisions shown) and numbered and the data read off and transferred to IBM cards, thus providing the raw data in a usable

"digitized" form. While this was of necessity done by hand, the instrumentation would be easily adapted to automatic digitizing by suitably designed tape

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apparatus with a "playback" feature to realize the scanning rate of 50/sec. corresponding to the hand calculation.

The analysis itself consists of two steps. First is the carrying out of the two x integratIons over each record, given by Equation (19), at a series of values of the "x-wave number" to give the Fourier sine and cosine transforms S and C as a function of w . These are in

y y

themselves a result and can be plotted. The second step Is the integration of the sum of the squares of these with the weighting function:

K (wy)

=

_{(z wv}

-i

in accordance with Equation (18), to give the wave resistance R for the tape analyzed at the speed V of the run. These results are non-dirnensionalized in terms of the resistance coefficient C and the Froude number Fr given by:

C =

-where S is the wetted surface and L the length of the model. The integrations are done by trapezoidal rule with steps of u given by:

b

where is a hypothetical effective tank width corresponding to the exact Eggers series-spectrum model of the same spacing. In the present case a choice of:

-

_(O

-

II (24)

-o

b

was ruade, thus giving steps

Lu

of O.1O.. The corresponding steps of w are not even and are given by:

(25)

The Fortran computer program designed to carry out the above

calculation is described in Appendix B. A cut-off corresponding to 51

was found to be sufficient in terms of no further contribution to the

resistance values beyond this point, although the spectrum itself continues, because of the strong effect of the weighting function K(w) in Equation (18); as shown in Table II extension to7 76 in the case of tape D-1 gave only

a 27. increase in the result. The calculations were run on an IBM-1130 computer. Analysis time for the final set of calculations was about 3 minutes per ship run.

Results

The resulting wave resistance coefficients from this set of tests are given in Table II and compared with the previous results3 using the X-Y Method, which are shown as a faired line, in Figure 4. Also included in this figure is the residual resistance coefficient C derived from a

r

previous total resistance test of the same model at Webb. It can be

seen that excellent agreement is present between the lateral slope results and the faired line at the lowest and middle speed tested, and that the

8.0

7.0

6.0

50

4.0

3.0

2.0

1.0

4: RESUL-rs o- LP.TERL \t\JAVESLOPE TESTS

o

ts

60 .50 B

o ca<.

M0

E L

e LTER/L

J'SLO

ESULTS ''65

FAtRED LNEtSt ON '&-Y

5ULTS, 963

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f.

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.30

_.40

.50

F=

V/L

a Froude number of 0.34 for these tests due to the XY cylinder being of too small a diameter. Moreover, it can be argued that the measured wave resistance should approach the residual in the range of high

Froude numbers. The basic validity of the X-Y Method has been further established by means of recent tests2 in Hamburg, Germany on a larger model. Thus from the limited number of results obtained so far the

lateral slope method appears to be a valid technique for measurement of wave resistance in the model tank, giving results as good as and perhaps better than the other methods used to date.

Several of the lateral wave slope spectrum results are shown in Figures 5(a)-(e). While these cannot be compared with any other established result, certain known features of such spectra can be examined. One is the presence of zeros in the ideal fluid case or near-zeros in the actual case of a real fluid of small viscosity.

Physical reasoning shows6 that these should be spaced, when plotted against a base of w , approximately at an interval of:

Lw=

¶//(

(26)

where=

lt can be seen in Figure 5(b) for the case of G-2

that this is approximately true. The trend of movement of these "zeros" away from the axis as the speeds are increased might be the result of worsening separation at the higher speeds. In addition the consistency of the resulting spectra can be noted for the highest speed tested in Figures 5(c)-(e); this is not too bad considering the accuracy of

measurement possible and the complexity of the analysis. An additional factor which might contribute to such variations is a port-starboard fluctuation of the wave system which has sometimes been noticed during

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