Application of ship resistance theories to the design of full forms

1. Introduction

The improvement of resistance characteristics is one of the important factors in the process of designing hull forms with better hydrodynamic performance. Quantitative prediction of ship resistance by use of existing wave resist-ance and viscous resistresist-ance theories is not completely

satis-factory due to the assumptions and sirnplifications

in-volved. However, the theories can provide useful qualita-tive information for hull form improvement.

Our efforts towards the improvement of resistance

characteristics aim to reduce pressure resistance rather than skin friction resistance, since the latter resistance component is insensitive to the change of hull form. The pressure resistance is composed of a wave resistance com-ponent and a viscous pressure resistance comcom-ponent. For fine hull forms for high speed vessels, the wave resistance component dominates in the pressure resistance, and for full forms for low speed vessels, the viscous pressure resistance dominates. For fine hull forths, the linearized wave resist-ance theories have played an important role in hull form improvement. A number of papers on this topic have been published and have been applied to the design of high speed liners and container ships. These studies are categorized into the following three major areas:

Minimization of Michell integral with proper side conditions

Effects of bow and stern bulbs on wave cancellation Application of wave pattern analysis to hull form improvement

One of the reasons for such prolific studies is the simpli-city of the linearized wave resistance theories Which are

expressed as a function ofship geometry.

With increase of fullness of ships, nonlinearity in the

mathematical formulation of the boundary conditions has to be taken into account, and at the same timeWave

break-ing around the bow and viscous flow separation around the stern present difficult problems. Due to these complexities of flow, progress in hydrodynamic treatment of full ship

forms has been slow when compared with those on the application of the linearized wave resistance theories.

To overcome the difficulties in analytical treatment, it is the recent trend to solve nonlinear differential equations directly by numerical methods for free-surface conditions and for ship viscous flow. These approaches are necessary, especially when dealing with full ship forms for which

non-*Dr. Eñg., Nagasaki Technical Institute, Technical Headquarters

MTB 166 January 1985

Application of Ship Resistance Theories to the

Design of Full Hull Forms

Eiichi Baba *

With increase of fullness of ships, the hydrodynamic approach becomes difficult due to the complexitiesofthe flow around ships.

In the Mitsubishi Nagasaki Experimental Tank efforts have been devoted in the past decade to the development of higher order wave and viscous resistance theories The former is called low speed wave resistance theory and it can take into account the non linear effect of free surface boundary conditions The latter is a higher order boundary layer theory which takes into_{account pressure varia} don across the boundary layer, so that it can predict ship viscous resistance. In the present paper, explanations are made on the efforts to apply these higher order theories to hull form improvement of full ships by introducing_{szmplzficatzons in the calculation} procedures, aimed at easy use in the early stage of hull form design.

linearities have to be considered. Progress in the numerical studies is remarkable and will be a useful tool in the near future for hull form design.

In parallel with the progress in such direct numerical

methods, there is a continuing need to get information on

hull form improvement through simpler ways involving

shorter periods of evaluation so as to meet with the urgent design tasks at hand.

Though it was an experiment oriented method, Taniguchi et al. (1966) developed a design method fOr full hull forms based on the so-called hull form separability concept which was derived by the analyses of a large number of test results on full ship modelsW. In this method, entrance, parallel

middle body and run parts are treated separately. Since it has been found that the entrance form is responsible mostly for the wave resistance, the entrance form is selected among ship types so as to achieve low wave resistance. Further, since it has been found that a well designed run part has al-most negligible effect on wave resistance, the run form is

selected among ship types so as to get better propulsive

performance with favourable stern flow and low viscous resistance. The parallel middle part is adjusted to satisfy the displacement volume required. An outline of the method is given in the Appendix. This design method has served as

a standard design method for full hull forms in Mitsubishi

Heavy Industries, Ltd. Entrance and run data have been renewed steadily by adding new hull form data.

Hull fotm itself, however, tends to change due to de-mands of the era. For instance, in recent years there are

trends of shallower draft and wider beam, hull fOrms with propellers of large diameter and so on. Theoretical guidance for the improvement of hull forms is highly desirable to

meet with such changing demands. In the Nagasaki Expeti-mental Tank, efforts have been devoted to developing im-proved wave and viscous resistance theories which can be

applied to full fOrms. For instance, a higher order wave

resistance theory which takes into account the free-surface nonlinear effects has been developed and it is called low

speed wave resistance theory. Also a higher order ship

boundary layer theory which takes into account pressure variation across the boundary layer has been developed so

that the theory predicts viscous resistance of ships.

In this paper, explanations are given on the efforts being

undertaken in the Nagasaki Experimental Tank for the

development and application of the low speed wave resist-This paper was presented at the Centenary Conference in Marine Propulsion of the North East Coast Institution of Engineers

ance theory to the improvement of entrance forms, and for the application of the higher order boundary layer theory

to the improvement of stern forms with respect to

resist-ance characteristics.

2. Near field flow measurement and improvement of ship wave and viscous resistance theories

The application of wave pattern analysis was once

con-sidered for the improvement of wave resistance theory,

because it was thought that accumulatel information must be mcluded in the wave system which has been generated by the complicated process of flow mechanism around a ship. It turned out, however, that the measured wave pattern resistance was much smaller thhn the wave resist-ance determined by use of the HughesProhaska method, and it was thought to be difficult to get information for the

improvement of the conventional theries. From wake surveys far behind ship models it was discovered that a large part of the flow energy imparted by a full ship dissipated into a disturbed free-surface phenomenoi, which isusually called wave breaking. It was also though that there was an interaction between wave system and ship viscous wake.

We had thus recognized that the far field information such as a wave pattern is inadequate for understanding the free-surface flow induced by a full ship form. We then turned our research direction to the observatibx and measurement of near field flow phenomena around a hip, aiming at the development of an appropriate mathemaical model of the flow. Various flow measurements by use of five-hole Pitot tubes were conducted around the bow of full ship models. As one of the results, it was found that the flow around a slow speed ship can be expressed fairly wll by double-body flow except the flow near the free-surfac as shown in Fig. 1. This finding became a turning point in ourunderstanding of the flow around a full ship, and motivated the present

author to use a flow model introduced earlier by Ogilvie (1968) based on his physical insight into the free-surface

phenomenon in low speed limit2'. Ogilvie conjectured that the free-surface flow induced by two-dimensional submerged body translating with low speed can be express-ed as the sum of double-body flow and surface layer flow which represents wave motion. A concppt analogous to

viscous boundary layer theory was thus introduced in the free-Surface problem. Encouraged by Ogilvie's basic study, the present author and Takekuma (1975) extendedOgilvie's

theory to a three-dimensional theory for suface piercing

bodies. The extended theory is now called low

speed

wave resistance theory. Since thedouble-body flow is used as zeroth-order approximation, the theory is applicable to even extremely full forms which are farbe'ond the applica-tion of the convenapplica-tional thin ship theory. [Fhedevelopment of this low speed theory was indeed a breakthroughtoward an analytical treatment of wave resistanc problems of full ship forms.

As mentioned above, the near field flow measurement played an important role on the development wave resist-ance theory. This approach was also succesfully applied to the development of ship viscous resistance theory. Bound-ary layer theory has been applied to the calculation of the

Full ship model 5-hole Pitot tube LI "/LI=cb(x,y.:)/U V/U = (x. y, /U=c&r (x.y.:)/U -100 "lu.V/U. /U -200 0 0.5 1.0 Velocitycomponents at x=-175mm, y=O from FP

u/U V/U,"lu

100

1.0 200

-200

0 0.5 1.0 0 05

Velocity components at Velocity components at

x-350mm, y=O from FP x0, y=400mm from FP Fig. I Flow measurement near the bow of full ship model

viscous flow field around a ship since Uberio's pioneering

work in l969. Improvement of theories has been steadily made in various directions. The conventional thin boundary layer theory is based on the assumptions that the boundary layer thickness is much smaller than the radius of curva-ture of the flow stream and the pressure is constant in the direction normal to the body surface and usually putequal to that on the body surface in the inviscid flow. Due to these

assumptions, only frictional resistance, which is not so

sensitive to hull form change, is calculated. Therefore, the conventional boundary layer calculation has not been used effectively in hull form design. However there has been continuous need to improve the theory to such an extent

that the theory could predict favorable hullform modifica-tions. With increase of fullness of ships, the thickness of the boundary layer increases toward the sternand becomes comparable with the radius of curvature to such a degree that the curvature effect can not be neglected anymore. Considering this fact, Nagamatsu (1979) developed ahigher order ship boundary layer theory which takes into account

the curvature effect, and made calculations of boundary layers for various ship forms5. In this higher order theory, the variation of pressure across the boundary layer canbe

predicted in terms of curvatures and thickness parameters

of the boundary layer. Calculated pressure variation was found to be in good corrlation with the measured results for various ship forms as shown in Fig. 2. This experiment-al and theoreticexperiment-al study encouraged us to movetowards the calculation of viscoUs resistance by integrating pressure and

tangential stress over the hull surface(6). By the develop-ment of this higher order theory, it became possible to find

a relationship between local hull geometry and viscous. resistance components.

Near field flow measurements conducted in the Nagasaki Experimental Tank thus contributed greatly to the

Bodyplan and measuring positions Pressure variationin boundary layer L8.00m 8/d 1.600 L/8=1O,00C0.444 LwL

IF!

Measuring positions L =S.00m B/d=2.355 L/B=6.468 CO571 Lwi (t L=8.00m B/d2.760 L/B6.00 CO.802 Lwi 121 Wigley model o Measurement Calculation s.s.I 0 00 0 0 50 100 150 200 01' _{0 0} I_S0000000 00 % 0 50 100 150 200 c s.s. 2i1 2 50 100 I0 101 5.5. l,2 1u l0 0.000°.--0.11. © 150 0 -5 - l0 lb lrnnl 200 5.5. 2 5.S. 2 W O2o 00

: :°°

0 100 200 0 100 200 0[ © 00 0 0 .0 0 _-0.2 0 0 D 0.3 0 100 200 Tanker model

Fig. 2 Pressure variation in boundary layer

Eq. (2) 'is written further in an asymptotic formula for low speed:

.sec2O f

exp {ivsec20(xcoso+ysino)} (6) Though this expression is simple, lengthy numerical calculation is necessary to calcu-late _{the wave resistance of conventional} ships. For the convenient use in the initial

design stage of lull form, siniplifications are

made of the wave resistance fOrmula as described in the next sectiOn.

3.2 Simplified wave resistance formula for full forms

For the calculation of wave resistance of given ships, the following silnplifications are introduced:

Full hull fOrms at low speed's are considered.

' The hull forms are represented by two-dimensional cylinders of infmite draft, whose wãterplane-area curves

are the same as the sectionalarea curves of the given hull forms.

These' simplifications are considered to be acceptable when we notice that the free-surface waves of low speed ships have small wave length and the effect of wave motions appears only near the free-surface.

Based on the above simplifications, the function D(x,y)

is calculated from the two-dimensional flow around the

cylihder by means of conformal mapping. The equation of the sectional-area curve is expressed by Fourier expansion through the relations:

x y

L/2 =cos/.9, L/2 2B0sin n/3

where, L is the length of a body, and B,, is the Fourier

coefficient. Then the amplitude functioh iS rewritten s:

A(8)₌ sec2O_{Ffl2f2,8 F(/)exp{i ecOW(fl o)}}

(7)

where, Fn _{LJ//E Froude nuniber and}

F(/3)=v.(l_u2_v2),

MI (fi, 0) rcosi9 cos0+( 2Bnsin9)sin 8

where, u and v are the velocity components in x and y coordinates respectively, and F) is determined by the

double body potential and is dependent on the body

geo-metry alone. The approximate values of v-and (l-u°-v) are

obtained by the use of a conformal mapping method

developed by Moriya for 2-dimensional wing sections9:

lot

MTB 166 January1985

Fig. 3 Coordinate system

3

ce © (41

ment of rnproved theories applicable to full hull forms. In the following sections, explanations are made of the efforts in applying these theories to the design of full hull forms.

3. Application of low speed wave resistance theory

3.1 Wave resistance fOnnula

The wave resistance formula based on the low speed wave resistance. theory is expressed as:

R=n.pU2fo2IA(8)I2cos3O dO (I)

where,

A (0) = -'-b-sec3off dxdy D(x,y) exp{ivsec2e(xcosO

+ysin8)} (2)

D(x,y)={d(x,y,O)(x,y)}+

{dy(x.y,O)r(x,y)}

(3)

5'7j-(4)

Ø(x,y,z) is the velocity potential for the double body

potential. The coordinate system is shown in Fig. 3. The author proved that this amplitude function A(0) is expressed as the sum of the following three components7:

A(0)=AS(0)+AL(0)+Aj-(8) (5)

where As(0) is due to the singularity distribution over the body surface. This term gives the wave resistance formula derived by.Havelock(8). AL(0) is due to the singularity

dis-tributions around the intersection between the body and the still water surface. This term is the so called line integral term. AF(0) is due to the free-surface disturbance

ex-pressed in terms of products of derivatives of double-body potential. It is a characteristic in the low speed theory

which contains this third term representing the nonlinear effect of free-surface condition..

0.0. I no '0 50 100 150 O,.lt h L - iF_i..o.. 0 00t0 50 10 150 © 00 0 50 5

Cargo shipmodel ln,n,)

0.I

i210

-0.1

0.1

vsiñiS(++nB') (nBcosn9).

1(--sin/3+nBesinn9)2

lu2v2=

_(de\2

_(d\2

\d9J

\d131

For full forms whose sectional area curves are convex, the integration in Eq. (7) with respect to 3 can be carried out by the stationary phase method. As a result, the wave resistance coefficient is writtenas(1O).

C=

_pU2L2 = .f7r/2FA(G)I2cos38d8 = Cei ± C'u2 + Cui

where, Cei =2Fn6fd8 {FC8i)}2cosO

IW"(i1,)I

2Fn6f'2d8{F(/92)}2 cos8

C2=

C,3 4Fn6f0'2dOF(/9i)F(l92)cos8 VIw"(191, 0) W"(/92,8)I W(92, 0)}±f(sgo{W"(/3i 0)} - sgn{1If"(9i, 0)))]

Coi is the contribution from the after bOdy, c from the:

fore body, and C'u.3 is due to the interaction between waves from fore body and after body. and j3 are the roots of

for 09i

ir/2 aft body 1 3/27r: fore body J In the present paper, considering that the entrance is

mostly responsible for the wave resistance, only the contri-butiOn from the fore body is considered. We have then

2Fn6fd8{FC8i)}2 cos6

c=

wheie, ><

{r(9, 8)

(8) (10) Fig. 5 0.010 0.005 0.30 - 0.35 0.40 - 0.45 0.50 Fe.= U//

Fig. 4 Wave resistance coefficients for various fullness of entrance

0.010 + 0.005 Measured Measured Calculated Calculated 0.30 0.35 0.40 0.45 0.50 FsU//ii

Wave resistance coefficients for different trend of entrance

expression U) is useful for this purpose.

3.3 Introduction .f empirica1 correction factor The sectional-area curve at the FP is usually not zero due to the protruding bulb forward of the FP. If we take into

account the protruding bulb, there exists a concave part around the FP. In this case, the present simplification of

the wave resistance formula becomes invalid because the stational phase method can not be applied to sUch forms. In the present calculation, therefore, only the sectional area curve between the FP and parallel middle body isutiized

Considering this simplification and at the same time

con-sidering the fact that the theory is developed under the

various simplifications and assumptions, a factor whichis

multiplied into Eq. (10) is determined empirically by

comparing calculated wave resistance values with those

experimeitally determined. We may say, therefore, that the

proposed wave resistance formula is a sort of empirica.l formula which is derived by use of the characteristics of wave resistance oflow speed ships.

Needless to say, in the stage Of detailed evaluation of hull form, the full three-dimensional forrñula may be used

for this purpose as shown by the author(). 3.4 Examples of wave resistance prediction

First, in order to examine whether the proposed simple formula can predict wave resistance reasonably, sectional-area curves of different fullness were selected asexamples and the wave resistance was calculated. The results were compared with experimental data of these hull fOrms Fig. 4 shows the results of calculation for different fullness of entrane thus selected. Reasonable agreement is observed both n the qualitative and quantitative sense. Fig. 5 shows another example of wave resistance coefficient curves for sectional-area curves of different trend but with constant

area. A reasonable correlation between calculated and

measured is observed. These results show that the siinpli-fied wave resistance formula is applicable to the prediction

tjl"(19, G)cosj9 coso_(4n2Be sin n9)sinO

For given sectional area curve we determine first the

Fourier coefficient B and then, seeking the roots of Eq. (9), we can calculate wave resistance b' Eq. (10). Since the following relations can be derived

(d\2 d2?

F(9) = - 2(1 + 2nBe)3 I

11+7 J

sin9 W"(9,8)sin2/9sin8 provided Mf'(fl, O) 0 d2i

d2

dO _d'?2

an alternative wave resistance formula is denved in terms of hull geometry as

dO 5 (dO\2 Ce=8Fn6( 1 + 2nBn)6f° d d

\d)

1 \a'ç)J

For the systematic study of the wave resistance

char-actenstics for various sectional area curves it is conveni

ent to introduce a mathematical expression of sectional-area curve in terms of pafameters which are closely related to the hull geometry. For instance, the Taylor-Gertler type

of wave resistance and at the same time to the improve-ment of entrance form in the early design stage.

4. Application of higher order boundary layer theory to

stem form iniprovement

4.1 Viscous resistance formula

Prediction of viscous resistance (form factor) in the

de-sign stage used to be made by employing' empirical for-mulae which were derived from statistical analysis of a large number of experimental data(1)(12). This prediction

method has high accuracy for certain specified types of ship, but its application is limited to the range of

experi-mental data. Furthermore, since only a few principal para-meters are included in these empirical formulae, the

relationship between viscous resistance and ship form is not

direct. Therefore, for further improvement of hull fortns

with respect to viscous resistance reduction, efficient use of theoretical means is desirable. Since it has been known em-pirically that the stern form has a significant influence on the viscous resistance characteristics, theoretical investi-gation into the effect of stern form is valuable for practical

application.

A higher order boundary layer theory, which takes into account the pressure variation across the boundary layer due to the surface curvatures, was developed by Nagamatsu(6) as mentioned in Section 2. The viscous resistance formula based on the theory is written as the sum of viscous

pres-sure resistance and frictional resistance RF. Viscous pressure resistace is obtained by integrating pressure Pw over the hull surface Sr

(14)

where, Po is the hull surface pressure in an inviscid fluid and Pe is pressure at the outer edge of the boundary layer. i is the x-component of the outward normal unit vectoron the hull surface. The first term on the right hand side of Eq.

(14)

_{represents the pressure resistance caused b' the}

pressure variation across the boundary layer and the second one caused by the displacement effect. In this calculation the change of wetted surface due to the wave formation is

neglected for the sake of simplicity.. This simplification might be accepted for slow speed ships.

Let us define pressure coefficients as

_p,p

P0P

CP

+pL/,

+pU,

Cpe

±pU2 where, p represents reference pressure.

The viscous resistance coefficiëntC is written as

R2

_{Cpi+ Cp2+ Cp3}

rpU S

cpi= .j_° df° (Cpe'

olda

c2=frf° dji°,,2{Cpocp}o1da

Cp3=-J

def°

{c. c}

1da

where,L2/4 .g dda corresponds to the surface element,a = 0 at the still waterline and a = -ir/2 at the bottom

centre-line. ç0 is the position of breakdown in numerical calcula-tion in the three-dimensional case and the posicalcula-tion of the

0.010 0.008 0.006 0.004 0.002 Rn=5X1Q =0 0 Z C=R/+pU'L z 0.002 0.004 0.006 0.00 C (exp.) Crp MTB 166 January 1985 Fig. 6

Viscous resistance coefficients calculated and measured for two-dimensional section shapes

0.0 10

5

zero shear stress in the two-dimensional case. Cp3 is a resistance component after the point of numerical

break-down or separation. This is estimated assuming that the pressu±e (c0 - c) is constant after that point. Nagamatsu derived the following results for pressure values:

Cpe Cp

2 (-.)2KI3( Oh) K23 822}

(I 6)

CP3=.t-.S; A(e)C(e)

(17)

f{c0c}dA

fdA at

=o

Ue is the potential velocity at the Outer edge of the

boundary layer, A() is a nondimensional sectional area at and X denotes arc length along the girth. K13 and.K23 are longitudinal and transverse curvatures in the streamline coordinate system on the hull surface respectively. Further, there are the following relations:

811=f t(l--_)d-,

822= 'j

7Tf d

where, is the coordinate normal to the hull surface,u, v are the velocity components in the direction of stream-line and cross flow respectively.

The frictional resistance coefficient C'F is expressed as

CF=

-

_{{f° df° Cf ada+J d f° Cj ada}}

(18)

where, Cf = r/(1/2pU2)and r

is the xcomponent of

wall shear stress r. The second term of the right hand side

of Eq. (18) can be omitted because its contribution is negligibly small.

It is

_{a characteristic of the present theory that the}

viscous resistance is calculated by integration of pressure and shear stress Over the hull surface. Because of this,we

can see the relation between local hull geometries and the components of viscous pr6ssure resistance.

Fig. 6 shows results of calculation by applying this

theory to NACA tivO,.dimensional symmetric aerofoil sec-tions of irarious thickness ratiOs after sOme modificasec-tions of the calculation formulae mentioned above for

tivo-dionen-sional use. Calculated drag coefficients agree well _with

NACA experimental data. In this calculatiOn, it is assumed

that the laminar boundary layer starts from the leading

edge and the Granville's transitiOn criterion was used in predicting the transition position(13). If laminar separation has occurred before transition, turbulent boundary layer is

Fig. 7 Viscous resistance coefficients calculated and measured for various ship models

Fig. 8 Examples of optimization of two-dimensional symmetric bodies

the increase of drag with increase of thickness ratio is

mostly due to the increase of pressure drag. It is also'noted that the frictional drag is rather insensitive to the thickness ratio. Fig. 7 shows results of calculation for conventional

ship forms. In this calculation, it is assumed' that the turbulent boundary layer develops from 2.5%L down-stream from the fore-end of a body. This corresponds to the flow condition of full-scale ships or ship models with a turbulence stimulator near the fore end. For fine forms of

Cb < 0.6, agreement with experiment is good. It is also noted in the three-dimensional case that the frictional resistance coefficient is almost the same among ship models used in the calculation. In the present calculation, Ca,, of

full 'forms is overestimated. This implies the difficulty in treating thick boundary layers, even though pressure

varia-tion across the boundary layer is

taken ihto account.

Though the present theory is not satisfactory in quantita-tivê prediction for full forms, it has been confirmed 'that this theory predicts the qualitative tendencyrather well for the resistance characteristics of various full forms. Due to this fact, its application to hull form improvement has been encouraged.

4.2 Optimization of two-dimensional symmetric body By use of the above mentioned viscous resistance theory, attempts were made to find ship forms of low viscous resistance. First, the two-dimensional problem was studied to examme the validity of the procedure of optimization.

In the present study, Hooke and Jeeves Direct Search Method('4) was' used to find minimum values where

coordi--+pf6L 0.005 0.010 Ca Ca ICPI (Transition point 50%L( (Trans. p. 33%L) (Trans. p. 32%L)

Fig. 9 Viscous resistance components for various two-dimensional section shapes

nates of several discrete points on 'the body surface are chosen as design variables describing the body shape.

The minimizat,ion problem was solved under the following constraints,

Chord length is constant. Sectional area is constant.

(.3) y-coordinate is positive everywhere.

(4) y-value increases' monotonously from the leading edge up to' the maximum value and then decreases mono-tonously to the trailing edge with increase' of x.

In this calculation a component of viscous resistance due

to the displacement thickness effect is neglected for the sake of simplicity; in other words, it is considered that the calculated results maintain validity qualitatively even if the boundary layer calculation was made without the displace-ment effect. The detailed explanations of theminimization procedure are reported in the author's recent paper(15). Examples of the optimum form with minimum viscous

resistance at Reynolds number of 106 and i07 are' shown in

Fig. 8 and compared with their original form, an elliptic

shape with length-beam ratio of 4. Thç optimum forms

ob-tained at each Reynolds number are considerably differ-ent from one another. The optimum foim at 'Rn = 106

looks like a conventional wing section. On the otherhand,

the optimum form at Rn

i07 looks like a so called

laminar wing section.

Fig. 9 shows' a comparison of components' of viscous

resistance of various section shapes and the optimum shape determined at Reynolds number of 106. FrOm this

comparison it is noticed that the optimum form (tic

0.29) has a smaller CF than that of NACA 0025 section .shape at Rn = 106. ThiS is due to the shift of transition point toward the trailing edge, and Cvpi value is markedly reduced and comparable with that for the section shape of much Smaller tic.. It is further noted that C0p3, the

corn-ponent due to the pressure resistance after theseparation point, increases with increase of C'vpo for the cases of

NACA section shapes at Rn = 5 x 100. It is then assumed that the variation of viscous pressure resistance with respect to hull form change is approximately represented by the variation of the component Cpi Fig. 10 shows a compari-son of the distribution of viscous resistance components for

NACA 0025. and the optimum form at Rn = l0. CF(S)

Low drag form

(r/c=0.29 Ru10' NACAOO25 Rn-5X 10' NACAOO18 Rn=5X 106 NACAOO12 (Trans. p. 32%L( C. L/B 0,085 0.007 0.832 Rn 1.2 X 10' Measured 0.57 6.47 Calculated Ca - C,r Meas. C 0.57 6.50 Cal, Ca Meas. C 0.60 7.50 Cal. Ca Meas. C 0.80 6.00 Cal. Ca Cr Meas. C 0.83 6.00 Cal Ca Cu C,'pi C,pa Ca Ca NACAO025 Rn=10' (Trans. p. 41%L( (Trans. p. 33%L) NACA0009 Rn 5 X 10' Original form U Rn 10' C=0.0475 Rn'lO' C;'=D.0366

Op5mum form _{Ca=0 00789}

1.2 1.0 0.8 UilU 0,6 0,4 0.2 0.0 X 1.0 0.5 0.008 - 0.004 Optimam shape R,, 10 1k/u 1.0 Re=10 0,/u 0.0 12 0.0'10 c'p 10008_c. - 0.005 0.04 0.O02 0.0 0.010 0.002 0.0 01.0 _{0 i:o'}

Fig. 10 Viscous resistance components for 'o-dimensional section shapes

and C,'-() are the integrated values up to the point

from the fore end. A large difference of. CO3,,' () between both section shapes is observed near the trailing edge. The

Optimum section shape has lower values of the Cpo component near the trailing edge. This reduction is due to the concave trend of section shape near the trailing edge.

Through the above mentioned study for a two-dimen-sional body, it is confirmed that the optimization technique used here gives reasonable results.

-4.3 . Optimization of conventional ship forms

In the optimization procedure, many iterative calcula-tions are necessary to seek the optimum form. Therefore,

the cost and time required for a single cycle of iteration must be minimized. For this purpose, a small cross flow assumption is employed to simplify the boundary layer calculation for conventional ship forms. Further, for the

convenience of computation, the potential flow around the

ship hull is' calculated by means of slender body theory

derived by Tuck and von Kerczek(16). Numerical

break-down is inevitable near the stern end in the boundary layer calculation around a ship by integral methods. To

minimize estimatiOn error of the Cvp3 term after the point of numerical breakdown for the original ship form and its modified forth, the boundary layer calculationwas stopped at 3.5% L before AP. Under these simplifications, optimiza,

tion of stern form of full ship was carried Out with the following constraints:

Displacement is constant.

Ship length, breadth and draft are constant.

Fore-body is unchanged. Only the longitudinal

distri-bution of sectional area of run part is varied without changing the shape of frame line.

Fig. 11 shows an example of the optimum run form

Fig. 11 Distribution of local viscous pressure resistance

for a three-dimensional ship form

0.6

0.4

0.0

Original ship ,, Optimum ship

Optimum ship Original ship 0.2 MTB 166 January 1985 r C, Ge

Fig. 12 Distribution of viscous resistance components

determined by the direct search method In this figure,

dis-tributions of the C,,o component along the girth are

shown 'and compared with those of the original hull form. In the case of the original ship form, a large amount of

vis-cous pressure resistance ( C0 component) is produced

near the stern end. On the other hand, the viscous pressure resistance of the optimum hull form increases a little near

the shoulder part and reduces remarkably near the stern

end. It 'is worthwhile to notice the negative pressure

resist-ance due to the concave bill surfae at the upper third of

draft near the stern end.

Fig. 12 shows a comparison of the longitudinal iiicre-ment of viscous resistance components. There is no appreci-able difference of the frictional resistance coefficient _CF'() between the' original and the optimum form. A consider-able difference is observed, on the other hand, in the curves' of viscous pressure resistance coefficient. _{c,,, '(s) of the}

optimum ship form increases steeply near station 2 and

decreases slightly near the stem end and finally becomes smaller than that of the original ship. This trend coincides

with athat shown in Fig. 10 for two-dimensional section shapes.

4.4 Simplified viscous resistance formula

In the results of optimization for two-dimensional sec-tion shapes and three-dimensional hull form it is observed commonly that a concave trend (negative curvature) of the

after part of the body contributes to the reduction of

viscous pressure resistance., From this result we may

con-sider a possibility, within the franiework of the present theory, to represent three-dimensional ship forms' by two-dimensional struts with the' same sectional-area curves for the simplification of calculation procedures as introduced

in the problem of wave resistance. There is, of _{course, a}

7 '.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 1.4 0.4 X 0.2 0.0 0,8 s.s 8 9' AP FP

1.6

-1.5

1.4

limitation of its applicability to detail design of hull form, since the effect of frame line forms on resistance

character-istics are not taken into account. However, as far as the

determination of favorable direction of the modification of sectional area curve is concerned such simplification may

be accepted when the trend of frame line shape is not

changed largely from that of the original form. When we

represent viscous pressure resistance by the component

Ci, the viscous pressure resistance is written as: Rrp

Ld rt

I U,\?

= K13 (6o'

where, d is the draft of ship, and it is assumed that pressure distnbution

is uniform m the direction of draft

This

expression coftesponds to the viscous pressure resistance due to the part of the strut. with span length of d. That, is, bottom effect is not considered. S is the wetted surface of the correspon4ing ship. In the two-dimensional case, the curvature K3 is written as

d2t7 dE2

KI3=

{ +

(d?7 )2 }3/2.

FOr further simplification, it is possible to approimäte the boundary layer thickness parameters

and 0j with

the expressions in terms of. potenal flow. Ue around a body and Reynolds number,, for instance, by use of the method due to Truckeñbrodt'7. Since the potential flow calculation can be carried out by means of conformal

mapping for a given sectional area curve 17(e), the' viscous pressure resistance formula can be. evaluated easily as in the case of. the simplified wave resistance formula.

For the practical use of such a formula in predicting vscous resistance in terms of form factor, an empirical

correction is required as employed in wave resistance cal-culation since the formula represents only a part of viscous 'pressure resistance The empirical correction factor can be

determined by statistical analysis of experimental data

06 ' 0.7 0.8

Fineness ratio !1/8 ) LilB'

Fig. 13

Form factois foi various fineness ratio of run

References

Taniguchi K., Watanabe K. and Tainura K., On .a New Method of Designing Hull Form of Large Full Ship, Based on the Separability Principle of Ship Form J Soc Nay Arch Japan VoL 120 (1966) 36-45

Ogilvie T.F., Wave Resistance: The' Low Speed Limit, Univ.

2.5 3.0

Beam draft ratio' 8/a

Fig. 14

Form factorsfor

various beam-draft ratios

3:5

through exploiting this simplified formula. Figs. 13 and 14 show some results of calculation by use of the simplifie4 formula for viscous resistance for different fineness of run

and for various Bid ratiOs. Qualitative trend of the (1 +

K) value is well predicted by the present formula. 5 Concluding remarks

In the present paper, explanations were made of the

efforts to improve full hull forms by use of the higher order wave and viscous resistance theories. For the sake of con-veniènce, in the early stage of hull form design, simplified calculation procedures were proposed whëre the hull forms were represented by their sectional-area curves. The wave resistance formula is used mainly for the improvement of entrance form and the viscous resistance formula is for the improvement of run form.

In the later stage of hull' form design, the full

three-dimensional calculations are used in deterniing detail hull forms as explained in a recent paper by Gadd('8. However, the present state of the art of ship resistance theories is not fully satisfactory in explaining the resistance characteristics of hull forms at the fore arid aft ends of the ship. T'he flow phenomena around the bow and the' stern of full forms are complicated and are accompanied by breaking waves, flow

separation and so on. In order to take into account these

effects on the resistance characteristics together with full three dimensional calculation experimental studies of the detailed flow phenomena are necessary so as to increase our understanding of them, as emphasized in Section 2. In the Nagasaki Experimental Tarik, the effect of the free-surface

shear layer developing around the blunt bow on wave' breaking is under investigation(09). It is expected that findings from this may influence the development of appro

priate analytical means to design the bow form. 'For the

better understanding of stem flow, detailed flow

measure-ments should continually be carried out so as to develop

sound analytical models of.the flow around the stern.

of Minhigan, Naval Architecture and. Marize Engineering Report No. 002 (1968)

Baba E and Takekuma K A Sudy on Free-Surface Flow

around Bow of Slowly Moving Full Forms, J. Soc. Nay. Axch

Japan VoL 137 (1975) 1.6

A (0)

A(9)

Hamburg (1972) 241-283

Granville P.S., The Calculation of the Viscous Drag of Bodies

of Revolution, David W. Taylor Model Basin Rep. No. 849 (1953)

Parsons M G Optimization Methods for Use in Computer

Aided Ship Design, Proc. of First Ship Technology and

Re-search Symposium (STAR), Walhington, D.C. (1975)

Nagamatsu T., SakamOto T. and Baba E., On the Mininiiza-tion of Ship Viscous Resistance, J. Soc. Nay. Arch. Japan Vol. 154 (1983)

Tuck E.O. and von Kerczek C., Streamlines and Pressure Distribution on Arbitrary Ship Hulls at Zero Froude Number, Journal of Ship Research Vol. 12 No. 3 (1968)

Schlichting H., Boundary-Layer Theory, McGraw-Hill Book Company (1968)

Gadd GE., The Use of Viscous and Wave Resistance Theory

as an Aid to Optinuzmg HUll Forms, International Symposium on Ship Hydrodynamics and Energy Saving, El

PardO (1983)

Kayo Y. and Takekuma K., Shear LayeE and Secondary Vortjcal Flow beneath Free-Surface around Bow of Full-Form Ship Models, Trans. West Japan Soc. Nay. Arch. No. 65 (1983) 17-25

design method for full forms

parability concept of hull forms

where Cm is the midship area coefficient, and He is an

effective length representing fineness of entrance defined by

He=(1

ce)Le

where Cpe is the prismatic coefficient of entrance part,_Le is the length of entrance. When a new ship form is designed,

a

proper bow form

_and a stern form are selected

independently from the stored data so as to achieve a desired propulsive performance.

As far as the bow form _{design is concerned, the} follow-ing method has been used in MHI. A ship like an oil tanker or bulk carrier usually serves both in full load and ballast load conditionswith almost the same freqUency. Therefore,

one must design the optimum entrance not only for full load condition but also for ballast condition. In the full

load condition it is thore economical to choose a shorter entrance under the design constraints. In the ballast condi-tion one of the measures to quantify the propulsive perfor-mance is the difference of service speeds between full and ballast load conditons with constant horsepower. To satisfy the required speed difference, the entrance length in ballast load condition is longer than that in full load conditionso

as to prevent enormous increase of wave resistance. Thus the optimum _{entrance length iii ballast load condition}

differs from that in

full load condition. TO meet this

requirement the MHI-Bow was invented. The key idea of MHI=Bow is to combine two entrance forms each of which is optimum in full and ballast load conditions respectively.

MTB 166January1985

Nomenclature

amplitude function of stationary ship-wave system _{AL (0)} : amplitude fUnction of ship wave system due to the amplitude function of ship wave system due to the: _{singularity distributions around the intersection} be-singularity distribution over the body surface _{tween the body and the still water surface}

9

Uberoi S.B.S., 'iscous Resistance of Ships and Ship Models, -Hydro-og Aerodynamic Laboratorium Report No. Hy-13 (1969)

Nagamatsu T., Comparison between Calculated and Measur-ed Results of TurbUlent Boundary Layers around Ship

Models, Mitsubishi Technical Bulletin No. 133 (1979)

Nagamatsu T, Calculation of Viscous Pressure Resistance of Ships Based on a Higher Order Boundary Layer Theory, J. Soc. Nay. Arch. Japan VOL 147(1980) 20-34

Bäba E. and Hara M., Numeribgl EvaluatiOn of a Wave

Resist-ance Theory for Slow Ships, Proc. 2nd International Confer-ence on Numerical Ship Hydrodynamics (1977) 17-29 Havelock T.H, The Theory of Wave Resistance, Proc Roy. Soc. Ser. A 138 (1932) 339-348

Moriya T., A Theory of an Arbitrary Wing Section, J: of Soc. Ae±onatitical Science of Nippon VoL 8 No. 78 (1941)

1054-1060

Baba E., Wave Resistance of Ships in Low Speed, Mitsubishi Technical BUlletinNo. 109 (1976)

Gertler M., A Reanalis of the Original Test Data for the Taylor Standard Series, TMB Report 806 (1954)

Gross A. and Watanabe K., Form Factor, Report of Perfor-mance Committee Appendix 4, Proc. 13th ITTC, Berlin!

Appendi - Outline of a

based on the se A design method for full forms was developed based on the following three experimental and practical indications which were derived from the analyses of towing test data of

more than 200 full forms'.

For a full ship with slow speed (U/J7T _{0.20, Cb}

0.80) with a well designed run part, the wa'e resist ance characteristics depend mainly on the geometry of the entrance part, and the contribution from the parallel part and run part is negligibly small in generatmg waves

The propulsion factors depend mainly on the geomet-ry of the run part. Practically, there exists a limitation of

fullness for the run part to prevent worse propulsive

performance and undesirable flow phenomena around the stem.

A reliable formula to estimate the form factor by use of given geometric parameters has been developed. In this method the bow forms are designed so as to

re-duce wave resistance, and stern forms are designed to ob-tain better propUlsive efficiency, and the parallel partsare

designed to satisfy the required displacement. So the ex-perimental data such as wave resistance and self propulsion factors are stored together with geometrical parameters of each part. For instance, the parameters for wave resistance data are

Frbude number:

FnB= UJjH

Breadth-draft ratio: Bid Fineness factor: Cm He/B

amplitude function of ship wave system due to the free surface disturbance expressed by nonlinear terms of double-body flow

B breadth Of ship B : Fourier coefficints

curve

C =

R/4pU2L2 : wave resistance coefficient based on ship

length

CB = R/3/2PU2B' wave resistance coefficient based on ship

breadth

RF/4PU2S2 : frictional resistance coefficient

r/Y2P U2 : local skin friction coefficient where tx isthe x-component of wall shear stress t

(pPJ/YiPU2 pressure coefficient C,L0 = (p0p)/Y2PU2 : pressurecoefficient Cpe (Pe_P=)/YZ0U2 pressure coefficient

C,, = R,/'/2PU2S : viscous resistance coefficient for

three-dimensional body and R/Vz PU2L for two-three-dimensional body

viscous pressure resisXance coefficient

resistance coefficient for viscous pressure resistance

component caused by the piessure variatiOn across the bundäry layer

C,,2 : resistance coefficient for viscous pressure resistancç component caused by the displacement effect

Cp3 : resistance coefficient for viscous pressure resistancç component after the point of numerical breakdown or separation

CF'() : integrated value of skin friction resistance coefficient up to the point from the fore end

C,,,i '(c) integrated value of viscous pressure component up to thepoint from the fore end

prismatic coefflcient of rufi part draft of ship

Fri a u/.T: Frouñde number based onshiplength FnB = U/V: Froünde number based onship breadth

g : acceleration of gravity

H, = (1 - C,,,)L, : effective length representing fineness of run AF(8)

CF Cf =

Cvp.

Cpi

in Fourier series of sectiOtial-area L

K13 longitudinal surface-curvature in the streamline

coordi-nate system

"23 trshsverse surface-curvature in the streamline coordi-natesystem

ship length (load waterline length)

x-component of the outward normal unit vector on the

hull surface

L, length of rUn

Pa hull sUrface pressure in inviscid fluid

Pw hull surface préssurein viscous fluid

PC pressure at outer edge of the boundary layer

P= reference pressure

RF frictiOnal resistance

Rn Reynolds number based on body length R viscous resistance

R,, viscous pressure resistance R wave resistance

S wetted hull surface

U ship speed

Ue potential velocity at outeredge of boundary layer

U, V velocity compOnents in x andy coordinates respectively (velocity components in the direction of streamline and

cross flow respectively in Chapter 4).

x, y, z Cartesian ëoordinate system shown in Fig. 3 6 : boundary layer thickness

* : displacement thickness of boundary layer , (x, y) : wave height due to double-body flow

y/(L/2): non-dimensional expression of ycoordinate momentum thickness along steaxiline

v = g/U2 : wave number

x/(L/2): non-dimensional expression of x-coordinate position of breakdown in numerical calculatiOn in the three-dimensional case and the position of the zero

shear stress in the two-dimensional case p : density of water

ør (x, y, z): velocity potential of double-bodyflow