(the david w.taylor model basin report 806) gertler, morton-reanalysis of the original test data for...

REANALYSIS OF THE ORIGINAL TEST DATA FOR THE TAYLOR STANDARD SERIES

US

NAVY DEPARTMENT

THE DAVID W TAYLOR MODEL BASIN

REPORT 806

Imperfections appearing on some pages in this book result from imperfections that

appear in the original manuscript

Reprinted 1998 By The Society of Naval Architects and Marine Engineers

Series Preface

n celebration of the centennial of the establishment of the first model basin in the United States of America by Rear Admiral David Watson Taylor, the Society of Naval Architects I and Marine Engineers is reprinting Rear Admiral Taylor’s Speed and Power of Ships. The original volume was published in

1910, revised in 1933, and revised again in 1943. Although the various editions have been out of print for some time, the Society chose to reprint the 1933 edition because it is the work of Rear Admiral Taylor that was revised and corrected by him from the 1910 edition. On the other hand, the 1943 edition was revised by a joint committee of members of the staffs of the Maritime Commission and the David W. Taylor Model Basin. Also reprinted is the David Taylor Model Basin Report 806, A Reanalysis of the Original Test Datafor the Taylor Standard Series, by Morton Gertler. This report presents the Taylor Standard Series in the form presently used today.

avid Watson Taylor was born on March 4, 1864, in Louisa County, Virginia. He graduated D from the U.S. Naval Academy in 1885 with the highest scholastic record up to that time. The U.S. Navy selected him to attend the Royal Naval College in Greenwich, England, for advance study from which he graduated in 1888, again, with the highest record up to that time. Throughout his life, he received many awards. He was an original member of the Society and was the first recipient of the Society’s Taylor Medal, which was named after him.

e had become interested in model basins while in England and persuaded the Navy and H Congress to build the Experimental Model Basin, which was dedicated in 1898 at the Washington Naval Yard. Because of the technical need for a larger facility and the deterioration of the foundation of the Experimental Model Basin, the Navy then constructed the David Taylor Model Basin at the Carderock site. The David Taylor Model Basin was dedicated on November 4, 1939.

Society, in the memory of Rear Admiral David Watson Taylor, trusts that this publication T t r l l be useful to naval architects all over the world.

William B. Morgan, Dr. Eng., NAE Chairman, Publications Committee

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

LIST OF ILLUSTRATIONS

PAGE

Lines and Offsets for the Parent F o r m of the Taylor Standard Series . . . . . . . . . . . . . . . . . . . . . . . . . Sketch Showing Relocation of Sections of a Parent F o r m to Produce a Derived F o r m Having a Different Longitudinal Pr i smat ic Coefficient . . . . . . . . . . . . . . Sectional-Area Curves for the Derived F o r m s of

Curves of Geometrical Parameters Used to De- fine Mathematically the Sectional-Area Curves for

Comparison of the Taylor Displacement -Length Ratio and W etted-Surface Coefficients with the Redefined Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . Contours of Volumetric Coefficient Versus Longitudinal Pr i smat ic Coefficient and Length-Beam Ratio for a Beam- Draft Ratio of 2 .25 . . . . . . . . . . . . . . . . . . . . . . . Contours of Volumetric Coefficient Versus Longitudinal Pr i smat ic Coefficient and Length-Beam Ratio for a Beam- Draft Ratio of 3.00 . . . . . . . . . . . . . . . . . . . . . . . Contours of Volumetric Coefficient Versus Longitudinal Pr i smat ic Coefficient and Length-Beam Ratio for a Beam- Draft Ratio of 3.75 . . . . . . . . . . . . . . . . . . . . . . . Effects of Longitudinal Pr i smat ic Coefficient Variation

the Taylor Standard Series . . . . . . . . . . . . . . . . . . .

the Taylor Standard Series . . . . . . . . . . . . . . . . . . .

on the Shapes of Derived F o r m s . . . . . . . . . . . . . . . .

3

4

6

8

9

10

11

12

14

xi

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Effects of Volumetric Coefficient Variation on the Shapes of Derived Forms . . . . . . . . . . . . . . . . . . . . . . . Effects of Beam-Draft Ratio Variation on the Shapes of Derived F o r m s . . . . . . . . . . . . . . . . . . . . . . . . . Water -Temperature in the U.S. Experimental Model Basin Versus Calendar Date for the Years of 1913 to 1918 . . . . . Curves of Residual -Resistance Coefficient Versus Speed - Length Ratio, Showing Typical Data Spots . . . . . . . . . . . Auxiliary Charts for Restricted Channel Corrections to the Taylor Series Models Tested in the U.S. Experimental Model Basin . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Curve of Residual-Resistance Coefficient Versus Speed - Length Ratio, Showing Restricted Channel Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . Curves of Residual-Resistance Coefficient Versus Longi- tudinal Pr i smat ic coefficient for Equal Values of Volu- met r ic Coefficient . . . . . . . . . . . . . . . . . . . . . . . Curves of Residual-Resistance Coefficient Versus Volu- met r ic Coefficient for Equal Values of Longitudinal Pr i smat ic Coefficient . . . . . . . . . . . . . . . . . . . . . Comparison of the Res idual-Re s istance Coefficient Obtained from Tes ts of New Taylor Ser ies Models with Values Read from Contours of Appendices 3 and 4 . . . . . .

Converting the Total -Resistance Coefficient E,"TEi:eB Resistance Coefficient . . . . . . . . . . . . . . Factors for Converting the Froude Number 3 to the @ Speed Coefficient . . . . . . . . . . . . . . . . . . . . . .

PAGE

15

15

22

23

26

28

29

30

31

34

35

PAGE Figure 2 1 Schoenherr Fr ic t ional-Resis tance Coefficients for a

400-Foot Vesse l Operating in Salt Water of 3.5 P e r c e n t Salinity and a Tempera ture of 59F. . . . . . . . . . . . . . . 36

The Variation in Effective Horsepower of Taylor S e r i e s Vessels with Change in Longitudinal P r i s m a t i c Coefficient

F igure 22

for a Volumetric Coefficient of 1 . 5 ~ 1 0 - ~ . . . . . . . . . . . 37

Figure 23 The Minimum Effective Horsepowers of Taylor S e r i e s Vesse ls of Various Lengths with a Volumetric Coefficient Equal to 1 . 5 ~ 1 0 - 3 . . . . . . . . . . . . . . . . . . . . . . . . 38

Comparison of the Effective Horsepower of a 650-Foot Passenger Vesse l with an EquivLlent Standard Ser ies V e s s e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Geometr ical P a r a m e t e r s for Taylor S e r i e s Comparisons . . 4 3

Figure 24

Figure 25 Effect of Sectional-Area Shape on the Selection of

F igure 26 Residual-Resistance Coefficient Correct ions . . . . . . . . . 44

xiii

LIST OF TABLES

PAGE

Table 1 - Ordinates of the Sectional-Area Curves for the Taylor Standard Series Expressed as Ratios to the Maximum A r e a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Sectional-Area Curves and Waterlines. . . . . . . . . . . . . 7

with a Nominal Beam-Draft Ratio of 2.25 (Series 22) . . . . . 18



f rom the Taylor Standard S e r i e s . . . . . . . . . . . . . . . . 39

Table 2 - Functions for Calculation of Mathematically Defined

Table 3 - Dimensions and Coefficients for Taylor Ser ies Models



Table 6 - Sample F o r m for the Calculation of Effective Horsepower

Table 7 - Part iculars for a 650-Foot Passenger Vesse l . . . . . . . . . 40

289733 0 - 54 - 2 xiv

PREFACE

The reanalysis of the original tes t data for the Taylor Standard Series was accomplished a t the David Taylor Model Basin during the period of 1948 to 1951. The work w a s originally prompted by the decision of the American Towing Tank Conference of 1947 to adopt the Schoenherr frictional-resistance formulation for use in predicting the effective horsepower of ships f rom model resistance tes t data. This decision, i t w a s realized, would resul t in effective horsepowers that were not directly comparable with those calculated f rom the original Taylor Standard Series contours using the procedures which a r e given in Taylor’s “Speed and Power of Ships.”

The differences in the calculated effective horsepowers result f rom two causes: the differences between the frictional res is tances obtained from the Schoenherr formula and those from the old Experimental Model Basin 20-foot plank data in the model range and the differences betwe en the c or r e sponding frictional res is tance s ob - tained from the Schoenherr formula and from the Tideman constants in the ship range. The former is reflected as a difference in residual resisi tance and thus would r e - quire that a lengthy correction be made to the Taylor residual-resistance per ton contours to make them comparable to modern data. The la t ter merely requires a substitution of the Schoenherr formula with the appropriate roughness allowance for the Tideman constants in the ship calculation procedure.

When the original Taylor Series contours were prepared, no effort w a s made to compensate for changes in resistance caused by the normal change in towing basin water temperature over the year , to insure that flow about the models w a s fully turbulent, o r to correct for possible res t r ic ted channel effects. In view of these considerations, i t w a s believed that a revision of the resul ts w a s warranted and could be accomplished only by reanalyzing the original tes t data and not by directly converting the existing fai ied residual resistance pe r ton contours.

In the reanalysis, the methods and procedures used were essentially the same as those currently used a t the Taylor Model Basin. A total-resistance coefficient for the model w a s computed, f rom which a Schoenherr frictional-resistance coefficient w a s deducted to give a residual-resistance coefficient which in turn formed the basis for expansion from model to full scale.

The aforementioned corrections were made to the Taylor Ser ies data as follows:

The Schoenherr frictional-resistance coefficient is a function of Reynolds number, so that compensation for differences in basin water temperature can be made by using the apprdpriate kinematic viscosity in the computation of Reynolds number. For the majority of the Taylor model tes ts , the basin water temperature w a s not recorded. A chart was therefore used of the day-by-day averages of the water temperature a t the U. S. Experi- mental Model Basin during the yea r s 1913 to 1918. In view of the variations shown by these records, the selected temperatures a r e , in most cases , believed to be accurate to within f 1 F of the temperatures actually prevailing at the t imes of model tes ts . A range of temperature f rom 53 to 80F w a s experienced during a given year of testing, representing a change in frictional res is tance of about 7 percent for a 20-foot model.

The method for correcting for the effects of t ran- sitional flow is based on the assumption that at low Froude numbers the residual-resistance coefficient, a s defined, is a constant. The original data showed that in general the residual-resistance coefficient decreased with decreasing speed so long as wavemaking resis tance w a s important. There was then a short range of speed for which the residual-resistance coefficient remained constant, after which, as the speed w a s still further reduced, the coefficient began to decrease again. This la t ter de- c rease has been attributed to transitional flow and has been ignored, the constant value of the coefficient being

xv

used for all lower Froude numbers. Although this procedure is not rigorous, a number of recent tes t s of 20-foot models which were towed with and without a turbulence- stimulating device indicate that in general such conditions obtain for models which experience only minor t ransi- tional effects at the lowest speeds. Good agreement between the residual-resistance coefficient curves f rom the experiments without turbulence stimulation, faired in the way described above, and those resulting f rom the tes t s with the turbulence device w a s attained in most of these cases. This seems to be especially t rue with forms having the Taylor Ser ies type of bow. Two new 20-foot Taylor Series models, having longitudinal pr ismatic coefficients of 0.613 and 0.746, were tested at the Taylor Model Basin in 1951. In both cases i t was found that turbulent stimulation w a s required only a t low speeds and that the aforementioned procedure gave reasonable agreement with the turbulent curve. The resu l t s of these tests a r e given in the 1951 Transactions for the Society of Naval Architects and Marine Engineers in a paper entitled, “A Proposed New Basis for the Design of Single Screw Merchant Ship Forms and Standard Series Lines,’’ by Dr. F. H. Todd and Captain F. X. Forest .

Corrections for res t r ic ted channel effects were made by using the formulas given in TMB Report 460 entitled,

“Tes ts of a Model in Restricted Channels,” by L. Land- weber, May 1939, with the appropriate model dimensions and the dimensions of the c r o s s section of the U. S. Experimental Model Basin. This correction was small in most cases , and even for the fullest model of the ser ies , it amounted to a decrease in res is tance of only 2 percent.

The resu l t s of the reanalysis of the Taylor Standard Series data a r e given in a form which employs a completely nondimensional representation. The faired resistance data a r e given as curves of residual-resistance coefficient versus Froude number. The major geometri- ca l parameters used a r e beam-draft ra t io , longitudinal pr ismatic coefficient, volumetric coefficient, and wetted- surface coefficient, the latter two being redefinitions of Taylor’s displacement-length rat io and wetted-surface coefficient.

The scope of the se r i e s has been enlarged to include a third beam-draft ratio of 3.00 in addition to the beam- draft ra t ios of 2.25 and 3.75 published in Taylor’s “Speed and Power of Ships.” These values were obtained by interpolation, using the reworked data for the hitherto unpublished Ser ies 20 which had a beam-draft ratio gf 2.92.

xvi

A Reanalysis of the Original Test Data for the

TAYLOR STANDARD SERIES

TABLE OF CONTENTS

v ............................................................................... iv

List of Appendices ........................................................................................................... vii Series Preface ...................................................................................................................... v

Acknowledgments .......................................................................................................... vm Notation ............................................................................................................................. ix Formulas ............................................................................................................................. x List of Illustrations ..................................................................................................... xi-xiii List of Tables .................................................................................................................... xiv Preface ............................................................................................................................... xv

...

History of the Taylor Standard Series ............................................................................. 1 Geometry of the Taylor Standard Series ......................................................................... 2

Configuration of Derived Forms .............................................................................. 13

Reduction of the Original Test Data .............................................................................. 17 Temperature Corrections ........................................................................................... 21

Characteristics of the Parent ........................................................................................ 4 Derivation of Series Forms from Parent .................................................................... 4

Characteristics of Actual Forms Tested .................................................................... 17

Transitional Flow Corrections .................................................................................. 21 Restricted Channel Corrections ................................................................................ 24

Cross-Fairing of Resistance Data ................................................................................... 27 Final Presentation of Data .............................................................................................. 30 Calculation of Effective Horsepower Using Revised Contours ............................... 37 Validity of Taylor Series Comparisons .......................................................................... 42 Use of the Revised Taylor Series Contours with Frictional-Resistance

Formulations Other Than Schoenherr ..................................................................... 44 References ......................................................................................................................... 45

iii

HISTORY OF THE TAYLOR STANDARD SERIES

The Taylor Standard Series was the first of the methodi- ca l s e r i e s of ship forms to receive wide attention and usage throughout the United States. The ser ies , as it is known today, was the result of an evolution of several parent forms before the final parent, which was used to develop the ser ies , w a s chosen,* The original parent w a s patterned after the Bri t ish armored cru iser LEVIATHAN of the Drake c lass (1900), a model of which w a s tested in the U. S. Experimental Model Basin a t Washington in 1902. The salient features of the LEVIATHAN consisted of a bulbous ram bow extended on a raised forefoot and a twin-screw cru iser type of stern. These features were retained in the f i r s t parent which w a s designed in 1906 using the sectional-area curve, load waterline, and bow and s te rn profiles of the LEVIATHAN together with mathematically derived body lines. The parent form so obtained w a s used to construct EMB Model 632, and, together with mathematically derived sectional-area curves, was used to develop the lines for the construction of 38 additional models. These models, designated Series 18, were tested to investigate the effect of changes in longitudinal pr ismatic coefficient and displacement-length rat io on the resistance of ship forms. The resulting data were prepared in reproducible f o r m a s contours of residual resistance pe r ton, for constant values of speed-length ratio, on gr ids of displacement-length ratio and longitudinal pr ismatic coefficient, but were not published.

The f i r s t parent w a s a l tered during 1906 by eliminating the bulbous ram bow. The resulting parent was used with the sectional-area curves of Ser ies 18 to develop and construct 25 additional models. These models were tested to determine the effect on resistance of the variation of the same form parameters as those of Series 18. The resul ts were similarly prepared for reproduction but were not published.

The final alteration to the parent consisted of dropping the forefoot to the baseline, adopting a 3 percent bulb, and moving the maximum section to midlength. The resulting parent became the basis of Ser ies 20, 21, and 22. These ser ies were developed using the sectional-area curves of Series 18 which were extended by the addition of mathematically derived sectional-area curves for longitudinal pr ismatic coefficients f rom 0.68 to 0.80 and by extrapola- tion to 0.86.

*References are listed on page 45.

Series 20 consisted of 38 models each having a constant beam-draft ra t io of 2.92 but with systematically varied values of longitudinal pr ismatic coefficient and displacement-length ratio. Ser ies 20 was tested during 1906 and 1907, and the resul ts of these tes t s appeared only briefly in publication as Figure 70 in Taylor's "Speed and Power of Ships" to i l lustrate the variation of residual resistance with midship section area.

Series 21 and 22, having beam-draft ra t ios of 3.75 and 2.25, respectively, were then formulated. These two se r i e s consisted of a total of 80 models, which were tested in 1907 and 1908. The resul ts of these model tes t s were published in the first edition of Taylor's "Speedand Power of Ships" in 1910. The scope of Ser ies 21 and 22 was extended during 1913 and 1914 by tests of 40 additional models. The augmented resu l t s of both se r i e s were published first in the 1933 edition andthenin the 1943 revised edition of Taylor's "Speed and Power of Ships." The data appeared as contours of residual resistance per tonplotted against longitudinal pr ismatic coefficient and displacement- length ratio, a presentation which is now familiar to the profession as the Taylor Standard Series contours.

The contours of residual resistance pe r ton which a r e found in the 1943 edition a r e based upon concepts which existed in 1910. No effort had been made, up to now, to a l te r the contours in accordance with the changes in analytical methods throughout the years . Thus, as in 1910, the data for the Experimental Model Basin 20-foot friction plane were used to reduce the model data to residual resistance , and the Tideman frictional - r e si stance constants were used in the prediction of the effective horsepower of the full-scale vessels .

In 1923, the U. S. Experimental Model Basin began to use the Gebers frictional-resistance formulation t o com- pute the frictional res is tance in both the model and full- scale ranges, and this practice was continuedat the Taylor Model Basin through 1947. During this period the Taylor Ser ies Contours continued t o be widely used. The validity of the Taylor Series comparisons was not altered, however, because it was the practice t o use an empirical factor, denoted G, with the Gebers formula to a r r ive a t effective horsepower values equal to those predicted by the EMB- Tideman method. Since 1947, in accordance with a decision

1

made by the American Towing Tank Conference, the Schoenherr frictional-resistance formulation has been in general use by all American towing tanks to predict the effective horsepower of ships f rom model tes t data. 2

The present use of the Schoenherr formula does not involve a rb i t ra ry factors solely for the purpose of seeking agreement with predictions obtained with past methods. However, since the basic Schoenherr coefficients apply to a hydraulically smooth surface, a roughness-allowance coefficient is normally added to allow for the deviation of the actual ship's hull surface f rom hydraulic smoothness. The roughness-allowance coefficient is generally based upon information obtained from correlation of ship t r ia l data with model tes t data and may vary with ship type, with bottom paints, and with construction details. The value of 0.0004 adopted for this roughness -allowance coefficient by the American Towing Tank Conference in 19472 appears to be a suitable figure for average merchant ships, a s borne out by recent merchant ship t r ia l s . The value of 0.0004 added to the Schoenherr coefficients coincidentally gives good agreement with the Froude coefficients for the range of ship lengths and speeds of average medium speed cargo vessels .

As the result of the introduction of the roughness allowance concept, the effective horsepower s calculated with the Schoenherr formula should differ f rom the values calculated by the EMB-Tideman method to a n extent which would depend upon the assigned value of roughness-allowance coefficient. If the Schoenherr formula plus the appropriate roughness-allowance coefficient were substituted for the Tideman values in the calculation procedure, the original Taylor Series contours could sti l l be used for comparative purposes i f there were no inherent e r r o r s in the contours themselves. This is on the assumption that the frictional- resistance coefficients versus Reynolds numbers obtained from the EMB plank resul ts and f rom the Schoenherr formula a r e equal so that the residual res is tances obtained with either would a l so be equal. The agreement is actually within 2 percent when they a r e compared upon the basis of a water temperature of 70F. The EMB plank w a s towed a t a temperature of 70F and the values so obtained were used, without adjustment, for the calculation of the residual res is tances of all of the se r i e s models regardless of the temperature of the water in which they were tested. This procedure introduced e r r o r s in the

Taylor contours which have not been corrected since their original presentation.

The present reanalysis of the original data for the Taylor Standard Series was started in 1948 for the purpose of correcting these e r r o r s . P r i o r to that t ime, consideration was given to the advisability of salvaging the original faired contours either by using them directly with the Schoenherr formula in the ship range or by the more complicated device of correcting the contours to account for the differences in residual res is tances that would resul t i f the Schoenherr formula were used instead of the EMB plank values. This approach w a s rejected because of the inherent e r r o r s in the contours which, a s mentioned previously, were due to failure to correct €or temperature, the probability that laminar flow existed in some cases since no effort was made to stimulate turbulence, and the restr ic ted channel effects as wi l l be subsequently, discussed. Since the original data were being reworked, the data for the hitherto unpublished Series 20 were included in the reanalysis. This w a s done to provide an intermediate value of beam-draft ra t io when i t appeared that Taylor 's assumption (that residual res is tance var ies linearly with beam-draft ratio) introduced further e r r o r in the interpolation between the values of beam-draft ra t io of 2.25 and 3.75.

GEOMETRY OF THE TAYLOR STANDARD SERIES

The geomety of any methodical se r ies should be completely and accurately defined. This prerequisite i s necessary when the resistance data a r e expressed a s functions of prescribed geometrical parameters since the resulting relationships apply, in the s t r ic tes t sense, only tothe particular parent which has been varied according to the specified procedure. It i s assumed, however, that within reasonable l imits , s imilar trends can be expected for offspring of other parent forms which have been derived by the same process. This assumptiondetermines the validity of the accepted use of the Taylor Standard Series a s a cri terion for the performance of specific shipdesigns. Ex- perience has shown that such a procedure is valid if the departures f rom the parent of the form being investigated a r e not too great.

The process of developing a ser ies covering a wide range of geometrical parameters f rom a single parent form will

2

FIGURE 1.--Lines and offsets for the parent form of the Taylor Standard Series This is the final parent form which was used to derive the forms of U S . Experimental Model Basin Series 20, 21, and 22. It incorporates the following changes to the original parent represented by EMB Model 632: The bulbous ram was eliminated, the forefoot was dropped to the baseline, a 3 percent bulh was adopted, and the maximum section was moved to midlength.

3

Stations 0.2

.048

.057

.074

.lo1

.132

.168

.209 ,308 .428 .561 .706 .a21 A83 .888 .839 .741 .602 .446 .285 .153 .067 .038

:.P.(Ext. 1 2 3 4 5 6 8

10 12 14 16 18 20 22 24 26 28 30 32 34 35 36 37 38 39 39 H A.P.

0.3

0.052 .063 .087 .122 .161 .207 .259 .379 .522 .663 .794 .885 .931 .937 .905 .833 .716 .568 .398 .237 .116 .068

0.036

__

0.1

1.036 .044 .054 .069

.114

.142 .205 .289 .399 .549 .691 .777 .779 .705 .568 .421 .274 .159 .079

1.037

-

.on9

-

1.4

1.012 .063 .141 .226 .314 .403 .488 .641 .764 .a53 .923 .966 1.988 1.000 1.991 .973 .937 .804 .a18 .734 .632 .567 .495 .416 .328 .222 .145 1.049

1.6

0.011 .084 .176 .267 .363 .449 .529 .665 .775 .858 .923 .966

0.988 1.000 0.991 .975 .944 .899 .836 .766 .662 .598 .524 .442 .353 .239 .161

0.056

-

0.4

.052

.067 .097 .138 .184 .237 .295 .434 .584 .731 .846 .921 .957 .964 .942 .889 .789 .658 ,495 .324 .175 .114 .063 1.032

-

.065 .112 ,167 .231 .298 .373 .528 .686 A17 .910 .962 .985 .996 .986 .958 .895 .799 ,677 .533 ,378 .299 .222 .143 .068

-

0.5

1.049 .067 .lo4 .151 2 0 3 ,262 .325 ,475 .628 .774 .a79 .942 .972 .981 .967 .925 .841 .725 .574 .405 .244 .172 . lo6 .053

1.026

-

-

.062

.114 ,174 .241 .314 .309 .551 .704 .830 .915 .964 .988 .998 .989 .964 .907 .a22 .709 .581 .437 .362 .284 .205 .120

0.034

-

0.6

.043 .066 .lo7 .159 .217 .282 .349 .506 .661 .a02 .a98 .953 .978 .989 .978 .947 .a74 .771 .636 .479 .316 .237 .162 .092 1.037

-

-

0.9

.029 .O 59 .117 .183 .252 .327 .407 .570 .718 .836 .918 .964 .988 .999 .990 .967 .916 ,839 .734 A16 ,487 .417 .342 .263 .178 ,082 .026

-

-

-WL 1 .o

1.023 .057 .119 .188 .262 .339 .422 .5R5 .729 .a41 .918 .964 .099 1.000 t.990 .967 .921 .a51 .755 .640 ,527 .462 .389 .314 .225 .123 1.068

-

-

1.2

1.015 .056 .127 .205 .285 .369 .455 .615 .749 A46 .921 .965

3.988 1.000 1.991 .970 .929 .867 .790 .699 .587 .524 .453 .375 .289 .184 .120

D.036

I 1.8

.009

.148

.263 ,356 .439 ,516 .583 .695 .789 .864 .925 .967 .988 .ooo .99 1 .978 .951 .913 .858 .782 .682 .619 .543 .458 .359 .246 .168 '.054

often resu l t in some f o r m s which a r e not useful for prac- t ical ship designs. Nevertheless , i t is des i rab le to know what these f o r m s a r e so that final evaluations can be made. Unfortunately, the offsets of the offspring forms of the Taylor Ser ies have not been published, and even the procedure for deriving these f o r m s has not been given in much detail.lP3 Consequently, an attempt is made h e r e t o explain the derivation procedures a s well as toprovide data where- by the individual resu l t ingforms of the s e r i e s can be easi ly reproduced.

CHARACTERISTICS OF THE PARENT

The his tor ical statement mentions that the final parent f o r m for the Taylor Ser ies was the resu l t of s e v e r a l suc- cessive modifications of the basic f o r m of the Br i t i sh c r u i s e r LEVIATHAN. The f o r m designated as the parent in Reference 1 incorporates all of these modifications except for the shift of the maximum section f r o m Station 19.2 to midships. It is not possible, therefore , to develop the off- spr ing f o r m s of the Taylor s e r i e s d i r e c t l y f r o m t h e offsets and charac te r i s t ics given therein. To fully es tabl ish the parent which was actually used t o develop the s e r i e s , the offsets were remeasured f r o m the original l ines drawings of the final parent. The l ines and nondimensional offsets and charac te r i s t ics der ived f r o m these measurements are shown by Figure 1.

The charac te r i s t ics of the final parent f o r m a r e similar in many respec ts to those of some of the modern higher speed ship types. The midship section is roughly rectangular except for a small deadr i se and a relatively l a r g e bilge radius. The forward sect ions, with the exception of the three percent bulb, a r e generally U-shaped whereas the ex t reme af ter sections are inclined to be somewhat V-shaped. The keel i s flat for the m a j o r par t of the length but r i s e s at the extreme s t e r n t o f o r m a centerline skeg whichis designed t o receive a single hinged type of rudder . It m a y be noted that the bow pr i smat ic coefficient is 0.574 whereas the s t e r n pr i smat ic coefficient is 0.532. Thus the longitudinal center of buoyancy is forward of midships. This charac- te r i s t ic was modified in the development of the s e r i e s proper by the selection of sect ional-area curves f o r the offspring which had equal forebody and afterbody pr i smat ic coefficients.

4

DERIVATION OF SERIES FORMS FROM PARENT

Most methodical s e r i e s make use of the procedure of effecting var ia t ions in f o r m p a r a m e t e r s one a t a t ime while other significant p a r a m e t e r s a r e kept constant. This ob- jective is usually achieved by resort ing t o s o m e sys tem for relating the geometr ical p a r a m e t e r s which a r e being inves - tigated. Various methods, both graphical and mathematical , have been ei ther used o r consideredfor this purpose. 4 ~ 5 It i s important t o know which sys tem was used to der ive a given series in addition t o the geometr ical p a r a m e t e r s for a. full understanding of the nature of the offspring which r e - sul t f r o m a given parent.

The method used t o der ive the Taylor S tandardser ies is esqentially a graphical process . The p a r a m e t e r s which were var ied a r e the longitudinal pr i smat ic coefficient C, , the beam-draft ra t io B/H, and the displacement-length ratio. The midship sect ion coefficient C, was held constant and the longitudinal cen ter of buoyancy was fixed at midships . The change in C is the only one of the three p a r a - m e t e r s which involve ccanges in the nondimensionally defined body sections whose offsets are the half breadths expressed as ra t ios t o the half maximumbeamand heights expressed as ra t ios t o the load waterline draf t .

The procedure f o r accomplishing changes in C, is i l lus- t ra ted by the sketch of F igure 2 . Curve A is the sectional- a r e a curve of the parent f o r m and curve B is the sectional- a r e a curve of the d e s i r e d offspring. Point e i s the point of intersect ion of any integrally numbered station ab with curve B. Point f is the horizontal projection of point e on curve A. Station cd , which is drawnthroughf , is the station of the parent f o r m which has the same body sect ion as that

FIGURE 2.--Sketch showing relocation of sections of a parent form to produce a derived form having a different longitudinal prismatic coefficient

of the der ived f o r m a t station ab. Thus, having the s e c - t ional-area and the var ious waterline c u r v e s of the parent f o r m together with the sect ional-area curve for the proposed offspring, the offsets for the la t ter can be completely de t e rmined.

Although this graphical process is relatively s imple , i t s execution for a la rge number of f o r m s becomes quite laborious. Unfortunately, a s e a r c h of the original f i les failed to produce the offsets for the individual f o r m s compris ing the Taylor Ser ies . Consequently, i t was considered advis- able t o redevelop the s e r i e s in the prescr ibed manner to reproduce these offsets in a f o r m enabling the i r d i rec t use. The result ing information is given in Appendix 1 as contours of the nondimensional half breadths plotted against longitudinal pr ismatic coefficient for each of a number of selected stations between and including the forward and af ter perpendiculars. Separate s e t s of contours a r e given for each waterline height in suitable increments to define completely the f o r m s covered by the s e r i e s . The contours for each waterline height a r e given separately for the bow and s t e r n s o that they can be used in combination t o produce changes in position of longitudinal center of buoyancy through the use of unequal bow and s t e r n pr i smat ic coefficients.

It i s apparent that the success of a methodical s e r i e s which is developed according t o the aforementioned p r o c e s s depends largely upon the sys tem used to vary the sectional- a r e a curves . Obviously, i f no at tempt is made to sys tema- t ize , l a rge distortions in f o r m would resu l t and any r e - semblance that the offspring might b e a r to the parent would be destroyed. As stated previously, the sect ional-area curves for the Taylor Ser ies were mathematically defined. Consequently, they were var ied systematically through a proper choice of the p a r a m e t e r s for the governing equations. The values of these p a r a m e t e r s were never published and a search of the original files produced values for only a l imited number of curves . The original delineations and nondimensional ordinates and a b s c i s s a s for all of the curves were available, and these a r e g iveninFigure 3 and Table 1, respectively. It is believed, however, that the mathematical approach is inherently m o r e prec ise and, in addition, provides a n excellent tool which can be used t o produce the a r e a curves for intermediate values of longitudinal pr i smat ic coefficient. Therefore , Taylor 's mathe -

J

1 A U L E 1.--Ordinates of the sectional-area curves for the Taylor Standard Series expressed as ratios to the maximum Area

T h e s e v a l u e s h a v e been reread from the original s ec t iona l -area curves.

matical approach is redeveloped here in , and the resul t ing equations have been applied t o obtain suitable fits to the sect ional-area c u r v e s defined by Figure 3 and Table 1. The notation originally used by Taylor in h is equations has been changed to avoid conflict with present day nomenclature and a n additional t e r m h a s been provided to take c a r e of special c a s e s .

5

Station Number

F.P. 0.2 0.4 0.6 1 2 4 6 8

10 12 14 16 18 20 22 24 26 28 30 32 34 36 37 38 39 A.P.

0.48

0.000 .022 .032 .038 .046 .066 .126 .211 .323 .450 .586 .724 .853

0.955 1.000 0.964 .870 .745 .604 .460 .322 .200 .lo3 .065 .034 .012 0.000

0.52

1.0000 .025 .035 .042 .050 .078 .152 .257 .382 .517 .655 .784 .893

1.971 1.000 1.974 .903 .797 .669 .528 .385 .251 .136 .090 .050 .018 1.000

Longitudinal Prismatic Coefficients

0.56

0.000 .026 .038 .045 .058 .092 .la4 .307 .446 .587 .722 .838 .929

0.983 1.000 0.984 .932 .846 .731 .596 .450 .304 .173 .115 .066 .027

0.000

- 0.60

D.000 .029 .042 .048 .065 .lo7 .221 .364 .514 .658 .786 .886 .955

0.992 1.000 0.991 .957 .891 .793 .667 .522 .368 .217 .149 .087 .037

1.000

0.64

0.000 .030 .045 .055 .074 .128 .264 .426 .587 .731 .846 .928 .976

3.997 1.000 1.996 .976 .928 .848 .737 .595 .433 .265 .185 .112 .048 1.000

- 0.68

0.000 .035 .050 .064 .087 .152 .316 .499 .668 .803 .go1 .961 .990

0.999 1.000 0.999

.988

.958 898 .804 .671 .506 .322 .231 .142 .060

0.000 -

0.74

0.000 .45 .064 .080 .110 .206 .423 .625 .784 .893 .957 .987

0.998 1.000 1.000 1.000 0.998

.987 .957 .895 .789 .631 .428 .315 .201 .092

0.000 -

0.80

0.000 .055 .083 . lo8 .159 .298 .564 .766 .892 .959 .988

0.998 1.000 1.000 1.000 1.000 1.000 0.998 ,992 .963 .898 .773 .567 .436 .290 .140 3.000 -

- 0.86

0.000 .074 .112 .153 .229 .423 .751 .913 .977 .995

0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999

3 8 7 .920 .745 .599 .414 .208

0.000 -

A.P. 38 36 34 32 30 28 26 24 22 2 0 18 16 14 12 10 8 6 4 2 F.P.

Stations

FIGURE 3.-Sectional-area curves for the derived forms of the Taylor Standard Series

Taylor's mathematical approach defines either the bow half or the s tern half of the sectional-area curve by the equation of a fifth degree parabola

The values of the c o e f f i c i e n t s ~ ~ a 2 , - - - - a r e determined in t e rms of prescribed geometrical parameters by imposing the end conditions and solving either by simultaneous equations or the determinant rule. The same results can be achieved more directly by the method of Reference 5. Thus Equation C1'1 can be restated in t e r m s of four basic polynomials of the fifth degree in x which involve the selected geometrical parameters as follows:

where

A y is the nondimensional ordinate z , a n d i s unity for the

x is the nondimensional abscissa & and is unity a t the maximum section when x is measured f rom the extremity of either the bow or s tern,

is the longitudinal prismatic coefficient for either the bow o r s tern,

section of maximum immersed a rea ,

c,

t is the tangent to the sectional-area curve a t either the forward o r after extremity, and

n is the second derivative of the sectional-area curve a t x = 1 and y = 1.

In Equation [2], W X ) , P ( x ) , T(x) , and , N ( x ) a r e independ- ent of the parameters C , , t , and n . Thus, general working tables can be prepared for a number of values of x for the bow or s te rn halves of the sectional-area curves. Since y

6

1.00

0.90 g 0.80 2 0.70 E 0.60 -g

3

0.50 :: I

0.40

0.30 c) c

- .- 0.20 $ 0.10 a 0

38 36 34 32 30 28 26 24 22 2 0 18 16 14 12 10 8 6 4 2 F.P.

Y C ( X )

1 . 0 0 0 . 0 0 0 0 0 0 .OE - .011217 .04 -. 0 4 1 8 6 5 .05 - .on7733

@ 8 -. 1 4 4 9 8 3 e l 0 -. 2 1 0140

. I 2 - . 280077 -14 -. 2520 0 1 - 1 6 - .423430 . I 8 - .402222 .73 - . ~ 5 6 4 n o

.22 - a 6 1 4 6 1 6 -24 -. E 6 5 2 9 9 - 2 6 7 0 7 4 5 2 .?8 e . 7 4 0 2 3 1 .30 -. 763020

P ( X I

9 . 0 0 0 3 0 3 e l l 2 2 5 8 9 .O 8 4 4 3 5 e l 7 9 4 0 6

2 9 9 0 1 6 - 4 3 7 4 0 0

.5 88 79 2 ~ 7 4 8 0 0 2 , 9 1 0 3 9 3

1.071659 1.22880 0

1.37809 9 1.51710 1 1.6 4350 9 1 .755759 1.85220 0

1.931 87 0 1.99407 2 2.0 3 8 4 3 2 2.064874 2.07360 0

2 .065065 2 0 3995 5 1 .999163 1.943765 1 87500 3

0 . 0 0 0 ~ 0 0 0 1 7 €94

.O 3 1 143

.040865

.o 4 7 2 4 4 a051030

0 5 2 337 .O 5 1 € 4 8 .049313 ,045653 e040960

,035496 - 0 2 9 4 9 9 -0 2 3 1 7 9 -0 1 6 7 2 2 .010290

s 0 0 4 025 -. 0 0 1 0 5 5 - .O 0 7 5 5 0 e.012679 -.017280

-. 0 7 1 306 - a 0 2 4 7 2 7 - 0 2 7 525 - a 0 2 9 € 9 6 -.031250

C ( X )

0 .a00 0 0 0 - . O O O 1 8 4 - . C O O 6 7 8 -.001400 - 002 2 7 5 - .003240

- 004 238 e.005 2 1 9 - - 0 0 6 1 4 2 - -006 971 -. 00 7 6 8 0

- 008 2 4 5 - .008 650

-. 008 941 - . O O B 8 2 0

-.ooaea(r

- - 0 0 8 5 2 3 - .008057 - .007432 - 006 6 6 1 - 0 0 0 5 7 6 0

- 0 0 0 4 7 4 7 - .003643 -.00246R - .00124€ -. 000 0 0 0

F ( X )

1.00000c - 9 8 8 6 2 8 .956¶30 - 9 0 8 327 e845967 .772740

m69l.285 .60 3 9 9 9 e513045 .4202€3 - 3 2 7 6 8 0

- 2 3 6 5 1 7 .148198 - 0 6 3 8 6 3

- a 0 1 5 5 2 7 - . 089180

-. 1 5 6 4 € 1 -. 2 i 6 a 8 7 -. 270 11 3 -.325928 - - 3 5 4 240

- 0 3 8 5 0 7 3 - .408557 -. 424901 -. 4 3 4 4 2 2 - 43750 0

- 7 2 - 5 2 0 0 5 5 .74 e 5 9 6 9 1 9 a 7 6 - 6 6 7 2 0 7 - 7 8 e73l .219 - 8 0 o 7 8 8 4 8 0

0 8 2 . 8 3 8 € 5 0 .84 . 8 8 1 9 3 7 .a6 - 9 1 7 1 1 6 . 8 8 . 9 4 5 5 3 6 - 9 0 e 9 6 7 1 4 0

a92 0 9 8 2 4 7 4 094 - 9 9 2 3 0 4 - 9 6 - 9 9 7 6 2 8 0 ? 8 a 9 9 9 6 9 2

1.00 1 .000000

1 .794245 1 . 7 0 2 ¶ ¶ 1 1.602822 1.4¶5Z¶2 1.382400

1.265568 1 . 1 4 € € 1 8 1.027249

. ? 0 ? 1 1 5 - 7 9 3 8 0 0

€ 8 2 7 9 5

.474085 3 8 8 6 9 5 30720 0

. 577477

. 2 3 5 2 8 € ,17340 8

1 2 1 7 6 8 e 0 8 0 2 9 0

0 4 6 6 0 0

0 2 6 0 0 1 - 0 1 1 4 5 1 ~ 0 0 3 5 3 9 . COO46 1 -. o o o o o o

T ( X I

, 001246 eOO2468 eOO3643 .a04747 a005760

, 0 0 6 6 6 1 .007432 e 0 0 8 0 5 7

00 8 5 2 3 eOO8820

. 0 0 8 9 4 1 ~ 0 0 8 8 8 4

0 0 8 6 5 0 0 0 8 2 4 5

- 0 0 7 6 8 0

0 0 6 9 7 1 0 0 0 6 1 4 2 ,005219 - 0 0 4 2 3 8 - 0 0 3 2 4 0

0 0 0 2 2 7 5 . 0 0 1 4 0 0 a 0 0 0 6 7 8 . 0 0 0 1 8 4

0 0 0 0 0 0 0 0

- 0 4 3 4 5 8 2 -. 4 2 6 1 7 6 - .412836 -. 3 9 5 1 5 6 -.373760

- 0 3 4 9 2 9 3 -. 3 2 2 4 1 2 - .2937 74 - . 264031 - .233820

- .203750 - 0 1 7 4 3 9 6 -. 1 4 6 2 9 1 -. 1 1 9 9 1 4 - .095660

- 0 0 7 3 9 3 6 -. 0 5 4 9 4 5 -oO38814 - 0 0 2 5 8 2 6 - 0 0 1 5 7 4 0

-oOO8475 - 0 0 0 3 7 5 5 - 0 0 4 1 1 6 7 -. 0 0 0 1 5 3 - 0 0 0 0 0 0 0

Table 2 - Functions for calculat ion of mathematically defined sectional-area curves and waterlines

The nondimensional abscissas x are given a s r a t i o s to the half length

(corrected, July 1972)

7

i s a l inear function of Cp t and n , the offsets of a particular sectional-area curve can be directly determined for any value of x by adding algebraically the products shown in Equation [ 2 3 .

It may be noted that Equation 1 2 1 i s not satisfiedwhen y + o a t x = o as is the case where a projected bulb or t r an - som s te rn is shownby the sectional-area curve. When these conditions exist, it is necessarytoredefine C, and y touse Equation [Z], as follows:

A --I

c31

and c41

where f i s the ratio of the bulb o r t ransom a r e a to the maximum a r e a.

This redefinition i s objectionable because it introduces numerical values for the C, and the a r e a ordinates which a r e not those desired in the end result and which may lead to confusion. This obstacle can be suitably overcome by the introduction of the t e r m fF(x) t o Equation [ 2 ]

Second Derivative n at 11x1 50COna urnyorive n or x = I

FIGURE b-Curves of geometrical parameters used to define mathematically the sectionalarea curves for the Taylor Standard Series

The parameter j is constant and equal to 0.03 for the bow and 0.00 for the stern

8

3.0 0 00

0.75

0"

$0.70 E c 0 s

QWI

0.SOj

-1.5 I

Tongent t ot x = 0 2.5 2.0 1.5 1.0 0.5

-1.0 -0.5 Socond Derivotive not x = i

Tonaent t ot x:O . - 0 2.0 1.5 1.0 0.5 0

am

0 -1.0 -a5 Second Derivative n at x =I

0.75

0 0

L)

0 - c

065 h

0.55

0.50

0

FIGURE 4.--Curves of geometrical parameters used to define mathematically the sectional-area curves for the Taylor Standard Series

The parameter f is constant and equal to 0.03 for the bow and 0.00 for the stern

8

Thus Equation [ZJ evolves to the more general expression

where the t e r m fF(x) drops out when the conditions for a bulb or transom do not exist.

Equation [5] can be directly used to define waterline curves by substituting the waterplane coefficient for C, and the half siding for f . Values of g x ) , P(x) , T ( x ) , N(x) , and F(x) a r e l isted in Table 2 for suitable incremental values of x to a s s i s t in the calculation of sectional-area o r waterline curves for prescribed values of the appropriate geometr ical parameters. The sectional-area curves for the Taylor Ser ies have been fitted with Equation [ 51, and the resulting values of t and n a r e plottedagainst Cp inFigure 4. The value of f is constant and equal to 0.03 for the bow and 0.00 for the stern. Thus, with these values and Equa- tion [ 51, any sectional-area curve belonging to the family for Taylor 's Ser ies can be easily reproduced.

The variation of C , for the Taylor Ser ies i s thus accomplished by the preceeding method involving the sectional- a r ea curves. The variation of the other defining parameters , namely the displacement -length ratio and beam-draft ratio is obtained simply by selecting the appropriate over-all proportions of beam to draft to length. This can be accomplished nondimensionally as follows:

2

($)= H

-bL The volumetric coefficient cp= - c 7 1

L 3 where 4 is the immersed volume and L i s the loadwater- line length, is used as a defining parameter in place of Taylor's familiar displacement-length ratio. This change was considered desirable since the displacement-length ratio is not nondimensional. Furthermore, since it has the dimension of density, i t s definition depends upon specified units and standards, such as the requirement that the displacement be given in tons calculated for a given draf t in salt water. However, the numerical values of displacement - length rat io have often been associated by members of the profession with certain types of vessels. To retainthis use,

9

the relationship between Cv and displacement -length ratio is shown by Figure 5.

The relationships of Equation [ 6 ' ] a r e shown in Figures 6 , 7 , and 8 for the range of parameters covered by the ser ies . Obviously, these curves apply to any form which has been derived from a parent having a midship section coefficient of 0.925, the value used for the Taylor Ser ies parent.

FIGURE S.--Comparison of the Taylor displacement-length ratio and wetted- surface coefficients with the redefined coefficients

Dbplacrm.nt-Langth !MI0

x) 20 4 0 60 80 100 420 I40 I60 180 200 220 240 260 280 410-3 a 0.0 2 84

2.82

ao 2.80

2.78

7.0. 2.76

8.0

* 5 t 's f 4.0 z

3.0

2.74

2.72 #

2.60

2.68

2.0 2.S6

2.54

to 2.12

e.so

e 48 OH.0 l4P 14.4 14.6 l46 IS0 lS.2 W.4 I86 I S 8 160 162 164 16.6 I 6 6

& T W ' s Wttd svtoce Mtkimt

0

E 0 0 m

I

Longitudinal Prismatic Coefficient

FIGURE 6.--Contours of volumetric coefficient versus longitudinal prismatic coefficient and length- beam ratio for a beam-draft ratio of 2.25

Values apply to all forms having a midship sect ion coefficient of 0.925

10

19

18

17

16

15

I c

10

9

7

6

5 0.48 0.52 0.56 0.60 0.6 4 0.68 0.72 0.76 0.80 0.84


FIGURE 6.-Contours of volumetric coefficient versus longitudinal prismatic coefficient and length- beam ratio for a beam-draft ratio of 2.25

Values apply to all forms having a midship section coefficient of 0.925

0

0

- c

a

I

Longitudino I P r i smotic Coefficient

FIGURE 7.-Contours of volumetric coefficient versus longitudinal prismatic coefficient and length- beam ratio for a beam-draft ratio of 3.00


11

FIGURE ?.-Contours of volumetric coefficient versus longitudinal prismatic coefficient and length- beam ratio for a beam-draft ratio of 3.00

Values apply to all forms havhr a midship section coefficient of 0.925

11

E

m 0 Q

Longitudinal Prismbtic Coefficient

FIGURE &--Contours of volumetric coefficient versus longitudinal prismatic coefficient and length- beam ratio for a beam-draft ratio of 3.75


12

18

I7

16

15

14

13

0 - c g 12

I

f 0 10

9

8

7

6

5

4- 0.48 0.52 0.56 0.60 0.64 0.6 8 0.72 0.76 0.8 0

FIGURE &--Contours of volumetric coefficient versus longitudinal prismatic coefficient and length- beam ratio for a beam-draft ratio of 3.75

Longitudinal Prismbtic Coefficient

Values apply to all fornu having a midship section coefficient of 0.925

0.84

12

As stated previously, the only variation whichaffects the nondimensional offsets of the individual forms is the longitudinal prismatic coefficient. The other variations must be shown either by selecting proportional scales on beam, draft, or length or by dimensionalizing to produce specific prototypes. The steps which a r e taken to determine any individual f o r m of the se r i e s may be summarized as follows:

1. Fo r the a s signed value of C, , read the nondimensional offsets f rom the waterline curves of Appendix 1.

2. For the assigned values of Cv and B/H, obtain the value of L / B f rom Formula [ 6 3 or Figures 6 , 7, and 8.

3 . Assign a characterist ic l inear dimension to the prototype such as length, beam, or draft. The other l inear dimensions can be readily obtained from the values of B / N and L / R which have already been determined.

4. Multiply the nondimensional values for the half breadths, heights, and station spacings by the dimensional values of beam, draft , and length, respectively.

When the values of C, , C+, and B/ l i a r e assignedfor any Taylor Series form, the corresponding wetted-surface coefficient can be determined. Unfortunately, as is the case with al l complex forms such as ship shapes, it is not possible to express the wetted-surface coefficient as a function of these three parameters in a form which i s capable of mathematical solution. This would be t rue evenif the lines themselves were mathematically defined. Consequently, it i s necessary to resor t to the usual numerical procedures for calculating the wetted surfaces of individual s e r i e s forms. If such calculations a r e made for a sufficient number of forms covering the desired range, the functional relationships can be shown graphically. The wetted sur - faces of the models comprising the Taylor Ser ies were originally calculated a t the time of the model tests. Since these calculations were carefully made and spot checks indicated their accuracy, it was not necessary to repeat them. The wetted surfaces were originally calculated using the trapezoidal rule with measured girths and applying corrections for obliquity. The over -all obliquity factor w a s small , amounting to not l e s s than 1.0015andnot more than 1.0070.l The original calculations were cross-faired to a r r ive a t the contours of wetted-surface coefficient given

289133 0 - 54 - .1 13

in Appendix 2. Contours a r e given for a B / B of 3.00 instead of the original 2.92 for ease of interpolation between the other two values of B / U . To achieve nondimensionality, the wetted- su r f ace coefficient

S Cs = C81

where S is the wetted-surface a r e a , -V is the immersed volume, and L is the load waterline length,

is used instead of the characterist ic Taylor wetted-surface coefficient. The numerical values of CS and Taylor's coefficient a r e related by Figure 5. It may be noted that the contours of Appendix 2 vary considerably in appearance in going from one value of B / H to the next. Unlike the contours for the other two values of B / H , the contours for a R / H of 2.25 have a distinct minimum a t C, = 0.66 . It is difficult to prove mathematically why this should be so. However, it would appear that a t values of B/H near 2.0, some of the resulting forms would approximate a half prolate spheroid. The la t ter has a c, = 0.667 and represents the type of form for which the wetted-surface coefficient for a given L / B i s the minimum value obtainable.

CONFIGURATION OF DERIVED FORMS

In planning a se r i e s of forms to be derived f rom a common parent, it may be difficult in some cases to visualize what the resulting forms will look like without going through the labor of the deriving process. F r o m this standpoint, it appears desirable to survey generally the effects of the geometrical variations on the individual forms of the Taylor Series. To accomplish this, the profile, the load waterline curve, and the midship section for selected forms a r e shown drawn to proportionate scales in Figures 9, 10, and 11. The most interesting effects result f rom the Cp variation which is illustrated for fixed values of B/H=3.00 and Cp.= 4.00 x 10-3 in Figure 9. It may be noted that in addition to the usual changes in fullness of the waterlines which a r e attendant with C, changes, there a r e distinct changes in s t e rn profile. Most apparent of these a r e the change of the overhang of the s te rn counterand the change in the r i se angle of the skeg. The variation in the overhang may lead to cer ta in ramifications in comparing the r e s i s t - ances of offspring of dissimilar parents, as will be ex- plained in a la te r section. The changes of form due to the

C = 0.48 B/H = 3.00 P

Cv= 4 x L/B = 6.08

B/H = 3.00 C = 0.64 P Cv= 4 x L/B = 1.08

Cp = 0.86 B/H = 3.00 Cv= 4 x L/B = 8.14

FIGURE %-Effects of longitudinal prismatic coefficient variation on the shapes of derived forms These forms have been derived for fixed values of beamdraft ratio 3.00 and volumetric coefficient : 4.0 x I@

14

C = 0.64 B/H = 3.00 P C v = 1 x L/E = 14.05

C - 0.64 B/H = .3.00 P - c,+ = 4 10-3 LIB = 7.03.

Cp = 0.64 .6/H = 3.00 cv = 7 10-3 L/B = 5.32

FIGURE 10.-Effects of volumetric coefficient variation on the shapes of derived forms

These farms have been derived f a fixed values of beamdraft ratio : 3.00 and loll(litudinn1 prismatic coefficient : 0.64

15

C - 0.64 B/H = 2.25 P - C+= 4 x lr3 L / B = 8.11

C =0.64 B/H = 3.00 P

Cv= 4 x lF3 L / B = 7.03

Cp = 0.64 B/H = 3.75

c y = 4 10-3 L/B = 6.30

FIGURE 11.-Effects of beam-draft ratio variation on the shapes of derived forms These forms have been derived for fired value. of volumetric coefficient I 4.0 x lo4 and longitudinal prismatic coefficient: 0.64

16

variation of C+are shown for fixed values of B/ I I = 3.00 and C p = 0.64 in Figure 10. Since effectively only L/R is being changed, the changes appear as proportionate changes of beam and draft , if length is held fixed. The changes in fo rm due to the variation in B/H a r e shown for fixed values of Cp= 4.00 x andCp=0.64 in Figure 11. Since L/Bvaries with WH, the changes a l so appear a s proportionate changes of beam and draft , if length is held fixed.

CHARACTERISTICS OF ACTUAL FORMS TESTED

The geometrical character is t ics of the individual forms which were actually tested to provide resistance data a r e listed for record purposes in Tables 3,4, and 5. The tabulated dimensions were originally obtained by actual measurements of the model template drawings, the drafts being corrected to the figure obtained by ballasting to the predetermined displacement. It may be noted that, due to human e r r o r , the individual model parameters differ to a small extent f rom the nominal values sought. These differences were taken into account in the fairing of the res i s t - ance data pursuant to the development of the final contours.

REDUCTION OF THE ORIGINAL TEST DATA

The original tes t data for the Taylor Ser ies models were recorded on U. S. Experimental Model Basin "Hull Resistance" data forms. The model res is tance in pounds and the change of level a t bow and s te rn in inches were listed for each of the various towing carr iage speeds. The displacement of the model in pounds and the agreement between actual and calculated drafts were also noted. In general, data values were listed for increments of approximately 0.1 to 0.2 knot to speeds up to 6.0 knots and a t increments not greater than 0.3 knot a t higher speeds.

The methods and procedures which were used to reduce the tabulated original data to nondimensional form a r e essentially the same as those currently used a t the Taylor Model Basin.6 The procedure is as follows:

The total-resistance coefficient, which is defined as

C9 1 R t

2 c t = -

So2

where C t is the total-resistance coefficient, R t i s the total resistance, P v is the speed,

is the mass density, and

is calculated for each of the tes t values of resistance versus speed. The frictional-resistance coefficient i s obtained from the Schoenherr formula

where Cf is the frictional-resistance coefficient, K, is the Reynolds number, equal to &. w is the speed, L v is the kinematic viscosity

V

is the waterline length, and

The frictional-resistance coefficients a r e subtracted f rom the total-resistance coefficients to obtain the r e - sidual-resistance coefficients , or

R r C t - C f = c, = - 2 sv2 C l l l 2

where C, is the residual-resistance coefficient and f i r is the residual resistance.

It should be observed that the frictional-resistance coefficients used in Formula [10].to obtain the defined residual - r e si stanc e coefficient apply to the equivalent f lat plates as derived by the Schoenherr formula. Con- sequently, it is possible that the residual resistance may include not only wavemaking and form resistance but a l s o the difference between the true frictional resistance of the vessel and that of the corresponding flat plate. If the frictional - r e si stance coefficient ve r s us Reynolds num - b e r curve for the actual vessel is parallel to that calculated for the flat plate, the comparisons based on the final predicted effective horsepowers will not be af- fe cted . However , when the r e sidual -resistance coefficients of two diss imilar vessels a r e directly compared, the possibility of discrepancies due to such differences should

17

TABLE 3.-Dimensions and coefficients for Taylor Series models with a nominal beam-draft ratio of 2.25 (Series 22)

Mode C, 2.565 2.574 2.58 2.586 2.593 2.594 2.606 2.558 2.563 2.570 2.575 2.581 2.585 2.595 2.554 2.559 2.561 2.565 2.569 2.575 2.585 2.545 2.549 2.553 5 5 7 2.561 2.564 2.575 2.542 2,547 2.548 2.552 2.555 2.559 2.569 2.540 2.545 2.547 2.551 2.555 2.558 2.569 2.549 2.553 2.557 2.561 2.564 2.574 2.559 2.564 2.568 2.569 2.579 2.588

2.572 2.578 2.587 2.593 2.598 2.605

815 809 803 797

1501 I501 14 93 816 810 804 798 792

1502 1494 817 811 805 799 793

1503 1495 818 812 806 800 794

I504 1496 819 813 807 801 795

1505 I497 820 8 14 808 8 02 796

I506 1498 8 69 868 867 866

1507 14 99 873 872 8 71 870

1508 1500

1492 1491 1490 1489 1488 1487 -

B/H 2.272 2.251 2.251 2.263 2.257 2.252 2.264 2.259 2.266 2 2 5 8 2.256 2.251 2.263 2.247 2.258 2.264 2.243 2.257 2.245 2.260 2.248 2.266 2.258 2.249 2.245 2.256 2.275 2.255 2.246 2.264 2 2 6 7 2 2 6 7 2.271 2.307 2.259 2.263 2.243 2.265 2.254 2.245 2.249 2.271 2.273 2 2 5 9 2.268 2.262 2.265 2257 2.256 2.257 2.250 2.248 2.26t 2.260

2.261 2.239 2.247 2.256 2.254 2.248

)ate Teste

27 Sep 07 20 Sep 07 24 Sep 07 21 Oct 07 17 Sep 07 18 Apr 14 15 Aug 13 27 Sep 07 19 Sep 07 23 Sep 07 19 Oct 07 19 Sep 07 17 Apr 14 15 Aug 13 28 Spe 07 20 Sep 07 24 Sep 07 10 Oct 07 26 Sep 07 20 Apr 14 14 Aug 13 27 Sep 07 27 Sep 07 24 Sep 07 24 Sep 07 26 Sep 07 12 Apr 14 16 Apr 14 22 Oct 07 25 Sep 07 21 Sep 07 19 Oct 07 18 Oct 07 21 Apr 14 24 Apr 14 21 Oct 07 25 Sep 07 25 Sep 07 26 Sep 07 18 Oct 07 20 Apr 14 24 Apr 14 29 Feb 08 27 Feb 08 27 Feb 08 26 Feb 08 20 Apr 14 18 A p 14 9 Mar 08

10 Mar 08 4 Mar 08 2 Mar 08

11 Apr 14 17 Apr 14

21 Jul 13 24 Jul 13 22 Jul 13 22 Jul 13 23 Jul 13 23 Jul 13

As 0.812 1.618 2.430 3.666 1.880 5.852 6.779 0.748 1.494 2.254 3.378 4.514 5.383 6.220 0.695 1.397 2.106 3.143 1.170 5.000 5.773 0.652 1.304 1.955 2.925 3.918 4.700 5.407 0.611 1.227 1.846 2.749 3.682 4.405 5.080 0.578 1.158 1.730 2.586 3.461 4.131 4.784 1.058 1.591 2.377 3.183 3.796 4.397 0.978 1.470 2.210 2.938 3.507 4.063

0.911 1.366 2.054 2.733 3.273 3.789

L A, s 17.05 32.91 23.96 46.71 29.33 57.31 35.93 70.39 41.47 81.50 45.31 89.22 49.00 96.42 17.38 32.82 24.60 46.56 30.00 57.16 36.72 70.13 42.34 81.17 46.43 88.85 50.04 96.01 17.72 32.77 25.09 46.44 30.50 56.96 37.44 69.82 43.18 80.77 47.34 88.47 50.83 95.56 17.95 32.65 25.46 46.32 31.12 56.75 38.15 69.63 43.96 180.55 48.23 88.07 51.85 95.17 18.27 32.61 25.91 46.22 31.72 56.64 38.78 03.49 44.75 80.34 48.90 87.90 52.06 94.95 18.52 32.59 26.12 46.20 32.15 56.65 39.21 69.49 45.18 80.35 49.40 87.87 53.39 94.95 26.79 46.21 32.75 56.70 40.14 69.55 46.18 80.41 50.48 88.07 54.32 95.16 27.13 46.44 33.25 56.94 40.62 69.8E 46.85 80.66 51.14 88.6C 55.20 95.6P

27.40 46.61 33.44 57.3: 40.85 70.4: 47.44 81.5: 51.90 89.3: 55.69 96.3:

20.52 20.52 20.52 20.52 20.50 20.49 20.51 20.52 20.52 20.53 20.53 20.53 20.52 20.52 20.53 20.53 20.53 2 0.52 20.52 20.52 20.49 20.52 2056 20.52 20.53 20.53 20.51 20.51 20.53 20.52 2052 20.53 20.53 20.51 20.51 20.52 20.53 20.53 20.53 20.54 20.51 20.51 2 0.5 1 20.51 20.51 20.52 2 0.5 1 20.51 20.5: 20.51 20.52 20.51 20.51 20.51

20.5 20.51 20.5 20.5 20.5 20.5

2.440 2.996 3.456 3.776 4.059 1.362 1.926 2.346 2.874 3.316 3.628 3.890 1.312 1.854 2.254 2.774 3.192 3.490 3.746 1.260 1.788 2.182 2.674 3.098 3.378 3.630 1.222 1.734 2.124 2.600 3.002 3.280 3 5 2 0 1.188 1.676 2.066 2.520 2,900 3.169 3.420 1.612 1.968 2.418 2.782 3.037 3.268 1.546 1.894 2.320 2.668 2.930 3.148

D i m e n s i o n s I i I i

1.084 1500 24.09 1.324 2250 36.11 1.531 3000 48.18 1.677 3587 57.54 1.793 4153 66.74 0.603 500 8.02 0.850 1000 16.06 1.039 1500 24.09 1.274 2250 36.11 1.473 3000 48.18 1.603 3587 57.54 1.731 4153 66.74 0.581 500 8.02 0.819 1000 16.06 1.005 1500 24.09 1.229 2250 36.11 1.422 3000 48.18 1.544 3587 57.54 1.666 4153 66.74 0.556 500 8.02 0.792 1000 16.05 0.970 1500 24.09 1.191 2250 36.11

11.373 3000 148.18 1.485 3587 57.54 1.610 4153 66.61 0544 500 8.02 0.766 1000 16.06 0.937 1500 24.09 1.147 2250 36.11 1.322 3000 48.15 1.422 3587 57.54 1.558 4153 66.61 0.525 500 8.02 0.747 1000 16.06 0.912 1500 24.09 1.118 2250 36.14 1.292 3000 48.15 1.409 3587 57.54 1.506 4153 66.61 0.709 1000 16.03 0.871 1500 24.05 1.066 2250 36.07 1.230 3000 48.09 1.341 3587 57.54 1.448 4153 66.66 0.685 1000 16.03 0.839 1500 24.05 1.031 2250 36.01 1.187 3000 48.09 1.292 3587 57.54 1.393 4153 66.6t

'B I H I A I V L i q q 5 - p 1.990 0.884 1000 16.06

- ' b -

1.4419 .4450 .4438 .4436 .4441 .4446 .4451 .4763 .4781 .4 797 .4805 .4 804 .4823 .4829 5 1 2 8 .5154 .5181 .5 162 5173 3203 5 2 1 8 5 5 8 1 5514 5546 .5522 5 5 1 7 S592 .5555 .5879 .5894 5900 5 8 9 8 .5909 .6015 5 9 2 1 .62 70 a 4 9 .622 8 .624 9 .6256 .62 83 .6302 .6839 . a 2 .6822 .6848 .6887 -6868 .7373 .7379 .7349 .74 03 .741( .7411

.800:

.798(

.7991

.800;

.797l

.799:

- cz -

1.9178 .9198 .9187 .924 1 .9223 .9241 .9219 .9117 .9126 .9249 .9227 .9242 .9256 .9237 .9117 .9203 .9298 .9220 .9187 .9278 .9250 .9306 .9209 .9235 .9184 .9210 .9370 3252 .9192 .9239 .9276 .9219 .9277 3445 .92 63 3 2 7 2 3249 .9182 .9180 .9236 .9252 3287 .925 7 3283 ,9220 .9301 .9320 .92 92 3235 .9251 .9239 .9276 3263 9 6 6

.930(

.92 71

.931(

.93Ol .9291 .930:

- cp 4817 4837 4832 4800 4818 4798 4800 5221 5238 5206 5207 5199 5207 5230 5624 5600 5571 5599 5630 5608 5 642 5998 5987 6004 6013 5 990 5969 6006 ,6400 .63 78 6360 63 98 63 70 ,6369 ,6393 ,6761 ,6756 ,6783 ,6807 ,6773 ,6791 6788 ,7387 73 70 ,7399 ,7363 ,7391 .7392 .7983 ,7976 .7953 ,7981 .799! .7999

860: .8601 .858; .a601 .858 .85%

- 7 P"

.7542

.7583

.7578

.7592 .7589 .7573 .7596 .7657 .7681 .7729 .7719 .7725 .7712 .776 .7791 .7815 .7860 .7848 .7847 .7852 .7848 .8041 .7961 .7979 .794 7 ,7982 .8014 .7960 .8073 .8091 .8106 .8118 .8139 .8274 2212 .8253 .823 1 .8216 .8243 .824 9 .8267 .8283 .844 1 .842 9 .8429 .8466 .8500 .8474 .8627 .8610 .8612 .8647 .8mE .8670

.891;

.8891 .8932 .890t .88M .89@

-

Coef clu - .5863 5 8 6 8 5857 5844 5852 S856 5885 .6218 .6225 .6229 .6224 .6219 .6238 .6269 .65 78 .6592 .6592 .6578 .6592 .6611 .6624 .694 1 .6926 .6951 .6950 ,6912 .6961 .6964 .7282 .7278 .7279 .7265 .7261 .7269 .7210 .7596 .7590 .7580 -75 78 .7584 .7600 ,7611 .8104 ,8114 .8094 .8093 .8104 .8104 .8547 A558 .8531 .8562 .851[ .8548

.897I

.8978

.8952

.8986 89 .8977 -

: ients

0 3 ~ C, - 0.928 1.859 2.788 4.180 5.593 6.688 7.735 0.928 1.859 2.784 4.173 5.568 6.660 7.725 0.927 1.856 2.784 4.179 5.576 6.660 7.758

1.847 2.788 4.173 5.5 68 6.669 7.720 0.927 1.859 2.788 4.173 5.565 6.669 7.720 0.92 8 1.856 2.784 4.176 5.560 6.668 7.720 1.858 2.787 4.181 5.566 6.668 7.726 1.853 2.787 4.175 5.5 74 6.669 7.726

1.863 2.794 4.197 5.588 6.681 7.747

~1.928

TABLE 3.-Dimensions and coefficients for Taylor Series models with a nomina 2% (Series 22)

L\ 0.01 L)

26.5 6 53.11 79.66

119.60 159.70 190.20 220.00 26.56 53.11 79.57

119.40 159.20 190.50 220.60 2 6.5 6 53.11 79.57

119.60 159.10 190.50 221.60 26.56 52.81 79.66

119.40 159.20 190.70 220.90

26.53 53.11 79.56

119.40 159.20 190.70 220.90

26.56 53.05 79.57

119.40 15Y.10 190.70 220.80 53.17 79.75

119.50 159.50 190.70 220.80

53.05 53.05

119.40 159.50 190.70 220.80

53.17 79.84

119.80 159.50 190.70 221.10

- L / B

14.470 10.310 8.410 6.849 5.932 5.426 5.053

15.070 10.650 8.751 7.143 6.191 5.659 5.275

15.650 11.070 9.108 7.397 6.428 5.880 5.470

16.2 90 11.500 9.404 7.678 6.627 6.072 5.650

16.800 11.830 9.661 7.896 6.838 6.253 5.827

17.270 12.250 9.937 8.147 7.083 6.472 5.997

12.720 10.420 8.482 7.376 6.753 6.276

13.280 10.830 8.845 7.687 7.00( 6.51!

13.78C 11.290 9.209 7.968 7.278 6.775

-

beam-draft ra ti0 of

TABLE 4.--Dimensions and coefficients for Taylor Series models with a nominal beam-draft ratio of 2.92 (Series 20)

19

Model

751 745 72 7 72 1 715 752 74 6 72 8 72 2 71 6 753 74 7 72 9 72 3 71 7 75 4 748 730 724 718 75 5 749 731 725 719

75 6 75 0 732 72 6 72 0

84 9 84 8 84 7 84 6

853 852 85 1 850

-

-

B/H

2.916 2.932 2.909 2.918 2.910 2.951 2.938 2.924 2.868 2.929 2.929 2.932 2.920 2.900 2.911 2.961 2.909 2.928 2.918 2.914 2.962 2.943 2.927 2.921 2.918

2.950 2.965 2.950 2.963 2.937

2.909 2.906 2.910 2.926

l a t e Testec

27 Apr 07 22 Apr 07 18 Apr 07 20 Mar 07 26 Feb 07 29 Apr 07 23 Air 07 18 Apr 07 20 Mar 07 11 Mar 07 29 Apr 07 23 Apr 07 19 Apr 07 22 Mar 07 25 Feb 07 30 Apr 07 24 Apr 07 20 Apr 07 21 Mar 07 11 Mar 07 30 Apr 07 24 Apr 07 20 Apr 07 22 Mar 07 12 Mar 07

1 May 07 25 Apr 07 22 Apr 07 23 Mar 07 18 Mar 07

24 Oct 07 24 Oct 07 30 Oct 07 31 Oct 07

2 Nov 07 1 Nov 07 28 Oct 07 29 Oct 07

L/B

12.800 9.004 7.391 5.729 4.687 13.220 9.408 7.699 6.039 4.860 13.790 9.730 7.999 6.215 5.062 14.110 10.070 8.230 6.362 5.232 14.620 10.400 8.503 6.599 5.386

15.150 10.650 8.731 6.770 5.537

11.220 9.178 7.404 5.787

L

20.52 20.51 20.51 20.51 20.51 20.52 20.51 20.51 20.51 20.51 20.52 20.51 20.51 20.51 20.51 20.52 20.51 20.51 20.51 20.51 20.53 20.51 20.51 20.51 20.51

20.51 20.50 20.50 20.54 20.51

20.53 20.54 20.53 20.52

20.54 l0.53 20.53 20.53

-

-

B

1.604 2.278 2.775 3.580 4.37E 1.552 2.180 2.664 3.396 4.220 1.488 2.108 2.564 3.300 4.052 1.454 2.036 2.492 3.224 3.920 1.404 1.972 2.412 3.108 3.808

1.354 1.924 2.348 3.034 3.704

1.830 l.238 2.890 3.546

1.756 !.162 !.782 3 .4 14 -

H

0.55C 0.777 0.954 1.227 1.504 0.526 0.742 0.911 1.184 1.441 0.508 0.719 0.878 1.138 1.392 0.491 0.700 0.851 1.105 1.345 0.474 0.670 0.824 1.064 1.305

0.459 3.649 0.796 1.024 1.261

3.629 3.770 1.993 1.212

1.606 1.740 1.953 1.167 -

Dimensions - A

501 1001 150( 250( 3 75( 5 O( loo( 150( 250( 375( 5 O( loo( 150( 250( 375C 50C l0OC 150C 250C 3 75c 50C 1000 1500 2500 3750

5 00 1000 I500 2500 3750

1000 1500 !500 1750

1000 1500 !500 1750

-

-

V

8.0: 16.01 24.0f 40.0I 60.11 8.03 16.04 24.0t 40.0f 6C.11 8.0: 16.04 24.0t 40.08 60.11 8.03 16.04 24.06 40.08 60.1 1 8.03 16.04 24.06

60.11

8.03 16.04 24.06 40.08 60.11

16.05 24.08 40.10 60.15

16.04 24.06 40.10 60.15

-

40.08

Ax - 0.811 1.62s 2.444 4.074 6.10( 0.751 1.49! 2.254 3.785 5.656 0.69f 1.396 2.085 3.501 5.220

1.313 1.959 3.290 4.892 0.610 1.226 1.836 3.073 4.586

0.5 78 1.153 1.728 2.888 4.320

1.062 1.589 2.666 3.977

0.977 1.468 2.446 3.666

0.658

A W - 19.34 27.2f 33.3i 43.12 53.1C 19.8; 27.85 34.21 43.53 53.96 20.30 28.62 35.24 45.05 54.79 20.64 29.08 35.48 45.79 55.98 20.92 39.5 36.18 46.49 57.03

21.06 29.88 36.50 47.15 57.73

30.32 37.10 47.86 58.79

30.83 37.86 18.77 59.78 -

S

32.1E 45.73 55.95 72.73 88.68 32.25 45.73 55.99 72.73 88.72 32.30 45.81 56.07 72.82 88.97 32.36 45.89 56.16 72.90 89.26 32.45 45.98 56.33 73.07 89.60

32.56 46.06 56.48 73.49 90.00

46.45 5 7.08 73.80 90.83

16.79 57.45 74.3 9 91.67

-

-

‘b

D.4431 .4415 .4431 .444f .4453 .4791 .4833 .4833 .4860 .4820 .5174 .5159 .5211 .5204 .5195 .5478 .5488 .5531 .54 84 5561 .5875 .5921 S904 .5909 .5899

.62 94

.6266 ,6280 .62 80 .62 75

.6792

.6804

.6806

.682 1

.7341

.7324

.73 67 1.7354

-

-

0% - 0.922; .9203 .9233 .92 74 .9268 .9211 .9265 .9287 .9413 .9301 .9234 .9208 .9263 .9324 .9255 .9217 .9214 .9236 .9234 .92 79 .9159 .9281 .9240 .92 92 .9229

.9300

.9230

.9245

.9295

.9249

.9227

.9222

.9289

.9253

.9182

.9175

.922 7 1.9202 -

-

‘P - 5.4803 .4801 .4800 .4796 .4805 .5201 .5218 .5204 .5163 .5 182 5604 .5603 .5627 .5581 .5613 .5944 .5956 .5988 .5940 .5993 .6410 .63 78 .63 89 .6359 .63 90

.6772

.6785

.6793

.6757

.6784

.7362

.7377

.7327

.7370

.7992 ,7983 .7985 .7992 -

- C

pv

1.7544 .75 73 .7559 .7575 .7527 .7678 .7762 .7719 .7776 .7730 .7782 .7794 .7776 .7817 .7881 .7919 .7878 .7970 .792 1 .7984 .8093 .8105 .&I071 .8102 A077

.8277

.8272

.82 60

.8286

.8245

.a91 .8403 .8418 .8422

,8567 .8566 ,8606 .8600

-

-

COl - ‘W -

3.587; .583! .586: .587: .5911 .623! .6231 .6261 -6251 .6231 .664! .661! ..6701 .6651 .65 98 .691; .69@ .694i .692! .6961 .725: .730: .7314 .7291 .7301

.7582

.7576

.75 8C

.75 65

.7599

,80 74 8073 3066 .8081

,8548 ,8529 ,8540 .8531 -

icients

I O ~ ~ C ,

3.927 0.927 2.788 4.645 6.967 0.929 1.859 2.788 4.644 6.967 0.929 1.859 2.788 4.644 6.967 0.927 1.859 2.788 4.644 6.968 0.927 1.859 2.788 4.644 6.967

0.930 1.862 2.793 4.625 6.967

1.855 2.779 4.634 6.692

1.851 2.781 4.634 6.951

A (0.01 L ) 26.52 53.17 79.76 132.9 199.4 26.55 53.17 79.76 132.9 199.4 26.55 53.17 79.76 132.9 199.4 2 6.55 53.17 79.76

132.9 199.4 2 6.52 53.17 79.76

132.9 199.4

26.58 53.24 79.85 132.3 199.4

53.05 79.39 132.6 199.2

52.93 79.5 7 132.6 199.0

c s - 2 50’ 2.521 2.519 2.537 2.526 2.512 2.521 2.521 2.537 2.527 2.516 2.525 2.525 2.540 2.534 2.520 2.530 2.529 2.543 2.542 2.527 2.535 2.536 2 .549 2.552

2.538 l.541 !.543 l.562 !.563

l.559 !.567 !.572 l.586

2.578 F.584 !.593 !.609 -


Model

20.53 20.53 20.50 20.50 20.49 20.51 20.51 20.51 20.53 20.51

20.51 20.51 20.51 20.52 20.52 20.53 20.52 20.51 20.51 20.51 20.51 20.54 20.54 20.47 20.52 20.49 20.51 20.53 20.53 20.53 20.52 20.52 20.51 20.51

20.53 20.53 20.52

20.51 20.49 20.51

20.51 20.51 20.52 20.50 20.51 20.52 20.51 20.51 20.52

20.51

20.51 20.51 20.51 20.51 20.51

20.50

20.52

20.52

20.52

20.51

782 776 770 764 758

1523 1515 783 777 771 765 759

1524 1516 784 778 772 776 760

1525 1517 78 5 779 773 767 761

1526 1518 786 780 774 768 752

1527 1519

787 78 1 775 769 763

1528 1520 861 860 859 858

1529 1521 865 8 64 8 63 862

1530 1522 1514 1513 1512 1511 1510 1509

L B H

1.824 0.48t 2.568 0.684 3.148 0.837 3.856 1.024 4.448 1.182 4.869 1.295 5.232 1.392 1.750 0.468 2.452 659 3.014 .804

4.276 1.134 4.670 1.246 5.023 1.343 1.654 0.451 2.384 .633 2.322 .771 3.568 0.952 4.118 1.093 4.496 1.193 4.850 1.294 1.628 0.43F 2.300 .617 2.822 .753 3.428 0.921 3.975 1.051 4.370 1.165 4.680 1.248 1.576 0.424 2.238 .595 2.736 .72i 3.342 0.889 3.846 1.025 4.224 1.121 4.537 1.20:

1.536 0.413 2.164 .576 2.652 .705

3.740 0.995 4.106 1.086 4.400 1.172

2.546 .671 3.112 .828 3.596 0.957 3.949 1.053 4.220 1.126 1.992 0.529 2.448 .645 2.992 .793 3.450 .920

4.058 1.083

2.352 .627 2.881 .767 3.334 .a86 3.650 0.968

13.9081 1.044

3.698 0.984

3.240 .a61

2.078 0.552

3.768 0.999

1 . 9 ~ 0.509

late Teste

' b

.4396 .4456 .4460 .4464 .4472 .4457 ,4468 .4782 .4843 .4847 .4844 .4842 .4827 .4824 .5248 .5191 .5170 .5184 .5219 .5237 .5183 .5492 .5512 .5518 .5592 .5589 .5523 .5568 .5856 .5875 .5900 .5928 .5956 .5925 .5929

.6171

.6282

.6278

.6313

.6313

.6298

.6308

.6810

.6866

.6825

.6811

.6750

.6845

.7411

.7425

.7411

.7384

.7450

.7400

.a018

.7953

.7963

.7956

.7954 1.7975

7 Aug 07 15 Jul 07 9 Jul 07

27 Jun 07 22 Jun 07 1 Sep 14 2 Sep 14 6 Aug 07

15 Jul 07 11 Jul 07 27 Jurt 07 22 Jun 07 27 bay 14 1 Sep 14 7 Aug 07

16 Jul 07

15 Jul 07 24 Jun 07 27 May 14

2 Sep 14 8 Aug 07

15 Jul 07 9 Jul 07

28 Jun 07 24 Jun 07 26 May 14 19 Sep 14 8 Aug 07

18 Jul 07 10 Jul 07 1 Jul 07

26 Jun 07 17 Apr 14 19 Sep 14

9 Aug 07 22 Jul 07 12 Jul 07 26 Jun 07 26 Jun 07 16 Apr 14 31 Aug 14 21 Feb 08 21 Feb 08 20 Feb 08 20 Feb 08 16 Apr 14 31 Aug 14 25 Feb 08 25 Feb 08 24 Feb 08 24 Feb 06 11 Apr 14 29 Aug 14 21 Jul 13 21 Apr 14 21 Apr 14 24 Jul 13 25 Jul 13 25 Jul 13

11 JUI 07

', 0.9168 0.9263

.9290

.9301

.9260 .9256 .9278 .9194 .9257 .9307 .9300 .9305 .9246 .9290 .9316 .9320 .9229 .9255 .9375 .9308 .9255 .9158 .9196 .9162 .9322 .9339 .9204 .9291 .9151 .9204 .9236 .9253 .9335 .9257 .9247

0.9095 .9270 .9246 .9287 .9272 .9269 .9269 .9233 .9315 .9239 .9233 .9122 .9266 .9265 .9278 .9300 .9256 .9291 .9249 .9354 .9295 .9286 .9242 .9250

0.9274

cP .4797 .4818 .4801 .4800 .4829 .4816 .4816 .5209 ,5233 .5183 .5210 .5206

.5192 .5634 ,5570 .5601 .5602 .5567 .5626 .5600 .6001 .5995 .6024 .6000 S985 .6000 .5993 .6402 .6384 .6388 .6406 .6380 .6400 .6413

.6781

.6790

.6797

.CEO9

.6794

.6807

.7377

.7371 ,7387 .!377 .7400 .7390 .8000 .a003 .7968 .7976 .8018 .8001 A571 A556 .8574 .a610 .a599 1.8599

.5zzi

Dimensions

'P"

0.7542 .7598 .7424 .7594 .7651 .7610 .7619 .7682 .7733 .7731 .7735 .7834

.7717

.7971

.7873

.7849

.7859

.7984

.7845 .7856 .7940 .7943 .7932 .8042 A030 .7908 .8004 A070 A071 .El13 .El26 .El52 A142 .El41

.8056

.a258

.a302

.8295

.8286 A277 A424 A547 A433 .8428 A340 .8465 A660 .a682 A675 A642 .a712 .a653 .8923 .8865 .8872 .8865 .8858

0.8874

.7843

.a271

' w

1.5825 .5865 .5990 .5879 .5845 .5857 .5864 .6227 .6265 .6240 .6263 .6184 .6185 .6250 .6589 .6592 .6584 .6597 .6537 .6675 .6596 .6922 .6941 .6958 .6953 .6961 .6984 .6956 .7250 .7282 .7270 .7294 .7306 .7277 .7283

.7659

.7594

.7604

.7605

.7609

.7600

.7624

.8084 A029 .a093 .8080 A094 .a090 .8559

Coefficients

1 0 3 ~

0.928 1.857 2.796 4.195 5.600 6.681 7.735 0.931 1.857 2792 4.195 5.584 6.677 7.735 0.930 1.860 2.784 4.183 5.584 6.677 7.731 0.931 1.854 2.780 4.213 5.617 6.697 7.731 0.928 1.857 2.784 4.182 5.576 6.669 7.731

0.928 1.857 2.788 4.183 5.584 6.689 7.735 1.855 2.787 4.180 5.566 6.679

I 1.855 I 7.735

-

21.81 30.92 38.77 46.47 53.27 58.49 62.93 22.35 31.54 38.58 47.48 54.23 58.95 64.39 22.36 32.25 39.50 48.30 55.21 61.55 65.61 23.11 32.79 40.33 48.79 56.76 62.53 66.77 23.46 33.46

4 s

32.58 46.24 56.24 69.14 80.18 88.39 94.70 32.60 46.32 57.09 69.35 80.65 88.68 94.95 32.76 46.41 56.94 69.78 80.98 89.02 95.25 32.90 46.70 57.29 69.81 81.48 89.25 95.67 33.17 46.91

's

2.537 2.546 2.531 2.540 2.552 2.571 2.559 2.539 2.545 2.547 2.550 2.565 2.579 2.568 2.551 2.556 2.560 2.563 2.576 2.589 2.576 2.562 2.570 2.576 2.567 2.592 2.598 2.586 2.583

2.589 2.593 2.607 2.617 2.601

2.595 2.605 2.605 2.612 2.623 2.629 2.619 2.626 2.629 2.642 2.653 2.656 2.650 2.653 2.656 2.673 2.681 2.688 2.684 2.679 2.684 2.703 2.706 2.714 2.721

2.584

A

0.01 L )

26.52 53.05 79.86

119.80 160.05 190.80 220.80

26.58 53.06 79.76

119.80 159.50 190.70 220.80

26.55 53.11 79.58

119.50 159.40 190.70 220.80 26.58 52.92 79.39

120.30 159.30 191.30 220.80 26.52 53.05 79.57

119.50 159.20 190.70

26.52 53.05 79.66

119.50 159.20 191.40 220.80

53.11 79.76

119.70 159.30 191.00 220.90

53.11 79.76

119.70 159.30 19 1 .oo 220.90 53.12 79.72

119.70 159.40 190.70 220.90

-

220.80

6

H

3.738 3.754 3.761 3.766 3.763 3.760 3.759 3.739 3.721 3.749 3.758 3.771 3.748 3.750 3.667 3.766 3.761 3.748 3.768 3.769 3.748 3.717 3.728 3.747 3.722 3.760 3.751 3.750 3.717

3.763 3.759 3.752 3.768 3.753

3.719 3.757 3.762 3.763 3.i59 3.781 3.754 3.764 3.794 3.758 3.758 3.750 3.748 3.766 3.795 3.773 3.750 3.772 3.747 3.772 3.751 3.756 3.762 3.770 3.743

3.761

h

500 1000 1500 2250 3000 3587 4153 500

1000 1500 2250 3000 3587 4153 500

1000 1500 2250 3000 3587

500 1000 1500 2250 3000

4153 500

1000 1500 2250 3000 3587 4153

500 1000 1500 2250 3000 3587

1000 1500 2250 3008 3587 4153 1000

4153

3587

4153

- L

B 1.260 7.983 6.494 5.316 4.607 4.212 3.920 1.720 8.373 6.805 5.544 4.797 4.392

2.410 8.607 7.026 5.751 4.981 4.562 4.229 2.600 8.930 7.279 5.971 5.164 4.689 4.382 3.030 9.173 7.504 6.140 5.335 4.8 56 4.521

3.3 70 9.487 7.738 6.333 5.484 4.990 4.661 9.875 8.056 6.591 5.706 5.191 4.860 0.300 8.378 6.8 55 5.948 5.446 5.054 0.680 8.720 7.119 6.151 5.619 5.248

-

4.083

-

P A,

8.03 0.81f 16.07 1.627 24.09 2.448 36.14 3.676 48.18 4.869 57.64 5.836 66.74 6.751 8.03 0.753

16.07 1.406 24.09 2.255 36.14 3.384 48.18 4.512 57.62 5.380 66.74 6.261 8.03 0.695

16.07 1.406 24.09 2.095 36.14 3.144 48.18 4.220 57.61 4.993

8.03 0.653 16.07 1.305 24.09 1.947 36.14 2.943 48.18 3.923

66.70 5.427 8.03 0.611

16.07 1.22f 24.09 1.831 36.14 2.74: 48.18 3.68[ 57.54 4.382 66.70 5.072

8.03 0.577 16.07 1.155 24.09 1.729 36.14 2.591 48.18 3.450 57.54 4.133

16.03 1.059 24.05 1.591 36.07 2.381 48.09 3.177 57.54 3.793 66.74 4.403 16.03 0.976

66.70 5.808

57.61 4.686

66.74 4.7ac


63.04 67.77

24.15 33.14 41.38 50.56 58.37 63.94 68.80 34.47 41.33 51.66 59.62 65.52 70.02 34.99 42.95 52.43 60.49 66.12 71.17 35.38 43.29 53.04 61.38 67.22 72.04

89.90 96.21

33.34 47.31 57.90 71.12 82.45 90.26 96.89 47.63 58.39 71.86 83.34 91.25 98.02 48.13

-58.98 72.70 84.22 9239 99.29 48.63 59.61 73.54 85.11 93.31

100.67

be considered, although it is present ly believed that such discrepancies would be small.

Since the aforementioned procedure for calculating Cr had to be performed on data f r o m t e s t s of 158 models, it was deemed pract ical to per form the calculations on a high-speed computing machine. The individual constants were calculated using desk machines and these constants were supplied to a n IBM card-punch computing machine. A sys tem of cross-checking was instituted to insure the accuracy of individual data points.

TEMPERATURE CORRECTIONS

One of the la rges t e r r o r s which is prevalent in the original Taylor Ser ies contours resplted f r o m the fai lure to account for the effects of changes in basin water temperature . Since the frictional - r e si stance coefficient is a function of Reynolds number, the temperature of the water must be accurately known to determine the appropriate kinematic viscosity to use in the computation of the Reynolds number. The basin water tempera tures had not been recorded for the majori ty of Taylor 's model tes ts . T o est imate these tempera tures , the char t of Figure 12 was prepared showing the water temperature in the Experimental Model Basin versus calendar date. The char t covers a five-year period (1913-1918) begin- ning with the t ime whenthe temperature was f i r s t recorded at the Experimental Model Basin and e m b r a c e s the temperature values recorded for the l a t e r Taylor Ser ies models. The char t indicates the maximum and minimum tempera tures by width of line. Although there was con- s iderable fluctuation in the air tempera tures and a var ia - tion of f r o m 53 to 80F in water temperature over the course of a given year , i t w a s quite surpr i s ing to find that the five-year averages of water tempera tures a t any given calendar date were generally repeated to within 2 to 3F. In most c a s e s , the assignedvalues of temperature used in the new calculations a r e believed to be accurate to within 1F.

The temperature differential of 53 to 80F w i l l , a c - cording to the Schoenherr formula, change the frictional res is tance on a 20-foot model by approximately 7 percent. When it is considered that the calculated frictional r e s i s t - ance of slow-speed vessels amounts to approximately 80 percent of the total res is tance, it canbe readily seen that

if the temperature correct ions were neglected, a n at tempt to c ross - fa i r the relatively small remaining residual res is tance would become complicated and large distortions might resul t .

The temperature of the basin water a l s o tends to change the nature of the flow about the model, that is, whether o r not appreciable laminar flow would exist. This is discussed in connection with the correct ions for the effect of transit ional flow upon resis tance.

TRANSITIONAL FLOW CORRECTIONS

At the t ime when Taylor 's model t e s t s were conducted, modern turbulence stimulation techniques were unknown and their need was not anticipated. Therefore , when the data were originally reduced for the preparation of the original Taylor Ser ies contours, no consideration was given to the problem of whether adequate turbulence had been established in the boundary layer of any given model. Even at a comparatively recent date, it was believed that with a 20-foot model and with typical basin water t e m p e r - a t u r e s , the Reynolds numbers were high enough to insure adequate natural turbulence in the boundary layer . Recent studies have shown, however, that the resis tance of cer ta in types of f o r m s is affected by transit ional flow even for 30-foot models, especially at Reynolds numbers below about 6.3 x l o 6 . In reanalyzing the original data, therefore , a n attempt was made to c o r r e c t for the transit ional flow effects.

The procedure used to accomplish this was as follows: Reference is made to the typical plot of C r versus speed- length rat io shown in Figure 13. The assumption is then made that at low Froude numbers (or speed-length ra t ios ) the residual-resis tance coefficient, as defined, is a constant. If the C r curve is t raced f r o m high to low speed-length ra t ios , it may be seen that the C r d e c r e a s e s with decreasing speed-length ra t io so long a s wavemaking resis tance is important. There is then a short range of speed for which the residual-resis tance coefficient remains constant, after which, as the speed-length rat io is s t i l l fu r ther reduced, the coefficient begins to decrease again. In the reanalysis , this la t te r decrease , which has been attributed to transit ional flow, has been ignored and the constant value of the coefficient used for all lower Froude numbers.

21

LL I)

t 50 I t Jonuary February March April MW June

July August Septemk Octokr December

FIGURE 12.-Water temperature in the U.S. Experimental Model Basin versus calendar date for the years of 1913 to 1918

The width of the line indicates the variation in temperature from year to year

On the above basis , curve A of Figure 13 apparently needs no alteration since it continues to be constant a t low speed-length ratios. This indicates that turbulent flow was probably attained in this case owing to one o r more of the following variables: the higher water temperature , the shape of the model, the surface finish of the model, and the initial turbulence in the basin. Curve B, however, drops off considerably below a speed-length ratio of 0.55, which corresponded to a Reynolds number of 8 .3 x 1.O6 for the tes t . Consequently, applying the aforementioned procedure , the constant value of the coefficient i s extrapolated a s shown by the broken line.

Although this procedure for correct ing for the effects of transit ional flow i s not r igorous , a number of recent t es t s of 20-foot models which were towed with and without a turbulence -stimulating device indicated that in general such conditions obtain for models which experience only

minor transit ional effects a t the lowest speeds. In such ca se s , the res idual- res is tance coefficient curves f romthe model exper iments without turbulence stimulation, fa i red according to this procedure, have agreed reasonably well with those result ing f rom the t e s t s with present types of turbulence devices. It i s realized that suchagreement does not completely establish the validity of the aforementioned procedure fo r correct ion of transit ional flow effects since the problem of turbulence stimulation on ship models i s not fully understood a t the p resen t t ime. However, it i s believed that the correct ions made in this work according to this procedure will prove to be reasonably good a s - sumptions.

Experience has shown fur thermore that 20-foot models with the Taylor Ser ies type of bow, i.e., a bow with a ver t ical s t em profile and with pronounced "U" sections, a r e l e s s susceptible to laminar flow than bow shapes

FIGURE 13.-Curves of residual-resistance coefficient versus speed-length ratio, showing typical data spots

Curve A i s for a Taylor Series v e s s e l with Cp : 0.56, B / H : 2.25,and C+c : 7.76 x 10.~. C w e B i s for a Taylor Series v e s s e l with Cp : 0.56, B / N s 2.25,andGp.r 5 .58 x

23

>

- - - - - - - - - - -3 - - - - - - - .

0

/

0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Speed-Length Rotio TMB- 40129

,

#. -

0.000

-1.0

- - a -

3--

2.000

1.000

n

a -

which involve a raked s tem and "V" sections. This effect i s believed to be related to the pressure distribution character is t ics associated with flow around the bows of the models. The degree to which models with U-type bows a r e affected by transitional flow w a s demonstrated by two 20-foot Taylor Ser ies models,, having longitudinal p r i s - matic coefficients of 0.613 and 0.746, which were tested at the Taylor Model Basin in 1951.4 In both cases , it was found that turbulence stimulation was required only a t low speeds and that the assumption of constancy of the residual-resistance coefficient a t these low speeds gave reasonable agreement with the turbulent curve.

At longitudinal prismatic coefficients above 0.74, i t be - came increasingly more difficult to establish the "plateau" in the residual-resistance coefficient curves of the Taylor Ser ies models. This was essentially due to two reasons. F i r s t , the existence of more pronounced stabilizing p res - sure gradients for the fuller models caused the laminar flow a t the bow to pers is t to higher Reynolds numbers. Second, the establishment of wavemaking humps a t low Froude numbers made i t more difficult to a s s e s s the speed at which laminar flow became unimportant. F o r - tunately, since the major par t of the tes t data did not suffer f rom these defects, it was possible, by the process of cross-fairing on the related parameters for the ser ies , to deduce reasonable values in these more difficult cases. However, in a few cases the highest values of longitudinal prismatic coefficient occur with the highest values of volumetric coefficient to become end points in the cross-fairing and therefore should be viewed with suspicion. These cases generally occur in a regime which i s little used in actual ship design and consequently their accurate determination is somewhat academic.

RESTRICTED CHANNEL CORRECTIONS

It had been suspected for a long time that the c r o s s - section of the U. S. Experimental Model Basin was not large enough to tow full-bodied 20-foot models without some restricted channel effect being present. Calcula- tions based on existing restr ic ted channel formulations tended to verify this suspicion.7 Furthermore, cor re la - tion tes ts of several 20- and 30-foot models, which were originally tested a t the Experimental Model Basin, were made a t the Taylor Model Basin in 1940 and 1941. These comparative tes t s exhibited trends for the la rger and

fuller models a t high speeds which could reasonably be due to the rest r ic ted channel effect. In view of the foregoing, i t was considered desirable to incorporate res t r ic ted channel corrections in the reanalysis of the Taylor Standard Series.

The method selected to compensate for the rest r ic ted channel effects was the semi-empirical method of Ref- erence 7. Since this procedure was developed for the general case , it i s somewhat cumbersome for application to a large mass of data. The existence of certain fixed parameters in the present case has enabled the development of the following new procedure which greatly sim- plifies the rest r ic ted channel corrections in calculations of this kind. To avoid conflicts with existing notation, the superscripts prime ( I ) and double pr ime ( ' I ) a r e used in this section to denote quantities which apply to speed in a restr ic ted channel and to "Schlichting's intermediate speed," respectively. The remaining quantities without the pr imes a r e the values sought for the unlimited case.

The procedure for deducing the corrections to be made for res t r ic ted channel effect is based upon two assumptions which may be summarized as follow^:^

1. The theoretical assumption that the wavemaking resistance a t "Schlichting's intermediate speed" vr r is equal to the wavemaking resistance a t a corresponding speed in deep water V . The relation between V " and v is given from wave theory by the formula

($) = tanh ($) where d is the depth of the channel and ,z is the acceleration due to gravity.

2. The empirical assumption that the change in displacement flow around the ship or model hull due to limitations in depth or width of the channel necessitates a correction to the intermediate speed vr ' to give the speed v ' of the ship o r model relative to the channel. This relationship is derived f rom systematic tes t s in res t r ic ted channels and is given as an empirical curve of the form7

C131

24

but has been changed herein to the more inclusive relationship

C18dl

$ = "(JT) JA,L= where C , is the residual-resistance coefficient, C, is the [14] frictional-resistance coefficient, and the same convention

is applied to the superscripts.

Equation [17] can now be restated as follows:

K Y ~ C , = K ( Y ' ) ~ C ; t K(v I ) 2 c; -K(v'1)2C;

where Fd is the displacement-flow parameter , '4% is the midship section a r e a of the vessel, L is the waterline length of the vessel , and

C19 1 r is the hydraulic radius and is equal to

c15 1 and rearranging

where w is the width of the channel and G is the wetted girth of the vessel a t the midship section.

or The relationships between the various components of

resistance which a r i s e f rom the two foregoing assumptions a r e as follows:

or

[16 ] thus

1171

Let where Rt is the total resistance,

R, K , is the residual resistance.

is the frictional resistance, and

The superscripts or lack thereof denote the appropriate Then speeds for which the quantities a r e determined either by test measurement or by calculation using the stated formulas or empirical relationships.

cr=c;(i2)2 + c;(f) -Cr 'I($) 2 c20 I

The dimensional quantities given in Equation 11 71 can be To deduce the values of C, versus speed-length ratio in an unrestricted channel f rom the corresponding values in a restr ic ted channels it has been found convenient to

[18a] resor t to the three se t s of auxiliary curves given in 2 Figure 14. Each set of curves consists of contours of

equal values of the displacement-flow parameter Fci plotted [1&] on the common abscissa of speed-length ratio in a

, 2 rest r ic ted channel. The ordinate in each case is a dependent variable whose numerical value can be used directly with

? Formula C241 to a r r ive a t the unrestricted channel values 2 of C r and the appropriate speed-length ratios. The curves

converted to coefficient form as follows:

R , = -. P Sv2 C , = K v ~ C ,

K(v 7°C: R * P =--S(vI )2ci =

[18c] R ; = - S ( V ' ) ~ C' f = K(v y2 C;

25

P a 0

FIGURE 14.-Auxiliary charts for restricted channel corrections to the Taylor Series Models tested in the U. S. Experimental Model Basin

26

1.00 2.1

0.99

0.98

0.97 - m

0.96

0.95

0.94

0.93

lCf3x 0.04

4

(0

0.03

0

0 5 - 0.02 0' a

2 0

1.9

I .e

I .7

1.6 - 0

0 u)

c

1.5 a s

>=Ik I .4

1.3

1.2

1.1

0.01 1.0

~

0.04 005 0.06 0.07 0.08 0.03 1.0 1.1 1.2 1.3 1.4 I .5 1.6 1.7 1.8 1.9 2.0

6 FIGURE 14.-Auxiliary charts for restricted channel corrections to the Taylor Series Models tested

in the U. S. Experimental Model Basin

- v;

were derived using the speed relationships denoted by Equations C13] and 1141, the corresponding Reynolds numbers, and the resulting Schoenherr frictional- r e s is t - ance coefficients. Curve se t p is used to determine A C r which is substracted from C, to give a net value based on the speed in a restricted channel. Curve set B is used

to obtain (:)2 which, when multiplied by the net value,

gives the C, for an unrestricted channel. Curve set C is used to convert the speed-length ratio f rom the restricted to unrestricted values. It may be noted that curve C is a single curve in the range of values required for the rest r ic ted channel corrections for the Taylor Series.

The procedure for applying the restr ic ted channel correction to the faired Cr versus speed-length ratio data for a typical Taylor Series model is shown by the following numerical example:

Reference is made to the curve in Figure 15. For illustrative purposes, assume the speed-length ratio in a restr ic ted channel to be 1.20. Then the uncorrected C, taken from Figure 15 is 6.750 X The value of Fd for this model operating in the Experimental Model Basin is calculated from Formula C141 and is equal to 0.335. Entering Figure 15 with this value of Fd, AC, =0.0290 x from curve set A and Z- - 0.9890 from curve set B. Then

f rom Formula C241, the value of C r in an unrestricted channel is

(J2-

CC: - A C,1(:)2= (6.750 X - 0.0290 X 0.9890 = 6.647 X

Entering curve set C with the assumed value of 1.20 for the speed-length ratio in a restr is ted channel gives, in this case, the unchanged value of 1.20 for the rest r ic ted channel. These values a r e plotted with other values similarly obtained to give the corrected curve shown by the broken line in Figure 15.

Since the U. S. Experimental Model Basin does not have a rectangular c ross section, the characterist ic depth d. which w a s used in the calculations w a s taken as the per - pendicular distance from the centerline of the basin a t the water surface to the nearest side of the basin. This gave a value of 13.6 feet as compared to the maximum depth on centerline of 14.0 feet.

It should be mentioned that the restricted channel corrections which' were applied to the Taylor Series data were generally found to be quite small. The largest correction amounted to a decrease of approximately 2 percent of the predicted effective horsepower for a geometrically similar 400-foot vessel operating in salt water at a temperature of 59F. It i s believed, however, that the corrections were significant and should not have been neglected since the c ros s-fairing of the residual-resistance coefficient curves w a s improved thereby and many ihconsistencies eliminated.

CROSS-FAIRING OF RESISTANCE DATA

After the residual-resistance coefficient versus speed- length ratio curves were initially faired and the corrections for transitional flow and restr ic ted channel effect applied, it remained 'to cross-fair the C r against the geometrical parameters C p , C w and B / H . There were only three variations of B/H in the ser ies a s compared to a minimum of six variations of C v Conse- quently, the la t ter two parameters were given the most consideration in the cross-fairing process.

and eight variations of c,, .

The fairing procedure was as follows: F rom the faired curves of c, versus speed-length ratio, for constant values of B/H and speed-length ratio, values of C, were read and plotted against C, for each of the given values of C p . When faired, these data formed a se t of contours of the type shown in Figure 16. In general, the curves of this type characteristically indicated a minimum value of C, some- where between C, values of 0.52 to 0.66. Again a t fixed values of B/I1 and speed-length ratio, c ross curves of C, versus C p were prepared for even values of C p using the faired values which were read from curves of the type illustrated in Figure 16. The resulting faired contours a r e exemplified by Figure 17. These contours showed a pro- gressive increase of C, with Cv and were in most cases very nearly linear. The se t s of contours showing the var ia- tion of C r with C, were then cross-faired with the sets of contours showing the variation of C, with Cp. This was accomplished by making minor adjustments in each of the sets until faired curves were obtained which satisfied the condition that each point on the Cr versus c, contours was equal to the corresponding point on the C r versus 4. contours. At this stage, the faired values f rom these two se ts of contours were plotted against B / H for fixed values of C, , C v , and speed-length ratio in incremental steps

27

J O 1.3

Speed- Length Ratio

FIGURE 15.4ample curve of residual-resistance coefficient versus speed-length ratio, showing restricted channel corrections

28

0.4 0.5

Without Correction

6.0

With Correction

5.0 L 0

E P) 0 I .- Y- Y-

4.0 0" 0 0 0 c 0 cn cn

c - 3.4 p

1

2.0

I .o

0.6 0.7 0.8 0.9 1.0 1 . 1 1.2 Speed- Length Ratio

FIGURE 15.-&unple curve of residual-resistance coefficient versus speed-length ratio, showing restricted channel corrections

28

0 1.3

8.0 x I 0-3

c 0

Longitudinal Pr ismat ic Coef f ic ient

'FIGURE 16.--Curves of residual-resistance coefficient versus longitudinal prismatic coefficient for equal values of volumetric coefficient

29

The curves are for fixed values of beam-drah ratio of 2.25 and speed-length ratio of 1.0

7.0

6.0

0) 0 0 0) 0

4.0 6 c ua u) 0)

I

3

tn Q) U

.-

a 3.0

0

2 .o

1.0

0 0.72 0.76 0.8 0 0.84 0.8 8 0.48 0.52 0.5 6 0.6 0 0.64 0.68


FIGURE 1 6 . ~ u r v e a of residual-resistance coefficient versus longitudinal prismatic coefficient for equal values of volumetric coefficient

The curves me for fixed values of beam-drah ratio of 2.25 and speed-lewth ratio of 1.0

29 209733 o - 54 - 4

FIGURE 17.--Curves of residual-resistance coefficient versus volumetric coe ficient for equal values of longitudinal prismatic coefficient

The curves are for fixed values of beam-draft ratio of 2.25 and speed-length ratio of 1.0

covering the range of the ser ies . Any inconsistencies indicated in the R/Fl variation curves were similarly rectified by making adjustments i n the other two se t s of contours. Finally, the cross-faired values were replotted back on the original curves of c, versus speed-length ratio. This w a s done not only to a s su re fa i rness in this view but to observe whether, in the cross-fairing process , significant departures were made f rom the original data spots.

It should be mentioned that, during the fairing process , special effort w a s made to adhere as strictly as possible to the original data which were corrected according to the aforementioned procedures. The humps and hollows normally found in res is tance curves were retained. It was

gratifying to find that the data above speed-length rat ios of 0.6 were excellent, even judging by present day stand- a rds . It w a s observed in this regime that, excluding a few obviously wild points, deviations of individual original data spots f rom the faired curves were l e s s than 1 percent of the total resistance for C , values up to 0.68 and l e s s than 3 percent beyond this range. At the lower speed-length ratios, since the C , is such a small percentage of the total-resistance coefficient in all cases , it is believed that through the combination of the corrections applied and the cross-fairing, very good standards of accuracy have been maintained.

During the cross-fairing process , curves of residual- resistance coefficient versus speed-length rat io were produced for a beam-draft ratio of 3.00 instead of the 2.92 used on the original models. This was done to provide an even value for interpolation purposes in the final pr e s e ntati on.

It would be of interest to compare the resul ts obtained from the reanalysis of the original Taylor Ser ies data with modern tes t results of Taylor Ser ies models. Fortunately, the resu l t s for two Taylor Series models, which were recently constructed and tested a t the Taylor Model Basin, a r e a ~ a i l a b l e . ~ The residual-resistance coefficients for these models a r e compared in Figure 18 with the values interpolated from Appendices 3 and 4. It may be seen that for the fuller model, there is close agreement up to a speed-length rat io of 0.8 which, in general, represents the practical range for vessels of such characterist ics. The maximum deviation in this range occurs a t a speed-length ratio of 0.72 and amounts to less than 2 percent of the total model resistance. F o r the finer model, almost perfect agreement is obtained above a speed-length rat io of 0.9 which represents the range of most interest for vessels of such characterist ics. The maximum deviation is obtained below this range at a speed-length rat io of 0.72 and amounts to approximately 3 percent of the total model resistance. It may also be noted that the wetted- surface coefficients which were calculated f rommeasure- ments of the new models agree with in 0.1 percent of the values read f rom the contours of Appendix 2.

FINAL PRESENTATION OF DATA The Taylor Standard Series was not only a comprehensive

undertaking but a lso one which involved a program of

30

7.0 10-3

6.0

4.0 2

3.0 1 0 e a

D

2 .o

I .O

0 I .o 2.0 3.0 4.0 5.0 6.0 7.0 8.01110'~

Volumetric Coefficient

model construction and testing which could be reproduced a t the present day only by large expenditures in t ime and money. Consequently, it seems only fitting that a special effort should be made to present the reanalyzed data in a form which will be most convenient and useful to the majority of the members of the profession. Obviously, the methods of presenting these data a r e numerous, and, while it seems worthwhile to present the same data in several different forms for the convenience of specialized groups, it is not always practical todo so. Therefore, the approach

used herein is to make one type of presentation of the data which can be universally applied, to suggest other methods of presentation, and to provide auxiliary curves to a s s i s t in rapid conversion of the generalized curves to some of the other desirable specialized curves.

The methods of presenting resistance data canbe classi- fied into two general types: one which facilitates the calculations for specific geometrically similar prototypes and a second which presents "merit" relationships, that is,

L 0

0 0

Speed- Length Ratio FIGURE 18.-Comparison of the residual-resistance coefficiente obtained from tests of new Taylor

Series Models with values read from contours of appendicks 3 and 4 Model 4322 bas the followi C p : 0.746, C+ : 6.26 I 103, and B / H : 2.50. Model 4333 ham the following choscterttica: Cp : 0.613, C, : 4 . 0 5 ~ lod, and B / H : 2.50

charactcristica:

31

C

From Revised Taylor Series Contours From TMB Model Test

-----

I I I

4 0 . 5 0 .6 0.7 0, I 0 3 I . 1 I Speed- Length Ratio

FIGURE 18.~omp&son of the residual-resistance coefficients obtained from testa of new Taylor Series Models with values read from contonra of appendicks 3 and 4

Model 4322 h a tbe follom Cp : 0.746, C+ t 6.26 I. lo? and B/ H = 2.50. Model 4333 has the followin# chracteriaticm: Cp = 0.613, Cy : 4 . 0 5 ~ lo3, and B / H = 2.50

charactwiatica:

31

4. o x 10-3

3.0

!.O

.o

either qualitative or quantitative changes in res is tance which would be expected to resul t f rom the variation of form parameters . A single method of presentation cannot efficiently combine these functions because the conflict in the dependency of components of res is tance on the Froude and Reynolds laws requires the assignment of specific dimensions and conditions to establish t rue mer i t comparisons. Consequently, for more. universal application, the pr imary method of presentationused here belongs to the f i r s t category.

The nondimensional coefficient which has been adopted to represent the resistance data is the residual-resistance coefficient. This coefficient is used in place of Taylor’s well-known r e sidual-re sis tance per ton of displacement. The speed-length ratio used by Taylor has been retained as the speed parameter , although it i s not nondimensional both from the standpoint of the lack of G i n the denomina- tor and the use of specified English units in i ts definition. An alternate scale of Froude number is provided, however, to facilitate international use.

The format used in the presentation also differs f rom that given in Taylor’s “Speed and Power of Ships.” It consists of curves of residual-rksistance coefficient versus speed-length ratio (Froude number scale added) for various even values of volumetric coefficient. Separate families of curves a r e given for each longitudinal prismatic coefficient between 0.48 and 0.86 in increments of 0.01 for each of the three beam-draft ra t ios of 2.25, 3.00, and 3.75. F o r each family of curves , ranges of speed-length ratios of 0.5 to 1.0 and 1.0 to 2.0 a r e given on adjacent pages. The scale divisions, for both ordinate and abscissa , at the lower speed-length ratios a r e twice those of the higher range to permit accuracy of reading. To achieve the most efficient use of the gr id , supplementary tables a r e provided f o r longitudinal pr ismatic coefficients above 0.80 where some of the residual-resistance coefficient curves a r e not constant below a speed-length rat io of 0.50. On each of the se t s of curves , a decimal scale is used both for the low and high speed-length ratios to minimize e r r o r s of reading. The greatest magnification has been obtained by avoiding the use of overlapping scales . This has resulted in a small gap between the low and high speed-length ratio curves for a few of the higher values of longitudinal prismatic coefficient. This slight incon- venience is considered tolerable, however, in light of the improved scales fo r the remaining curves.

32

The use of the residual-resistance coefficient with the aforementioned format possesses several distinct ad- vantages over the Taylor method:

1. Reference to only two pa i rs of pages is needed for a given case when an interpolation on B/ l I is required and only one pair of pages when such interpolation is not required.

2. The increments of longitudinal pr ismatic coefficient a r e small enough to give an accuracy in reading to the neares t five in the third significant figure without interpolation.

3. Interpolation on a given set of contours is along the ordinate and not normal to the contours, as in the original Taylor Ser ies contours.

4. The values of the residual-resistance coefficient contours a r e read on the fine grid of the ordinate, per- mitting closer readings.

5. The shape of the residual-resistance coefficient versus speed-length rat io curves can be seen directly, showing significant features such as humps and hollows.

6 . Values of residual-resistance coefficient can be read directly at uneven speed-length rat ios or Froude numbers which correspond to even speed on the full-scale vessel . This permits computation of a single point, such as at the design speed, which is often all that is needed.

7. The residual-resistance coefficient curves of other vessels can be compared directly with those of the equivalent Standard Series vesse ls , an attribute which is desirable fo r preliminary analytical purposes.

An objection that has often been raised to the use of the residual-resistance coefficient as a parameter for residual res is tance is that i t contains the quantity wetted surface instead of the two-thirds power of the volume in the de- nominator. This objection is based on the premise that the residual res is tance is a function of the immersed volume or displacement and not the wetted surface. This appears to be somewhat academic, however, when it is realized that neither the wetted surface nor the volume t e r m implies any physical significance to the coefficient alone, but only se rves to render the coefficient dimen- sionless. It can be fur ther shown, by reference to the wetted-surface coefficients in Appendix 2 , that the contours of residual-resistance coefficient for the various values of volumetric coefficient do not converge when C r is redefined on the basis of volume to the two-thirds power. On the other hand, the present definition of the C r is desirable

since it eliminates a step in the calculation of the effective horsepower. In this calculation, the C, can be added directly to values of Cf, which can be represented by a single function of Reynolds number. A C , based onvolume to the two-thirds power would require either a conversion back to the definition based on wetted surface o r separate calculations for values of a newly defined Cf for vessels of different immersed volumes but having equal Reynolds numbers .

Variations in the format for presenting. the residual- resistance coefficients a r e shown in Figures 16 and 17. In addition, contours of C , on C p and C p f o r even speed- length rat ios , a format similar to the original Taylor Series contours, could be prepared. These a r e not considered desirable since small changes have a tendency to entirely change the appearance of such contours f rom one speed-length ratio to the next.

Merit relationships can be presented through the use of the so-called “circle coefficient” system which is widely used. The most popular presentation of this type consists of plots of a v e r s u s 0. Fortunately, @is simply related to the total-resistance coefficient Ct and can be readily obtained as follows:

1000 O=m@Ct c251

where @ is the total-resistance coefficient,circle system,

@ is the wetted-surface coefficient, c i rc le system, C t is the total-resistance coefficient,

equal to S + 2 / 3 ’

is the wetted surface, and s j+ is the immersed volume.

Since

1261

where C , is the wetted-surface coefficient used in the reanalysis of Taylor’s Ser ies , is the waterline length, and 1,

CJL is the volumetric coefficient,

then innnc-

C271

where B is considered to be a function of CS and C+. The calculation of B f rom the contours of C, versus C+of Appendix 2 requires the use of tables of fractional powers or a log-log slide rule , facilities which a r e not always readily available. Consequently, i t was considered desir - able to prepare the specialized chart of Figure 19 which gives values of R for the range of values of C S and CJL‘ which a r e covered in the Taylor Series.

The @coefficient is simply related to the Froude number by the equation:

where @ is the speed coefficient, is the speed,

3 is the Froude number, g is the acceleration due to gravity, and 0 is a function of C y .

C2 81

Again to avoid calculations involving fractional powers , a specialized chart is provided by Figure 2 0 for values of D in the range covered by the Taylor Ser ies data. If it is desired to convert directly f rom values of speed-length rat io to D , the values in Figure 20 must be multiplied by 0.2978, a factor based on a value of g for the North At- lantic Ocean.

When the plots of @versus @are used for mer i t comparisons, it is customary to base the calculations on stand- a rd conditions which apply to a 400-foot prototype vessel operating in salt water of 3.5 percent salinity and a temperature of 59F. This permits the preparation of a chart of Sc hoenhe r r frictional resistance coefficient versus 0, Froude number, or speed-length ratio, as shown inFigure 21. These values can be added directly to the values of r e s idual- r e s is tance c oeff ic ient obtained from Appendice s 3 , 4 , and 5 for the appropriate case to give the total- resistance coefficients. Then, by use of Formulas C271 and [28] and Figures 19 and 20, the values of a v e r s u s @are readily obtained.

33

360

340

320

260

2 40

220

V Vo lumetr ic Coeff ic ient C+= -

L3

FIGURE 19.--Factors for convertingthe total-resistance coefficient Ct to the @resistance coefficient

34

360

340

320

7 6 5 3 3 2 V

Volumetric Coefficient C - - 4- L,

FIGURE 19.-Factors for convertingthe totalqesistance coefficient Ct to the @resistance coefficient

260

240

220 I

34

0

Y Volumetric Coefficient Cy=

3

FIGURE 2O.-Factors for converting the Froude number 3 to the @ speed coefficient

35

I

5 4 3 2 v

Volumetric Coefflclent Cy= 3

0

9

0 I

FIGURE 2O.-Factors for converting the Froude number 3 to the @ speed coefficient

35

Froude Number

(u

C + .- SJ a

FIGURE 21.-Schoenherr frictional-resistance coefficients for a 400-foot vessel operating in salt water of 3.5 percent salinity and a temperature of 59F

36

0 0.05

'f

2.271 2.058 1.945 l.871 1815 1.772 1.737 1.707 1.681 1.659 1.638 l.621 1.605 1.590 1.576 1.563 1.552 1.541 1.531 1.522

Froude Number a55

0.10 , , 0.!5 . . . . 0.20. . . . 0.25 I I I a 0.30, I I , O F I , I I O.?O I I I I 0.?5 I I I I O?Ol I I I I I I I I

'k

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

' 39 40

- 'k

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

-

Schoenhea Frictional-Resistance Coefficient versus Speed-Lengtb Ratio and Froude Number for a 400-Foot Vessel Operating in

Salt Water of 3.5 Percent Salinity and 59 F Temperature

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65

0.75 0.80 0.85 0.90 0.95 1.00

0.70

3 0.0149 0.0298 0.0447 0.0596 0.0745 0.0893 0.1042 0.1191 0.1134 0.1489 0.1638 0.1787 0.1936 0.2085 0.2234 0.2382 0.2531 0.2680 0.2829 0.2978

V k m

1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40. 1.45 l.50 L55 1.60 1.65

1.75 1.80 1.85 1.90 1.95 2.00

1.70

3 0.3127 0.3276 0.3425 0.3574 0.3723 0.3871 0.4020 0.4169 0.4318 0.4467 0.4616 0.4765 0.4914 0.5063 0.5212

0.5509 0.5658 0.5807 0.5956

0.53m

'f l.511 1.503 1.496 1.487 1.480 1.473 1467 1.459 1.453 1.448 1.443 1.437 1.432 1.427 1.422 1.4 16 1.412 1.408 1.403 1.399

20

1.9

c

0 u .- .-

I.* H 0

::

E a

0 + .-

1.7 4 C

V .- c .- I=

1.6

1.5

1.4 0 0.1 0.2 a 3 0.4 0 5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Speed- Length Rotio

FIGURE 21.-Schoenherr frictional-resistance coefficients for a POO-foot vessel operating in salt water of 3.5 percent salinity and a temperature of 59F

36

Another device for showing m e r i t relationships is exemplified in Figure 22. Here the effect of the variation in C , for constant C y is shown for each of the three values of R / l i at each of s e v e r a l even speed-length ratios. The changes a r e shown as ra t ios of the effective horsepowers to the minimum effective horsepower for each set of curves . The effective horsepowers used to determine these rat ios were calculated for the s tandard conditions of the 400-foot vesse l operating in salt water of a temperature of 59F. The numerical values of these ra t ios apply reasonably to the second decimal place, however, to vessels ranging f r o m 100 to 1000 feet in length.

The merit curves of the type shown in F igure 22 a r e a l so useful in the prel iminary design s tage for quickly reproducing the effective horsepowers of a vesse l of prescr ibed geometrical parameters if they a r e used in conjunction with curves of the type given in F igure 23. These are curves of effective horsepower for each of the minima shown in F igure 2 2 plotted against length for each of a number of even speed-length ratios. The effective horsepowers for the des i red vessel can be obtained by multiplying the effective horsepowers f r o m Figure 23 by the suitable ra t ios of F igure 22.

CALCULATION OF EFFECTIVE HORSEPOWER USING REVISED CONTOURS

Several methods of presenting the res i s tance data for the r e 4 s e d Taylor Ser ies were suggested in the preceding section, and the method whichuses the residual-resis tance coefficient versus speed-length rat io curves was adopted for general application. The following discussion will demonstrate the use of these curves for specific types of calculations. The two most frequent u s e s of the original Taylor Ser ies contours have been to es t imate the effective horsepowers of prel iminary designs where only the over- all dimensions and coefficients a r e known and to establish c r i t e r i a for the performance of proposed and existing designs where either model o r full-scale data a r e available. The method for accomplishing these types of calculations using the revised contours is shown in Table 6 by the sample f o r m which briefly indicates the step-by-step procedure. The method is fur ther i l lustrated by the following numerical example which involves a typical passenger vesse l having the dimensions and coefficients given in Table 7.

Longitudinol Prisrnotic Coefficient

FIGURE 22.--lhe variation in effective horsepower of laylor Series vessels with change in longitudinal prismatic coefficient for a volumetric coefficient

of 1.5 The effect ive horsepowers pertain to a 400-foot ship operating in salt water of 3.5 percent sal inity at a temperature of 59F and are expressed a s ratios to the minimum EHP for each set of curves

37

Another device for showing mer i t relationships is exemplified in Figure 22. Here the effect of the variation in C, for constant C y is shown for each of the three values of R / N a t each of several even speed-length ratios. The changes a r e shown as rat ios of the effective horsepowers to the minimum effective horsepower for each set of curves. The effective horsepowers used to determine these rat ios were calculated for the standard conditions of the 400-foot vessel operating in sal t water of a temperature of 59F. The numerical values of these rat ios apply reasonably to the second decimal place, however, to vessels ranging from 100 to 1000 feet in length.

The mer i t curves of the type shown in Figure 22 a r e a lso useful in the preliminary design stage for quickly reproducing the effective horsepowers of a vessel of prescribed geometrical parameters if they a r e used in conjunction with curves of the type given in Figure 23. These a r e curves of effective horsepower for each of the minima shown in Figure 22 plotted against length for each of a number of even speed-length ratios. The effective horsepowers for the desired vessel can be obtained by multiplying the effective horsepowers f rom Figure 23 by the suitable ratios of Figure 22.

\

\- SpeLd-Lenglh Ratio! 2.00

CALCULATION OF EFFECTIVE HORSEPOWER USING REVISED CONTOURS

1.30

Several methods of presenting the resistance data for the reVised Taylor Ser ies were suggested in the preceding section, and the method whichuses the residual-resistance coefficient versus speed-length ratio curves was adopted for general application. The following discussion will demonstrate the use of these curves for specific types of calculations. The two most frequent uses of the original Taylor Ser ies contours have been to estimate the effective horsepowers of preliminary designs where only the over- all dimensions and coefficients a r e known and to establish c r i te r ia for the performance of proposed and existing designs where either model or full-scale data a r e available. The method for accomplishing these types of calculations using the revised contours is shown in Table 6 by the sample form which briefly indicates the step-by-step procedure. The method is further illustrated by the following numerical example which involves a typical passenger vessel having the dimensions and coefficients given in Table 7.

37


FIGURE 22.--l’he variation in effective horsepower of Taylor Series vessels with change in longitudinal prismatic coefficient for a volumetric coefficient

of 1 . 5 ~ 1 0 - ~ The effective horsepowers pertain to a 400-foot ship operating in salt water 01 3.5 percent salinity at a temperature of 59F and are expressed a s ratios to the minimum M P for each set of curves

1,000 100,000

800 80,000

600

500

400 350

300

250

200

L e 150

91 b r 0

$ 100

E, 80

c iz E .- C .-

60

50

40 35

30

25

20

15

‘400

60,000

50,000

40,000 35,000

30,000

25,000

20,000

9 15,000 &

% b I e

l0,OOO g r ”- W

8000 E

E 3

E 6000

5000

4000 3500

3000

2500

1500

I50 200 300 400 500 600 I00 150 200 300 400 500 600 800 1000

TMB- 40136 Ship, Length In feet Ship Length in feet

FIGURE B.-The minimum effective horsepowers of Taylor Series Vessels of various length with a volumetric coefficient equal to 1.5 x

38

B 3.00 --F

COLUMN PROCEDURE

1 From Forniula 10 using speeds in Col. 17 with factor 1.689 to convert from knots to Wsec

Fmm Formula 11 using speeds in Col. 17. (Need not be computed i f Col. 1 i s used.)

Fmm contours in Pppendices 1, 2, and 3, using given values of cP and Ctr: (Only Col. 4 with either Col. 3 or Col. 5 is needed when linear interpolation IS

2

5 used.)

6 B

B Col. 4 -Col. 3 when 7 > 3.M) or

Col. 5 -Col. 4 when < 3.00

’> 7 K X C o I . 6

8 Col. 4 + Col. 7

(‘)Prime indicates that the Quantity for the given s h i p is numerically different than that for the Standard Series vessel.

COEFFl Cl ENTS

BeamDraft Ratio,! - - - - - - - - - - - - - - - - - Longitudinal Prismatic Coefficient, cp - - - - - - - .Volumetric Coefficient, CY - - - - - - - - - - - - - Wetted-Surface Coefficient, cs ,, - - - - - - - - - - - Wetted Surface, Std Series, Cs

Wetted Surface, Ship cs’ RoughnessAllowlnce Coefficient, A c, - - - - - -

- - - - - - - - - - - - -

COLUMN PROCEWJRF:

9

10

11 Col. 10 + Ac, 12 Col. 8 x,

From Formula9 using speeds in Col. 17 with factor 1.639 to convert fmm

From T a l e s of Appendix 5 using values of R= i n Col. 9

knots to fVsec

c

CS ’ 13

14

15 16

17

When Standard Series MP is cmpered with that computed for a given ;hip from model data, Columns, 9. 10, 11, 14, 15, and 17 have already ieen computed.

Col. 11 + Col. 12

Cube of speeds in Col. 17 A fromFormla 12 \* Col. 17

Col. 13 X Col. 15(Formula 12) Assumed even ship speeds (knots)

(2)

TABLE 6.--Sample forni for the calculation of effective horsepower from the Taylor Standard Series

39

If it is assumed for illustrative purposes that the desired value of effective horsepower is for a speed of 25.00 knots, then the speed-length rat io for the given vessel is

a.00 2738 0.923 1892

63.64

F r o m Table 7 , the rounded-off values of longitudinal prismatic coefficient Cp and volumetric coefficient cv a r e 0.62 and 3 . 7 8 ~ 1 0-3, respectively. Since the beam-draft ra t io R / H is 2 . 9 7 , the residual-resistance coefficients must be interpolated between the contours for B/H = 2.25 and 3.00 in Appendices 3 and 4. Thus reading f rom these contours a t - vk = 0.980, c , = O . 6 2 , and c f ~ = 3.78X10m3 for each U / H Ji:

650.0 88.98 30.00 29690 67220

c,, = 1.950XlO" for B/W = 2.25

-3 for R/H = 3.00 C r , = 1.910~10

and c,, - c,, = 0.040 x 10"

Midship Section Coefficient cz Beam-Draft Ratio BJH FIGURE 24.--Comparison of the effective horsepower of a 6mfoot passenger

vessel with an equivalent standard series vessel

The interpolation factor on B / H is

0.972 2.966

3.00 - 2.97 - 0.03 - 0.04 3.00 - 2.25 0.75

Then the C, for the Taylor Series vesselwitha B/H = 2.97 is

1.910X10" -t 0.04(0.04x lo-') = 1.91ox 10''

where the amount involved in interpolation in this case is negligible.

The Reynolds number for a 650-foot vessel operating in salt water of 59F at a speed of 25.00 knots is

where 1.689 is the conversion factor f rom knots to feet per

I Dimension I Model 1 Prototype

Length, feet Beam, feet Draft, feet Displacement, pounds, tons Wetted Surface, feet

Coefficient

S P M In hmts

Longitudinal Prismatic Coefficient C, Volumetric Coefficient Cp Wetted-Surface Coefficient C,

The effective horsepowers are calculated separately from model test data, the revised Taylor Series contours; nnd the original Tarlor Series contours. TABLE 7.-Particulars for a 6mfoot passenger vessel

40

40,000

36,000

32,000

28.000

24,000

6 = 20,000 .: ; i * w

16.000

11,000

8000

4000

'6

Orlplnol

Revised Phontom

~ ~ 1 s . d wototw

Predicted from Model Tests

R w i u d PhMlom

1.00 f 0

w . 0.90 s

~

8 10 12 14 16 18 eo e2 24 e6 S m In kmcs

second and 1.2817~10-5 is the kinematic viscosity which is read f rom Appendix 7 for salt water a t a temperature of 59F.

The Schoenherr frictional-resistance coefficient for a Reynolds number of 2 . 1 4 2 ~ 1 0 ~ i s , f rom the tables of Ap- pendix 6 ,

C f = 1.397 X

If the roughness-allowance coefficient A C f is assumed to be 0.400~10-3, then

C , = C r + C f + A C f

= (1.910 t 1.397 0.400) low3

= 3.707 x

Since

Power = resistance x speed, P c, x z s v 3

EHP = 550 ft-16 /sec

1991 3.707 X X X 65720 X (25.00)~ X (1.689)3

550 - -

= 33190

where 1.991 is the density in slugs per cubic foot which is read from Appendix 7 for salt water a t 59F. The wetted surface of the Taylor Ser ies vessel is computed from the wetted-surface coefficient which is interpolated from the contours of Appendix 2 using the assigned values of C,, Cp and E / i l .

The preceding numerical example i l lustrates the direct computation of the effective horsepower for a Taylor Series prototype. In the past , the effective horsepower of the so-called "phantom vessel" was used in Taylor Series comparisons. The phantom vessel may be defined as one which has a residual resistance equal to that of the com-

parative Taylor Ser ies vessel but with a frictional r e s i s - tance equal to that of the subject vessel. This concept was probably originated to compensate for the fact that the wetted-surface coefficients of modern vessels a r e , in general , slightly higher than those of the comparable Taylor Ser ies vessels. The phantom vessel procedure implies that the divisian of the total resistance into the components of residual and frictional resistance is an exact one, a fact which few would be willing to concede. Fur thermore , if the wetted-coefficient used for the model differs f rom that for the full scale , a small e r r o r is introduced due to the difference in frictional-re sis tance coefficient between model and full scale , commonly designated as the Df. This e r r o r , although small, leads to inconsistencies in comparisons because i ts magnitude depends on scale and choice of roughness-allowance coefficient.

F o r the aforementioned reasons, it is believed that the "phantom vessel" concept should be discontinued. How- ever , the present calculation technique may be simply adapted to suit this concept if suchis considered desirable. This may be accomplished either by separately calculating the residual and frictional horsepowers or by conversion of the residual-resistance coefficient as follows:

The value of C, based on the wetted surface of the Taylor Ser ies form which is given in the numerical example is converted to C, , based on the wetted surface of the subject vessel , o r

2.537 - 1.867 x x --

2.595 C,' = C, x - c s = 1.910 X

c s ' where the prime indicates that the wetted surface of the subject vessel was used to calculate the coefficient. Then

c,= C, 't Cf + A "= (1.867 + 1.397 + 0.400)

= 3.664xlO-3

1.991 and EHP 3.6~i4XlO-~ x 2 x z96w (25.00)3(1.689)3

550 - -

= 33550

41

where 29690 is the wetted surface of the subject vessel in square feet.

The sample form of Table 6 is se t up for the phantom vessel approach. The f i r s t approach can be used if the form is modified as follows:

1. Eliminate Column 12. 2. Add Column 8 to Column 11 to give Column 13. 3. Calculate Column 15 using a value of A based on the

4. Then Column 13 t imes Column 15 gives the desired wetted surface of the Taylor Ser ies vessel S.

EHP in Column 16.

When model tes t data a r e available for the subject vessel , it i s likely that some of the calculations required a r e already available. The sample form indicates the steps that can be eliminated in these circumstances.

The effective horsepower curve for the subject vessel is shown in Figure 24 together with the effective horsepower curves derived from the original and revised Taylor Ser ies contours. Ratios of the effective horsepowers to those of the subject vessel a r e a l so given. It may be noted that a t speeds up to 20 knots, there is considerable divergence between the predictions obtained with the original and with revised contours. At speeds below 14.0 knots, a t least half of th,e discrepancy resul ts f rom predicting with the EMB- Tideman method instead of the Schoenherr formula plus a roughness-allowance coefficient of 0.0004.6 At 18.0 knots, however, both EMB-Tideman and Schoenherr plus 0.0004 should give the same answer and yet there is a difference of approximately 6 percent. This difference is twofold in origin: f i r s t , the more refined interpolation between a B/II of 2.25 and 3.00 (in this case no interpolation) instead of the original l inear interpolation between a B/!l of 2.25 and 3.75 and second, the refairing process combining the corrections for water temperature, laminar flow, and restr ic ted channel effects.

It may be further noted that for the case i l lustrated, the prediction based on a Taylor Ser ies prototype gives an effective horsepower which is approximately 1.5 percent lower than that for the phantom vessel. The revised resul ts for both the Taylor Ser ies prototype and the phantom vessel characterist ically show a lower effective horsepower than the subject vessel a t some speeds a t the expense of a higher effective horsepower a t other speeds.

VALIDITY OF TAYLOR SERIES COMPARISONS

As previously stated, the results of methodical s e r i e s a r e applicable, in the s t r ic tes t sense, only toforms derived f rom the common parent. The common usage of s e r i e s data to evaluate the performance of specific ship designs assumes , therefore, that the incremental changes in res i s - tance with the given geometrical parameters will apply reasonably to offspring of other parents. Thus, if several competitive designs of a given shiptype a r e to be compared, the Taylor Ser ies data maybe usedtodetermine the r e s i s - tance changes due todifferences in C , CF, and R/II among them, such that the remaining differences in resistance can be attributed to other variances in form. One method of accomplishing such comparisons is through the use of ratio curves similar to those shown in Figure 24. With this device, the effective horsepowers of each designbeing studied a r e expressed a s ratios to i ts equivalent Taylor Ser ies vessel , and the ratios a r e compared instead of the absolute values of effective horsepower.

Direct comparisons with the Taylor Ser ies prototype a r e permissible where the hull type of the subject vessel is similar to the equivalent s e r i e s form. In such cases , the Taylor Ser ies form may be considered as one of the possible alternative designs, and the effective horsepowers can be directly compared without resorting to the intermediary ratios, If, however, it is desired to make direct comparisons o r estimates of effective horsepower f rom Taylor's Series when the subject vessel is of entirely different type, a certain amount of interpretation may be involved. Significant departures in form which may affect the validity of s e r i e s comparisons a r e often manifested by the comparable sectional-area curves. Fea tures pertaining to the extremities of cer ta in types of vessels such as t ransoms, pronounced counters, bulbous bows, and acutely raked s tems, may significantly affect the load waterline length with very little change in immersed volume. As a consequence of this, the magnitude ofthe over -all longitudinal prismatic coefficient is affected by local changes, as may be seen from the sketches in Figure 25. The value of C, for the Taylor Ser ies variation, however, can only be altered by an over-all change in sectional-area curve. One possible solution to this problem would be to intro- duce an "effective" length to be used in the calculation of C,, Cp, and C, for the comparative Taylor Ser ies vessel.

42

This concept can be illustrated by a numerical example involving Figure 25a. The sectional-area curve showing the t ransom is taken to represent a modern 664-foot cruiser . If the actual length of the subject vessel L , is used to define the geometrical parameters , Cp = 0.613, C ~ = 1 . 9 5 x 1 0 ' ~ , B/H = 3.00, and C, =2.641. If, however, the sectional-area curve for the c ru iser is extended to form an ending similar to that of the Taylor Ser ies sectional-area curve as shown by the broken line, then the effective length L , is determined. The geometrical parameters based on L 2 a r e cp = 0.595, Cv = 1 . 7 8 ~ 1 0 - ~ , B/H= 3.00, and C, = 2.641. The predictedeffective horsepowers for a speed of 30 knots and standard conditions a re :

f rom model tes t s of subject vessel f rom Taylor's Series using L , from Taylor's Ser ies using L ,

36940 40070 36420

It can be seen that the se r i e s prediction based on L , is approximately 8 percent higher and that based on L , is 1 percent lower than the model tes t prediction ofthe subject vessel. The use of the effective length concept for prediction of the effective horsepower of diss imilar ship types does not appear unreasonable upon further consideration. If the s te rn of the longer Taylor Ser ies vessel , length= L , , is cut off to form a t ransom and the resulting length is L , , experiments have shown that, within reasonable l imits, the change in total resistance would be small. Thus the closer prediction using the effective length appears to be valid.

Obviously the use of an effective length does not always reduce the Taylor Series predictions. If the subject vessel has an acutely raked s tem, no bulb, or a relatively large counter, the effective length would be reduced with co r re - spondingly higher values of Cp and Cp. Fur thermore , Figure 22 shows that if Cp. is reduced beyond a certain minimum, a further reduction in Cp would actually result in an increase in the predicted effective horsepower.

Direct comparisons with Taylor Series will a lso be affected if the midship coefficients o r shapes of the subject vessel differ radically f rom that of the ser ies . However, Taylor has previously shown that relatively small changes of resistance resul t f rom large changes in midship coefficients when the other parameters a r e held fixed.l

43

To summarize, the Taylor Series data provide excellent c r i te r ia for selecting the over-all coefficients of various ship forms. They also provide a good intermediary for evaluating the performance of competitive forms of a given ship type. The data may be used for absolute comparisons o r predictions of hull types similar to those of Taylor Ser ies , but for widely dissimilar types, i t may be necessary to redefine the geometrical parameters on the basis of an effective length.

Counter Stern

Tronsom Stern

Roked Stem

FIGURE E.-Effect of sectional-area shape on the selection of geometrical parameters for 'I aylor Series comparisons

The broken line indicates the sectional-area curve for a Taylor Series ves se l of an equivalent Cp based on load waterline length

USE OF THE REVISED TAYLOR SERIES CONTOURS

OTHER THAN SCHOENHERR WITH FRIC TIONAL-RESISTANC E FORMULATIONS

When dealing with residual resistances, the consequences of a change in frictional-resistance formulationare always a source of concern. The question i s whether the laborious task of reducing the original data would have to be repeated. This question can be answered in the negative especially if the proposed formulation i s a single function of Reynolds number. Such a change could be accomplished simply because of the existence of certain fixed parameters. Since all of the tests were conducted with models of the same length, namely 20.51 feet, the difference between the frictional- resistance coefficients can be expressed as functions of speed-length ratio and basin water temperature. This may be illustrated by considering the possibility of the use of Gebers formula which gives values for the frictional-resistance coefficient in the model range considerably different than those obtained with the Schoenherr formula. Since the differences in frictional- resistance coefficient a r e reflected as a change in residual-resistance coefficient, this quantity will be denoted AC,. A set of curves of A c, versus speed-length ratio and basin water temperature i s shown inFigure 2 6 . Thus, i f it were desired to determine the effective horsepower predicted from the Gebers formula, the procedure for calculation would be essentially the same as that giveninthe preceding section, except that A C , would be added to C , and c f would be deduced from Gebers formula.

Similar charts could be prepared for establishments using other frictional resistance formulations o r , i f it is desired, the conversions can be made on the total effective horsepowers by the methods giveninReferences 6 and 8.

Speed-Length Ratio

FIGOHL 26.--Residual-resistance coefficient corrections These values are added to the residual-resistance coefficients of the revised Taylor Series to convert from Cr based on the use of the Schoenherr Formula to C, based on the use of the Gebers Formula

44

REFERENCES

l .Taylor, D. W. , "The Speed and Power of Ships," Third Edition, U. S. Government Printing Office, 1943.

2. "Minutes of the Seventh Meeting of the American Towing Tank Conference," 7-8 October 1947.

3.Rossel1, H. E. and Chapman, L. B., "Principles of Naval Architecture," Volume 2, Society of Naval Architects and Marine Engineers, 1939.

4.Todd, F. H. and Fores t , F. X., "A Proposed New Basis fo r the Design of Single-Screw Merchant Ship F o r m s and Standard Series Lines," Transactions of the Society of Naval Architects and Marine Engineers, 1951.

5. Landweber, L. and Gert ler , M., "Mathematical Formula- tion of Bodies of Revolution," TMB Report 719, September 1950.

6. Gert ler , M., "The Prediction of the Effective Horsepower of Ships by Methods in Use a t the David Taylor Model Basin," TMB Report 576, Second Edition, December 1947.

7. Landweber, L., "Tests of aModelinRestrictedChannels," TMB Report 460, May 1939.

8.Gertler, M., "A Method for Converting the British@ Coefficient Based on the Froude "0" Values to a@Coef- ficient Based on the Schoenherr Formula," TMB Report 657, Second Edition, June 1949.

289733 0 - 54 - 5 45

APPENDIX 1

CURVES O F HALF-BREADTH AND WATERLINE ENDINGS VERSUS LONGITUDINAL PRISMATIC COEFFICIENT FOR DERIVED FORMS OF THE TAYLOR STANDARD SERIES

The half breadths are expressed as ratios to the half maximum beam at the load waterline.

1-1


1-2

39

38

37

36

35

34

0.50 0.55- 0.60 0.65 070 0.75 0.80 0.85 33 Longitudinal Prismatic Goeff icient

1-2

I

P b 2 L

0

Longitudinal Prismtic Coefficient Lcqitudinol Privnatic Goefficbnt

0.1 WL

d

P P

P Ib Q L

0

1-3

0.1 WL

1.0

0.9

0.8

0.7

$0.6

3 .-

3

t 90.: f

m

.Q c B a3

1.0

0.9

0.8

a7

a61 P P

0.5

! g P

a3

a2 a2

ai QI

O 0.50 0.55 0.60 0.65 030 0.75 0.80 a85 0.50 a s Q60 0.68 010 075 0.80 a= O Longitudinal Prismatic Coefficient Laqitudinol RiMlalic CoefficM

1 -3

0.2 WL

1.0

c m

P 9

f b

0 Longitudinal Prismatic Coefficient

E 8 m

Longitudinal Prismtic Coefficient

1-4

0.2 WL

1.0 1.0

0.9

0.8

0.9

0.8

Stotion 20 22

Stotion 20

I?'

16

24

14

0.7 0.7 26

) I 2 0.6 $

.- 5 s"

$0.6 .- i = I" 2 0.5

28

0.5 ' r m $ 10

= a4

0

= I" a4 r

30, 8

0.3 0.3

32 0.2 0.2

ai 2 I

F. P. 36 ZL 3 8

0.55 0.60 0.65 070 0.75 0.80 Longitudinol Prirmotic Coafficient

0.50 0.85 0 0.50 0.55 0.60 0.65 070 035 0.80 0.85

Longitudinol Prismatic Coefficient

1-4

0.3 WL

1.0

0.9

0.8

0.7

1.0

0.9

0.0

0.7

0.6 2 .- I"

= I" 9 0.5 f

? m

2 0.4

0

c

0 .- c B 0.3

0.2

ai

0.50

0.3

a2

~ 10 0 55 060 0 65 070 0 75 080 0 85 0 50 0 55 060 0 65 070 075- ~ 080 0 85

Longitudinal Pnsmhc Coefficnnt Longitudinal Pnsmhc Coefficnnt

1-5

0.4 WL

I .o S t a h 20

2 2

24.

26

1.0

0.9

0.8

Station 20

16 18

0.9

0.8

14

12

28

0.7 0.7

!i m 10 E

0.6 .- 9

f 0.6 .- 30

32 = a4 I"

8 .Q

a c

6

0. z 0.3

3 4 4 a2 a2

2 ai ai 36 37

3- F. P.

0.50 0.55 0.W 0:65 010 0.75 0.80 0.85 Lofqitudinal Prismatic Coefficient

0 0.50 0.55 am 0.65 070 0.75 0.80 0.85

Lonqitudinol Prismatic Coefficient

1-6

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