WAVE RESISTANCE DETERMINATION The prediction of resistance for an arbitrarily given ship hull form is an important problem in the design of ships which are defined by ship lines or offsets. The investigation of the so-called residuary resistance as a function of hull parameters and speed are carried out by model tests. However, the number of form parameters which influence the resistance characteristics is too large to enable us to find out an optimum form by the experimental means alone. Progress in this domain can only be expected when both theoretical and experimental methods are adopted simultaneously. An attempt has been made to show how far one can predict the wave resistance for both analytical and empirically developed ship forms with the help of presently available theoretical knowledge. The first part deals with the wave resistance calculation of empirically developed models of the different series, namely from DTMB, BSBA and the Goteberg Experimental Basin by Weinblums polynomial method for Michell integral.This part is not discussed here as more and more systematic series have been developed for a range of vessels till to date. As such these databases could be used to determine resistance with a reasonable accuracy. The second part considers ships of any arbitrary form which is represented by exact singularity distribution at zero speed whereas the last section is devoted to the more exact determination of the wave resistance for any given hull form at all speeds with the help of Guillotone refined method based on the process of finding the real hull for which the linearised computation of Michell theory is exact.

COMPUTATION OF WAVE RESISTANCE FOR NORMAL SHIP FORMS Small changes in ship forms, as have been found by the calculation of analytical hull forms and afterwards verified by experiments, can cause a considerable amount of difference in the wave resistance. The shipbuilding practice has always encouraged developing hull forms out of experience and aesthetic viewpoints. That is how the renowned empirical series forms have been designed and resistance tests have been conducted. The theory of wave resistance should have been utilized in the determination of the form parameters. The computation of wave resistance for empirical forms demands mathematical representation of the sectional area curve. setting up an expression for wave resistance for such analytically approximated sectional area curves. The least square method is mathematically the best for approximating the empirically designed section al area curves Two different sections of the sectional area curves (fore and after body) should be expressed by different polynomial equations. The characteristic parameters for the sectional area curve consist of the ordinates, length of entrance and run, and also the length and position of the parallel middle body. In addition to these the following geometrical quantities have been satisfied as secondary conditions. Prismatic coefficient. Longitudinal center of buoyancy location.

Taylors tangent values for fore and after body. The vanishing curvature of the entrance and run at both ends of the parallel middle body.

e) The maximum ordinate at both ends of the parallel middle body, i.e. = 1, when = F or -A

f) The same condition also at the midship section, i.e. = 1, when = 0.

g) The zero ordinates at both ends, i.e. = 0, when = + 1.

Fig. Asymmetrical Sectional area curve (non-dimensional) with parallel middle body. The non-dimensional sectional area curve can be represented as:

n

n

n

n aaaaa ++++= 112210 .........

So that when i is the approximate value at = i , the weighted least square is given by the form:

( )[ ]=

m

iiii g

1

2 Note: The mathematics here has been deliberately deleted to avoid the complexity or

confusion.

MICHELL INTEGRAL ACCORDING TO INUI In the range of small Froude numbers, the undulations of the wave resistance curves appear more frequently than at higher speeds. The calculations for wave resistance for normal ships should, therefore, be made at a closer interval of Froude numbers so that the crests and troughs of the curve can be properly located. An asymptotical expression of the Michell-Integral allows us to obtain the wave resistance in an easier manner.

As long ago as 1942 WIGLEY suggested that the wave resistance curve be divided into two parts, namely,

21225.0 WWW

W CCBVRC +==

where

CW1 (normally a monotonous part) = 221

5.0 BVRW

and CW2 (oscillating part) = 222

5.0 BVRW

According to Havelock, the expression of wave resistance for a singularity distribution on the center plane is given as:

RW = ( ) +2/

0

22222

cospi

pi

dUNMBV where

( )

( )( )

)()(2

1

2secexp1

seccos)(

and secsin)(

20

20

1

10

1

10

md

dF

LTK

KU

dmN

dmM

n

=

=

=

=

=

=

Since for the parallel middle body -A < < F, m=0

( ) +==2/

0

22222 cos

25.0

pi

pi

dUNMBV

RC WW

After partial integration for M and N and the asymptotic expansion according to INUI one obtains the expression for CW. From these expressions CW can be split up into CW1 and CW2 where CW1 is formed from the number 1 of the right-hand sides of the above expressions and CW2, which is oscillating in nature, contains all the other terms of the right-hand side expressions. For the oscillating part CW INUI has given two asymptotic expressions as follows:

A)

+

+2/

00

20

2/1

00

122

42cos

4)sec2cos(cos

pi pipi UdU m

B)

+

+2/

00

20

2/1

00

222

42sin

4)sec2sin(cos

pi pipi UdU m

Where U has been defined earlier and

( )KU 00 exp1 = For low and medium speeds Inui has given asymptotical expression for CW. The calculated results according to INUIs expression show a very good agreement with those obtained by the exact evaluation of the Michell integral by polynomial method of Weinblum.

EXTENSION OF THE INUIS ASYMPTOTIC METHOD FOR SHIPS WITH PARALLEL MIDDLE BODY The wave formation between the bow, stern and shoulder wave systems can be divided from the equation for CW in the following way: Bow waves Stern waves Forward shoulder waves After shoulder waves Interaction between bow and forward shoulder after stern forward shoulder and after shoulder stern after shoulder and stern The terms a) to d) form the fundamental term CW, whereas the rest of the terms go to form the oscillating part CW2 of the total resistance coefficient CW . The final expression of wave

resistance coefficient CW for a ship with parallel middle body is given as follows [1]:

( ){ } ( ){ } ( ){ } ( ){ }

gLVF

F

LTk

kU

dUUA

UAmmmUAmUAmmmUAmC

n

n

m

mm

W

=

=

=

=

=

++

+=

+

20

20

2/

0

122

20

22202111

202

020

22202111

202

01

21

length is L anddraft is T where,2)secexp(1

cos

1)1()1(21121)1()1(2112

pipipi

Non-dimensional sectional area curve is given by:

The resistance calculations for various sectional area curves of different methodical series have been made using the procedures as outlined above. Along with the increase of the size of bulk carriers, the extent and the fore and aft position of the parallel middle body play a very significant role in the determination fo a hull form. A study, both theoretical (as outlined above) and experimental, should be investigated to determine an optimum parallel middle body depending on the speed and the prismatic coefficient. Besides the effects of the parallel middle body, other characteristics of the sectional area curves, namely, the shape of the entrance and

21 =

2)1( ,2)1( ,22)1(2)1( ,22)1(2)1( ,2

, ,

111

00

00

23

3

12

2

0

==+=

+===

=+=+=

===

mmm

mm

mm

mdd

mdd

mdd

( ){ } ( ){ }( ){ } { }

gLVF

F

LTk

kU

dUUA

Ummmmmmmmmmmm

UmmmmmmmmC

n

n

mmm

W

=

=

=

=

=

+

+

+

+

=

+

20

20

2/

0

122

020

2/1

03003122130

001102

0

020

2/1

020021120002

02

21

length is L anddraft is T where,2)secexp(1

cos

42sin

41)1()1()1()1()1()1()1()1(1)1()1()1(14

42cos

41)1()1()1()1()1(1)1(14

pipi

pi

pipi

pi

pi

run of the sectional area curve, can be studied systematically when a computer program is available for calculating the wave resistance of any given sectional area curve. But this procedure does not allow us to consider the effect of the difference in station shapes of different hulls. However, it has been pointed out in various publications of Weinblum and others that the wave resistance is primarily dependent on the longitudinal distribution of displacement, i.e. on the sectional area curve and the vertical distribution of displacement has only a secondary effect. Example: Carry out an exercise to determine the wave resistance coefficient CW of a vessel with following parameters:

Length of waterline L 120 m

Breadth B 17.5 m

Draught T 7 m

Wetted surface area S 2950 m2

Service speed V 9 knots