Model Resistance Data

commonly used presentations.

7.2 The CrRn Presentation. In research problems concerned with the separation of resistance into its components, methods of extrapolation to the ship, model-ship correlation allowances and the like, the re- sistance coefficient (Section 2.3)

is usually used, plotted to a base of the logarithm of Reynolds number Rn = V L / v .

Curves of this kind have been used in earlier sections of this chapter. In any consistent system of units, both CT and Rn are nondimensional.

7.3 Design Presentations. For design purposes, a method is desired which will show the relative merits of different ship forms.

Ships are usually designed to carry a given displace- ment a t a specified speed. CT is not suitable for such cases, since it is based on wetted surface and not on displacement, and can lead to misleading presenta- tions. An obvious merit criterion is the resistance per Previous Page

RESISTANCE 63 unit displacement weight, RT/ W, which is nondimen-

sional when RT and Ware expressed in the same units.

This ratio is the basis of a number of presentations, which differ principally as regards the speed coefficient used as the base.

7.4 The

0-0

System. R. E. Froude (1888) adopted the RT/W criterion in his “constant” system of nota- tion.

In order to have a speed base which would also be nondimensional, he devised a coefficient @ which is the ratio of the ship’s speed to the speed of a wave having a length equal to one half of the side of a cube of the same volume of displacement as the ship. If this vol- ume is V, the wavelength i s 4V1’3, and the wave speed will be

Hence

If RT/ W is plotted directly to such a base, the values increase rapidly a t high speeds, and the curve becomes very steep, obscuring some of its important charac- teristics, such as the wave-making humps and hollows.

Froude therefore divided the ordinates by

02,

and introduced a factor 1000 to avoid small numerical val- ues. The resistance “constant” is then

Since at low speeds the resistance is mostly frictional and varies approximately as

V ,

the @curves are nearly horizontal in this region. Any increase in the rate of variation of RT with V is shown by a rise in the curve, and these changes are very valuable in a diagnostic sense when appraising the merits of a hull form. In the foregoing equations, both @ and @ are non-dimensional.

Since @ relates to the total resistance, its frictional component will vary with size of ship, and for presen- tation purposes it is usual to give the values of @ for a standard value. In the past this standard value was a length between perpendiculars of 121.92 m (400 ft).

The ITTC in 1969, however, decided to adopt in addition a standard ship displacement volume of V = 10,000 m3. I t was also decided a t that time that for the pre- sentation of resistance and propulsion data at least two additional curves for other ship sizes be shown.

For other ship sizes a correction must be applied.

This correction depends on the ship length (for the

calculation of the Reynolds number), the Froude num- ber value and the wetted surface @, where

- S

- -

wetted surface

@ =

(volume of displacementp V2/3 7.5 The R,/ W VS. Fn or R,/ W vs. Fn System. The wave-making pattern and its associated resistance are largely dependent on the Froude number Fn ⁼ V/m. For many purposes, therefore, especially for ships with an important wave-making resistance com- ponent, it is useful to plot ^{R T /} W against Fn.

Very often the results of so-called standard series of hull forms are presented in the R R / W vs. Fn form, where R,/ W is called the specific residual resistance coefficient. Use of a form factor will allow the deter- mination of the wave-making resistance coefficient, R,/W, in which case a R,/W vs. Fn representation can be given.

A comparison between the residuary or wave-mak- ing resistances of two alternative designs should be carried out with care, since it ignores differences in frictional or viscous resistance, and the total resistance has to be computed in all cases to make a proper eval- uation.

7.6 The R,/ W ^VS. Fn System. When curves of ^{R T /} W for a number of ships are plotted to a base of Fn for comparison, the relative merits of the designs at a given value of Fn will be shown by the order of the RT/ W curves. If we wish to introduce some function of speed into the ordinates to reduce the steepness of the curves and bring out the wave-making character- istics (which is one of the reasons for plotting on Fn), Telfer (1933) has shown that we can divide the ordinate RT/ W by ( V/\j2)2. To retain a non-dimensional quan- tity, however, it is possible to divide by ( V/,@)2 and obtain:

g R T L

CTL = -

W V

When plotted against V / a , this leads to what Telfer has called a “compatible” presentation, correctly pre- serving the relative merits of comparable hull forms.

7.7 Conversion Factors for Speed and Resistance Coefficients. In converting model resistance data from one form of presentation to another, the speed relationships given in Table 15 are useful. The factors for converting frequently used resistance coefficients are given in Table 16.

C

,

defined by:

The most important coefficients are:

CTL defined by Equation (64),

Table 15- Relationships for Converting Frequently Used Speed Coefficients

Fn (nondimensional)

(non-

0

dimensional)

v/ w%

(International units)

p1/6

v/w"G

s""m

V/&

(International (non- FnV

dimensional) FnV

FnV

units) V/$

&

JZ. v/&

&

Fn

(nondimensional) Fn

FnV

(nondimensional) Fn

0

(nondimensional) V /

w"G

(International units) V / &

(International units)

6

Fnv

d 4 7 ~ L / V ' ' ~ Fn

0

_in

v)

0 n

'

^Fn,

JzP

V / &

rri n

Fn = V/m

Fnv ⁼ V / w

@ = & V / J p

V ⁼ ship velocity (m / sec) V = displacement volume (m3)

W = displacement weight (kNewton); W = pgV p = mass density, k g / L (or t/m3)

g = acceleration due to gravity (m/sec2)

i c

rn a

NOTE: ^{p =} 0.999 k g / L and 1.0259 k g l L for fresh and salt water respectively at 15 deg C (59 deg F) and g = 9.81 m/sec2 (32 ft/sec2).

NOTE: These notes also apply to Table 16.

Table 16-Relationships for Converting Frequently Used Resistance Coefficients

(non- CT dimensional)

C T V

(non- dimensional)

cTV

s / v 2 / 3

c,

- 125 ^CTV

7T

(Fb)'

2

cTv

(p)

L

c,

x (4)"

9

c,

2

C T L

(non- dimensional) (non-

0

dimensional) -- a 0 125 s / v 2 / 3

7T

1250

0

- ₂₅₀ 7T (Fnv)'

0

($)

^@

(g3 ⁰

250 / 7~

2 5 0 / ~

(non- dimensional)

2 R T / W CT

(nondimensional) ^{C T} (Fn,)' S/V2/3

-.

S CT

v'/3

CTV (nondimensional)

0

(nondimensional)

125 S/V2/3 CT

7T

250 C T L

--

7T L A %

(nondimensional)

R T / W Fn'

(

^{- - L/V1/3 -2) (Fn,)' R W}

L S

(+) ^(F)

C T L

(nondimensional)

RT w"'3v.2 (International

units)

V ⁼ Ship velocity in m/sec.

V = Displacement volume in m3

W = Displacement weight (kNewton); = pgV p ⁼ Mass density, k g / L (or t/m3)

g = Acceleration due to gravity in m/sec.2

66 PRINCIPLES OF N A V A L ARCHITECTURE

g R T L G I . = w1/2

The well known (but outdated) admiralty constants by are defined bv

and

c,

^{= -.}

C, is related to the power-displacement coefficient C,, PE

C, is related to the resistance-displacement coefficient CTV by

Section 8