On the flate plate experiments of Kyushu University

With the Corn plinzents of the Author.

By

Keizo UENO, Michihiko TOKUNAGA

and Takeshi HARÁ

Reprinted from the Memoirs of the Faculty of Engineering

Kyushu University, Vol. XXIII. No. 3

FUKUOKA JAPAN 1964

Lab.

y. Scheepsbouwkunde

Technische Hogeschool

De Ift

By Keizo UENO Professor of Naval Architecture

and

Michihiko TOKUNAGA* and Takeshi HARA"

(Received December 11, 1963)

Abstract

In order to obtain the formulation of the frictional resistance of smooth plane surface in tur-bulence the towing experiments of vertical flat plates of nine lengths within the range from i ni. to 12 ni. were carried out in the range of Reynolds number from 3.6/10 to 2.9x107 in the Ship Model Experimental Tank of Kyushu University over the period from 1952 to 1954 that is almost the same as the period of Dr. G. Hughes's experiments of fiat plates and pontoons. Eliminating the laminar effect by using trip wires as the artificial turbulence stimulation device and also eliminating the edge effect by the experiments with the plates of two or three breadths or drafts for each length of plates, we derived a following plane friction formula in turbulence.

Cí=0.173/(Logio R, _2)2u5

This new friction line plotted in base Reynolds number has a steeper slope than any other hither-to derived friction line. We shall not, in the present state, touch the problem of what a role the new friction formula has in the model-ship correlation, which would be examined in future comparing with the other friction formulae.

I. Introduction

In the aims to obtain the formulation of

the frictional resistance of smooth plane sur-face in turbulence, the towing experiments

of flat plates and pontoons with various sizes

have been carried out from about one

cen-tury ago by many scientists such as Dr. W. Froude, Dr. F. Gebers, Dr. G. Kempf, Dr. Y.

1-liraga, Dr. K. E. Schoenherr and Dr. G.

Hughes etc.. Among those experiments, how-ever,

the early ones are not adequate to

the aims because they have more or less

some amounts of laminar effect or edge

ef-fect. Dr. G. Hughes ' succeeded for the first

time in deriving the basic friction formula

in two-dimensional flow of a plane surface, eliminating the laminar effect by using the artificial turbulence stimulation devices and

t This paper was originally presented as the Dis-cussions on the Subject of Resistance to the 10th International Towing Tank Conference held in Teddington, England, on September 1963, while the data in this paper are slightly different from the original one because the experimental results had been a little more minutely reanalysed

after-wards.

* Student, Department of Naval Architecture, Fa-culty of Engineering, Kyushu University Naval Architect, Japan Defence Agency

also eliminating the edge effect by carrying out the towing experiments of horizontal flat

plates and pontoons with various kinds of

length-breadth ratios. In order to accomplish

the same purpose, the towing experiments

of vertical flat plates of various sizes 2),3)4)

were carried out in the Ship Model

Experi-mental Tank of Kyushu University within

the range of time from 1952 to 1954 that is

almost the same as the period of Dr. G.

Hughes's experiments.

We took the

flat

plates of nine lengths within the range from

1 m. to 12 m. and towed them with the range

of speed from 0.49 m./sec. to 2.15 m./sec.

which corresponds to the range of Reynolds

number from 3.6x105 to 2.9x107 and derived

a plane friction formula in turbulence, elimi-nating the laminar effect by using trip wires

as the artificial turbulence stimulation device

and also eliminating the edge effect by the

experiments with the plates of two or three breadths or drafts for each length of plates.

The new friction line thus plotted in base Reynolds number has a steeper slope than

any other friction line, such as Dr. G. Hughes

line, Dr. K. E. Schoenherr line or I. T. T. C.

1957 friction line. In the present paper the above flate plate experiments carried out at the Kyushu University are stated in detail.

190 Keizo UENO, Michihiko TOKUNAGA and Takeshi II sis (Vol. XXIII,

2. Symbols Used

Rf = Frictional resistance of fully turbulent smooth plane surface in kg.

R' = Measured towing resistance of a fiat

plate in kg.

4R1oAmount of correction of frictional re-sistance of a plate for total allowance including both the trip wire effect and the laminar effect before the trip wire

position, in kg.

R10 = Frictional resistance of a flat plate,

corrected for both the trip wire effect

and the laminar effect before the trip

wire position, in kg.

Ríe = Amount of increase of frictional

resis-tance due to edge effect of a plate in

the towing experiments, in kg.

R ==Turbulent frictional resistance of a

plate within the range of length from

the leading edge to the trip wire posi-tion, in kg.

Rf(S = Laminar frictional resistance of a plate

within the range of length from the

leading edge to the trip wire position, in kg.

R = Total amount of allowances for two

effects of both the resistance increase

due to the proper resistance of trip

wire itself and the increase of boundary

layer thickness due to the existence of

trip wire, that should tend to zero at the limit of trip wire diameter would

tend to zero, in kg.

L ==Length of a plate in m.

¡ = Half girth length, breadth or draft of

a plate in rn.

S =Wetted surface area in m.2=21L

X = Distance from the leading edge to the

trip wire position in m.

d

=Diameter of trip wire with circular

section in mm.

B = Breadth of water of model basin in m.

a =Sectional area of a plate under water in m.2

A = Sectional area of model basin under

water in rn.2

m =Blockage of model test=a/A

V0 =Measured towing speed in m./sec.

4V =Amount of increase of mean relative

speed through water due to the boun-dary effect of model basin, in m./sec.

V = Relative speed through water after

cor-rection for the boundary effect of model

basin, in m./sec.

p = Density of water in kg. m.4 sec.2 = Coefficient of kinematic viscosity in

m.2J sec.

R5 =Reynolds number VLJv = Local Reynolds number Vx/v Non-dimensional factors:

C1= Rj/pILV2, C'joJ piL V2,

4C10= 4R10/p1LV2, Cjo=R4p1LV2, CfC= R1eJp1LV2, C1t = Rp,/plxV2,

Cfl= R1,/plxV2, C = R/plxV2. 3. Fiat Plates Tested and Methods of Tests

Particulars of test plates are represented

in Table 1. As seen in Table 1, we took nine lengths of plates such as i m., 1.81 rn,

2 m., 3 m., 4 m., 5 m., 6 m., 9 rn. and 12 m.,

among which for each of the former six

lengths from i m. to 5 m. two or three plates of different breadths were made and towed

in vertically and fully submerged conditions,

and for each of the last three lengths from

6 m. to 12 m. the plate built up by connecting

longitudinally the individual piece plates of

3m. each was used and towed in vertical

conditions of three different drafts, namely,

the upper edge of plate emerged above water.

Summarizing the above, we made 16 flat plates and tested 24 conditions from Test No.

1 to 24, as indicated in Table 1.

All plates were made of Formosan cypress,

well sheathed and finished into smooth sur-faces with varnish. As the turbulence stimu-lation devices piano wires of 1 mm. diameter

were fitted around the plate surfaces at the

stations of 100 mm. abaft the leading edges of the plates.

Details of longitudinal and transverse sec-tions of each of 1 rn., 1.81 m., 3m., 4 m. and 5 m. length plates arc represented in Fig. 1, Fig. 2, Fig. 3, Fig. 4 and Fig. 5 respectively. Each plate has a sharp nose and a sharp tail

in a longitudinal section and a sharp lower

edge with its upper edge gradually tapered,

being 6 mm. thick in a transverse section.

In the case of towing experiment each plate

was vertically supported at the 83mm. im-mersion of the upper edge beneath water surface by two brass frames as shown in

Test No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Length of plate (in m.) ± 4S

-4S--+-4S

_-H

Breadth Thickness or draft of plate of plate (in m.) (in mm.) 240 TRAt')SVERsg SECT/ON (2)

Fig. 6. As indicated in Fig. 6 the sections

of the brass frames have square shapes in

the above-water part, lens shapes in the un-der-water part and half-round shapes in the

Table 1. Temperature

of water (in 'C)

TAÌjSVEE SECTION

Remarks: In the third column, breadth refers to Test No. 1 to 15 and draft to Test No. 16 to 24. FLAT PLATES (I)

LON-ITUDINAL _ECT1oÑ

(FE & //FT SYMMETRY)

8.3 Jan. 26-Feb. 1 7.5 1952 9.0-9.5 8.8 9.1 7.3 7.3 8.0 8.0 7.0 7.0 9.4-97 8.5-9.0 9.4 19.0-.-19.1 18.8 18.1-18.6 24.1-24.7 24.6 -24.8 24.8-26.2 Fig. 1. Date of experiment Test condition Oct. 2-4, 1952 Oct. 4-6, 1952 Jan. 26-27, 1954 Jan. 28-29, 1954 Feb. 1, 1954 Jun. 9-10, 1954 Jun. 14, 1954 Jun. 15-16, 1954 Sep. 1-2, 1954 Sep. 2-3, 1954 Sep. 4-6, 1954 Fully submerged Upper edge emerged above water

lowest part where the upper part of

fiat

plate is fitted to the two brass legs by two

bolts of which heads were flushed over the half-round surfaces of brass legs.

1.00 0.090 8 1.00 0.240 8 1.81 0.239 6 1.81 0.309 6 1.81 0.375 6 3.00 0.090 10 3.00 0.240 15 4.00 0.090 10 4.00 0.240 18 5.00 0.090 10 5.00 0.240 20 6 00 0.200 30 6.00 0.170 30 6.00 0.140 30 9.00 0.200 30 9.00 0.170 30 9.00 0.140 30 12.00 0.200 30 12.00 0.170 30 12.00 0.140 30 8 20.0 8 21.2 8.5 21.2 8.5 21.2 Jan. 8-13, 1952 Fully Feb. 11, 1952 submerged Feb. 8, 1952 Feb. 6, 1952 1.00 0.090 LOO 0.240 2.00 0.090 2.00 0.240

192 Keizo UENO, Michihiko TOKUNAGA and Takeshi HRA (Vol. XXIII,

4S 4S

FLAT PLATES (II)

LCPJITLJDINAL SECTION

DEPTH OF FLAT PLATE

(1) 375 (2) 309 (3) 239 Fig. 2.

FLAT PLATES (III)

LoN-1TUDINAL SECTION (FORE & 4FT )

H

240

Lo

As shown in Fig. 7 transverse sections of

each of 6 rn, 9 m. and 12 m. length flat plates have 30 mm. thickness, blunt upper edges and round lower edges of 15 mm. radius. Adjacent

piece plates of 3 m. length are connected by longitudinal scarf joints and fitted mutually by 8 bolts of which the upper row of 4 bolts

are above water and the lower row of 4 bolts

having flush heads over the plate surfaces are under water, as represented in Fig. 7.

Each plate has a sharp nose and a sharp tail, as shown in Fig. 8. In the case of towing

experiments each plate was vertically

sup-Fig. 3. /4 q

// O

TRANS VE g-5E SECTJ ON

--

--p

(FORE 8 AFT SYIThiEJTKY.)

30o0

TI?ANSVESE SECTION (1)

'I )

ported by two duralmin thin plate legs, to the lower ends of which the upper part of the plate was fitted with 4 bolts well clear up over the water surface as shown in Fig.

9.

Next, the methods of towing experiment of flat plates will be explained as follows.

As shown in Fig. 10, the test plate was fitted

to the longitudinal and horizontal stiffener beam B with some number of supporting

legs mentioned before, while both fore and after ends of the beam were connected with the pivoting points p' to the swinging frames TRANSVERSE SECTION (z>

To ia

-FLAT PLATES(IV)

LONc-/TUDfNAL SECTION (FOF'E & liFT sy1r1ETRY

TI'ANSVERSE SECTJOP'J I) So s -f. Z4 o TR,INSL/ERSE SECTION () 3ZS---4O Z4 O Tk'MNSVEkSE SECT/ON ) I 1 o Fig. 4. FL/iT PLATES

(V)

LONCrITUDINAL SECTION (FO'E k AFT SYMMETRY)

-I

J _t

T'ANSVE'SE SECTION

(I)

Fig. 5.

194 Keizo UENO, Michihiko TOKUNAGA and Takeshi HARA (Vol. XXIII, SUPPORTIf'JQ FAr4E Fog 3)7L4t FLAT PLATES. MATERIAL SRSS UNIT /t 1.M

F which were supported by the pivoting

points p at the 'fore and after part of the

towing truck. Details of swinging frames

F having the pivoting points p and p' are

shown in Fig. 11. All swinging systems

including the test plate with its upper edge either set at 83 mm. below water surface or

emerged above water, the supporting legs,

the

stiffener beam B and the

swinging frames F were perfectly balanced by control-ling the counter weights w and w'. The test

plate was towed by the horizontal rod fitted

on the beam and connected to the lowest

point of the lower arm of the resistance

dynamometer to measure the resistance of

SECTION

A-4-SECTION. 5

SECTION

C-C-Fig. 6.

the plate. On towing, the longitudinal

move-ment of the plate was freely allowed, but

the transverse movement, especially the

trans-verse heeling trend which is apt to occur at

high towing speeds were strictly restricted

by the swinging frames.

4. Methods of Analysis of Flat Plate Test Results

First, the towing resistances of a flat plate R'10 were experimentally measured for its various towing speeds Vo. Next, the

fric-tional resistances of a smooth plane surface in turbulence Rf for relative flow speeds V

of R'ro and V0 after carrying out successively

the following corrections of three steps from

(a) to (c).

(a) Allowance for tank boundary effect

The Ship Model Experimental Tank of

Kyushu University has the following

dimen-CONNECTION OF FLAT PLATES (6m, 9m, 12m)

Fig. 7.

FORE & AFT SHAPE OF FLAT PLATES (6 m, 9 m, 12 m)

Fig. 8.

sions.

Length over all=76 m., Breadth of water=2.67 m., Depth of water=2.72 m.

The amount of increase of mean relative

flow speed around the test plate due to tank

400

o 20

196 Keizo UEN0, Mich ihiko TOIWNAGA and Takeshi HARÁ (Vol. XXIII,

boundary following

of Dr. K.

SUPPORTING FRAMES FOR FLAT PLATES (6m, 9m, 12m)

1ArERI/L DURALMIN

'a BOLT

effect AV was calculated by the

formula according to the method Taniguchi and Mr. K. Tamura5.

4V

V0=umW

where the above symbols are explained in

the Section 2. And we get the value of V,

the relative flow speed after correction as

follows:

V= V0 ± 4V.

(b) Allowance for the artificial turbulence Fig. 9.

W. L

stimulation dcvice etc.

As before mentioned, in order to stimulate

the artificial turbulence in the boundary layer flow we used piano wires of 1 mm. diameter around the test plates at the

sta-tions 100 mm. abaft the leading edges of the

plates. In order to obtain the value of R,-,

the frictional resistance of fully turbulent smooth plane surface, we must carry out

the following two corrections for the

mea-sured resistance of the plate R',-0. Namely,

amount of allowance for laminar part of the

boundary layer within the range from the

leading edge to the position of the trip wire

of the plate and subtract from the value of

R' by some amount of allowance for two

effects of both the resistance increase due to

the proper resistance of the trip wire own

and also the increase of the boundary layer

thickness due to the existence of the trip

wire. The total allowance including the

above two corrections can be calculated by the following formula of which deduction is

explained in detail in Appendix 1.

4R10= 0.00lpxlV2,

where the above symbols are explained in

the Section 2 and in the present case x=O.10 m. and we get the value of R10, the frictio-nal resistance of the plate corrected for both

the trip wire effect and the laminar effect

before the trip wire position, as follows:

R=R'± 4R1.

(c) Allowance for the edge effect of test

plate

As mentioned before, the towing

experi-ments from Test No. i to 15 were carried out

under fully submerged conditions of the plates and those from Test No. 16 to 24 under

the conditions of the plates, the upper edges

A

li

B BEAM r T TRIANGLE w F SWINGING FRAME k RRAIL S P PEN

p&p'

D RECORDING DRUM w' w MODEL Fig. 10. HORIZONTAL LEVER

WEIGHT OF SWINGING FRAME KNIFE EDGE

SPRING

PIVOTING POINT

COUNTER WEIGHT OF FRAME WEIGHT

of which were emerged above water. In

the former cases, therefore, the plates should suffer the extra-resistances due to the exis-tence of both lower and upper edges having some number of supporting legs, and also in

the latter cases the plates should suffer the

extra-resistances due

to the

existence of lower edges and free surface effects. Now

we shall call the total amount of those

ex-tra-resistances as a simple name "edge effect"

and express it by a letter "R1". Since the

value of R1, obtained in the preceding

para-graph (b) includes the edge effect R1,, the

value of R1, namely, the frictional resistance

of turbulent smooth plane surface will be

obtained by subtracting the value of R1, from the value of R10. Therefore

R1= - Rie

According to the method once used by Dr.

Y. Hiraga6 and also one of the authors7,

the subtraction of the edge effect R1, from the value of R10 was carried out as follows: As mentioned before, in Test No. 1 to 15 the values R10 of a definite fiat plate length on various speeds V were obtained for two

or three fiat plates with different breadths, that is, for two or three values of half girth

length I and also in Test No. 16 to 24 those were obtained for three different drafts, that

F

I?

i

V/AjG-Irn- FRIME W' COUNTER WEIwr 117' PIVOT/"16- PO/NT B STJFFNEP J1

N

is, for three values of half girth length i.

In the case of a definite flat plate length,

the values of R10 were first plotted for two

or three definite values of / on base the speed V, and the mean lines through the spots were drawn for each definite value of / respective-ly. The cross values of R10 read from

these mean lines were plotted for various definite values of V on base the half girth

length 1.

In this case, such mean lines

FLAT PLATE

Fig. 11.

through the spots for each definite value of V could be approximately drawn as the

ex-tensions of these mean lines would cut the negative axis of ¡ at some definite point.

Then it might be considered that the values R of these mean lines at /=0 would

approxi-mately give the values of edge effect R,

for the various corresponding speed V of the

plate. Thus we could obtain the values of R1 for various speed V by substracting these 198 Keizo Uiso, Michihiko TOEIJNAGA and Takeshi HARÁ (Vol. XXIII.

values of Rje from the values of R0.

Finally it should be noted that in the

pre-sent analysis of the flat plate test results

the allowance for the thickness of the plates

was not considered as the thickness effect

of our plates on frictional resistances might be negligibly small, according to the Dr. Y. Hiragas experimental results6 which

show-ed that the thickness effect on frictional

resistance of planks was negligibly small in

the case of smaller values of thickness length

ratio of the planks than 1/200 and smaller

speed than 3 m./sec..

Corrected Test

Results

Note: Values of R are given in units of 1 million. Values of C1 are given in units of 0.001.

5. Results of Analysis of the Flat Plate Tests

We obtained the values of R1 for various

speeds

V as the frictional resistances of

smooth plane surfaces in turbulence by carry-ing out such corrections as mentioned in detail

in the preceding Section 4 to the measured

values of the flat plate tests from Test No.

1 to 24 indicated in Table 1. Using these

values of R1 and V, we calculated the f riction-al resistance coefficient C1 and the Reynolds

number R,, and thus obtained values of both

R,, cf 0.720 5.567 0.740 1.208 0.985 4.786 0.888 4.648 1.133 5.032 1.016 4.990 0.828 5.042 1.124 4.906 0.653 4.778 1.218 5.021 1.294 4.869 1.553 4.724 1.444 4.847 1.339 5.052 1.308 5.043 1.167 5.423 1.246 4.999 1.648 4.940 1.844 4.567 1.817 4.578 1.708 4.732 1.598 4.710 1.940 4.516 2.049 4.549 1.981 4709 2.131 4.471 2.199 4.301 2.268 4.357 2.281 4.376 2.459 4.319 2.404 4.341 2.477 4.464 2.557 4.319 2.611 4.387 2.665 4.434 2.793 4.280 2.979 4.243 2.911 4M39 2.735 4.298 3.033 4.306 3.074 4.242 3.377 4.286 Table 2. (a) Test Number

i

2 3 1.639 4.556 1.857 4.511 1.734 4.595 1.779 4.524 1.821 4.655 1.894 4.575 1.690 4.645 0.962 5.047 1.074 4.915 1.042 5.131 1.153 4.792 1.144 4.788 1.217 4.811 1.093 4.872 1.295 5.347 1.253 4.915 1.366 4.703 1.217 4.940 1.430 4.625 1.333 4.964 1.480 4.587 1.399 4.515 1.537 4.665 1.472 4.460 1.615 4.573 1.566 4.640 1.700 4.566 Length (m.) 1.000 1.000 1.810 Breadth (m.) 0.090 0.240 0.239 Thickness (mm.) 8 8 6 Temperature ÇC) 7.5 9.0-9.5 R C; Cf 0.357 6.361 0.365 6.389 0.430 6.111 0.417 6.196 0.503 5.889 0.491 5.560 0.554 5.855 0.552 5.639 0.583 5.539 0.637 6.308 0.678 5.526 0.697 5.236 0.729 5.291 0.776 5.220 0.641 5.677 0.854 5.271 0.816 5.113 0.918 5.015 0.903 4.879 0.989 7.629

200 Keizo UESO, Michihiko TOWNAGA and Takeshi H.&RA P s q 1.810 1.810 0.309 0.375 6 6 8.8 9.1 4 5 Corrected Test Results R,C Cj R C 2. (b) 0.653 5.695 0.784 5.409 0.925 4.980 1.044 4.886 1.200 4.726 1.371 4.889 1.445 5.024 1.606 4.761 1.766 4.690 1.862 4.650 2.074 4.370 2.249 4.259 2.408 4.244 2.609 4.273 2.509 4.287 2.783 4.213 2.703 4.226 2.964 4.135 2.890 4.130 3.077 4.227

te: Values of R,C are given in units of 1 million. Values of C.r are given in units of 0.001.

0.652 0.772 0.919 1.040 1.215 1.337 1.499 1.669 1.823 1.976 2.106 2.275 2.390 2.525 2.633 2.795 2.889 2.958 3.160 3.228 R. 5.1 5. 5. 5.1 5.1 4. 4.1 4. 4. 4. 4.1 4.: 4. 4.1 4. 4.1 4.1 4.1 4.1 4.1 (Vol. XX III,

C1 and R for each test are represented in test are plotted logarithmically in base the

Table 2, and also the values of C1 for each values of R in Fig. 12 to Fig. 20, in which

f 98 47 18 52 61 59 57 01 34 70 82 27 77 60 02 46 08 72 06 80 o Ct Test Number 3 R Cj 3.168 4.281 2.870 4.252 3.405 4.143 3.527 4.082 3.311 4.130 3.661 4.132 3.688 4.501 4.146 3.973 4.011 4.075 3.899 4.595 Table Length (m.) 1.810 Breadth (m.) 0.239 Thickness (mm.) 6 Temperature (CC) 9.0C9.5 3.131 4.180 3.268 4.1 46 3.211 4.141 3.363 4.( 93 3.345 4.138 3.498 4.1 34 3.532 4.177 3.592 3. 70 3.613 4.138 3.876 4.1 23 3.412 4.145 3.781 4.( 12 3.748 4.720 3.916 4.1 18 3.653 4.266 3.997 4.( 88 3.680 4.324 4.105 4.1 45 3.613 4.266 4.254 3. 88 I. 000 0240 TEST NO 2 LENGTH OF PLAIE SNEADTH OF PLAIE '0 /.5 2 6 5 4 3 OO°0 T&ST NO I LENOTH OF PL4TE. 1.000 . 8REAOTH OF PLATE. 0090 2 I I I

11111

2/o 3 4 5 ¿ 7 8 N/a6

ihe approximate mean lines through the spots whiçh the K. E. Schoenherr friction line, the are drawn. All of the mean Cf----R,, lines of G. Hughes friction line and also the I. T. T. Test No. i to 24 are reproduced in Fig. 21, in C. 1957 friction line are represented. It is

Length (m.) Breadth (m.) Thickness (mm.) Temperature (CC) Test Number 6 7 8 Corrected Test Results

Note: Values of R are given in units of 1 million. Values of C1 are given in units of 0.001.

e ea e a i%.. 1.2.. <:c) R C1 R, Cf R,L Cf 1.086 5.175 1.027 4.657 1.459 4.310 1.305 4.423 1.341 4.401 1.719 4.145 1.369 4.511 1.464 4.377 2.008 3.964 1.615 4.231 1.617 4.441 2.346 3.689 1.804 3.986 1.774 4.283 2.586 3.616 1.999 4.098 1.990 4.025 2.889 3.545 2.281 3.936 2.196 4.033 3.062 3.520 2.387 4.038 2.421 4.029 3.409 3.462 2.483 4.064 2.648 3.736 3.727 3.368 2.589 3.849 2.888 3.783 4.016 3.308 2.737 3.704 3.206 3.671 4.305 3.437 2.950 3.855 3.334 3.633 4.594 3.235 3.332 3.703 3.588 3.656 4.882 3.195 3.247 3.583 3.737 3.649 5.287 3.135 3.480 3.667 3.970 3.555 5.431 3.050 3.968 3.626 4.098 3.504 5.634 3.113 3.820 3.735 4.289 3.453 5.922 3.072 3.629 3.537 4.458 3.421 6.096 3.039 4.117 3.623 4.671 3.466 6.385 3.255 4.329 3.549 4.925 3.413 6.760 3.160 4.541 3.404 5.222 3.383 4.753 3.508 5.095 3.438 5.072 3.346 5.369 3.427 Table 2. (c) 3.000 3.000 4.000 0.090 0.240 0.090 10 15 10 7.3 7.3 8.0 TEST NO /2 LEN(,TI1 OF PLATE. /000 BREADTH OF PLATE 0 090 m. e a (e e

202 Keizo UEo, Michihiko ToiujroAo. and Takeshi HARÁ (Vol. XXIII, seen in Fig. 21 that the C1 values obtained

from the present experiments indicate higher

values than the I. T. T. C. 1957 friction line

within the range of i m. and 1.81 m. plate

Length (m.) Breadth (m.) Thickness (mm.) Temperature (CC) Corrected Test Results Table 2. (d) 4.000 5.000 5.000 0.240 0.090 0.240 18 10 20 8.0 7.0 7.0 R, Cf R, C1 R C1

Note: Values of Ra are given in units of 1 million. Values of C1 are given in units of 0.001.

s

a®

o

o

LEN&TH 0F PLATE. /000 in.

SFEAOTH OF PLATE. 0.24-O in.

Rn.

1ig. 12. (d)

length, almost the same values as the I. T.

T. C. 1957 friction line in the range of 2 m.

and 3 m. plate length, nearly the same values

as the K. E. Schoenherr friction line in the

TEST NO 13 1.572 4.044 1.822 4.200 1.773 3.712 1.837 3.841 2.102 3.800 2.057 3.638 2.121 3.806 2.435 3.478 2.442 3.472 2.340 3.769 2.901 3.116 2.779 3.490 2.586 3.572 3.182 3.340 3.087 3.373 2.791 3.695 3.504 3.400 3.402 4.417 3.036 3.550 3.889 3.181 3.823 3.301 3.325 3.458 4.205 3.049 4.138 3.316 3.643 3.392 4.520 3.201 4.559 3.192 3.990 3.297 4.836 3.044 4.979 3.115 4.307 3.242 5.116 3.019 5.400 3.145 4.654 3.273 5.396 3.096 5.680 3.111 4.943 3.237 5.782 2.969 6.065 3.056 5.290 3.198 6.237 2.805 6.521 3.003 5.521 3.150 6.517 2.826 6.661 3.043 5.781 3.130 6.833 2.771 7.012 2.965 6.099 3.060 7.113 2.956 7.365 2.884 6.359 3.075 7.358 2.712 7.716 2.875 6.792 3.026 7.919 2.746 7.996 2.854 7.168 3.048 7.639 2.669 8.206 2.820 6.994 3.433 8.199 2.685 8.347 2.817 6.474 3.037 8.340 2.605 8.697 2.667 8.690 2.578 8.767 2.912 8.865 2.656 Test Number 9 10 11

range of 4 m. plate length and also about the same values as the G. Hughes friction

line in the range of larger plate lengths than

5 m. with some exceptions such as Test Nos. 16, 17 and 18.

If we omit all abnormal

parts of the mean C1-R,, lines in Fig. 21 which might be considered to arise from various causes such as the under- or

over-estimations of the turbulence stimulation

ef-fects, the edge effects or the thickness effects

of the plates tentatively neglected as small,

we should be able to draw an approximate

mean line through the values of C1 obtained

Table

Corrected Test

Results

by the present tests within the test range

of R,, from 3.6x105 to 2.9x107 in Fig. 21. Assuming this mean line as expressible by the following form,

C3=A/(Logi0R,, -2)",

we obtained A=0.173 and n=2.55 from the

values of _{'j-R,, indicated in Fig. 21.} The

C1 values calculated by the formula C1= 0.173/ (Log10 R,,- 2)2.55

are represented by the curve named "New friction line" in Fig. 22. It is apparently

seen in Fig. 22 that the new friction line

2. (e) R,, Cj R,, C1 R,, C1 R,, C1 0.511 4.446 0.383 0.587 5.692 0.445 0.373 7.566 0.540 0.426 7.088 0.594 0.643 5.403 0.619 0.692 5.081 0.676 0.730 5.056 0.710 0.766 5.423 0.989 0.828 5.307 0.785 0.869 5.924 0.844 0.927 5.713 0.923 0.979 5.317 0.953 1.023 5.191 0.998 1.080 5.163 1.052 1.117 5.304 1.111 1.165 5.196 1.172 1.237 5.078 1.208 1.335 5.352 1.254 1.445 4.544 1.329 1.394 4.392 1.393 1.512 4.583 1.427 1.620 4.447 1.487 1.657 4.562 1.097 1.716 4.497 1.059 1.781 4.169 1.592 1.854 4.353 1.652 1.923 4.284 1.720 1.956 4.725 1.765 1.970 4.651 1.878 2.043 4.835 1.992 2.147 4.649 2.050 2.264 4.490 2.127 2.251 4.721 2.201 2.343 4.713 2.235 2.321 4.553 2.321 2.190 4.676 2.388 2.554 4.532 2.448 2.438 4.498 2.560 2.486

Note: Values of R,, are given in units of i million. Values of C1 are given in Units of 0.001.

5.908 0.726 6.269 0.792 5.052 7.124 0.845 5.593 0.939 4.318 5.480 1.085 5.050 1.067 4.455 5.127 1.124 5.015 1.171 4.241 5.619 1.261 4.371 1.263 4.151 5.316 1.321 4.552 1.380 4.151 5.409 1.423 4.146 1.430 4.240 1.485 1.520 3.793 1.487 4.319 5.035 1.637 3.856 1.618 4.134 4.976 1.715 4.091 1.719 4.116 4.525 1.829 4.376 1.844 4.080 5.052 1.959 4.037 1.955 4.128 4.778 2.069 3.810 2.006 4.029 4.681 2.102 4.019 2.246 3.912 5.630 2.236 3.958 2.072 4.038 5.420 2.314 3.954 2.322 3.942 5.375 2.394 3.576 2.457 3.935 5.221 2.494 3.486 2.535 3.774 4.629 2.601 3.471 2.722 3.712 5.086 2.800 3.552 2.837 3.775 5.068 2.966 3.767 2.914 3.839 4.972 3.154 3.571 2.972 3.749 5.397 2.746 3.702 3.154 3.759 5.686 3.335 3.597 3.050 3.743 4.833 3.499 3.528 3.251 3.739 4.819 3.671 3.600 3.349 3.827 4.814 3.899 3.543 3.421 3.812 4.872 4.008 3.547 3.548 3.813 4.732 4.106 3.506 3.772 3.802 4.560 4.120 3.516 3.815 3.807 4.610 4.336 3.541 _{3.936 3.762} 4.654 4.434 3.410 4.040 3.763 4.564 4.600 3.523 4.176 3.741 4.545 4.867 3.246 4.338 3.721 4.540 4.762 3.420 4.410 3.777 4.555 4.906 3.397 4.701 3.640 4.551 5.197 3.513 4.483 3.714 4.551 5.048 3.495 4.871 3.690 4.478 4.968 3.629 5.045 3.644 5.136 3.676 1.000 2.000 2.000 0.240 0.090 0.240 8 8.5 8.5 21.2 21.2 21.2 Test Number 12 13 14 15 Length (m.) 1.000 Breadth (m.) 0.090 Thickness (mm.) 8 Temperature (oC) 20.0

has a steeper slope than any other friction

line, and gives higher C1 values than the I.

T. T. C. 1957 friction line in the range of

smaller R, than about 4x106, cut the I. T. T. C. 1957 friction line at R=about 4x106, the

K. E. Schoenherr friction line at R= about 7.5x106 and the G. Hughes friction line at

R=about 5x107 and gives lower C1 values than the G. Hughes friction line in the range

of larger R.., than about .5 X 10v.

204 Keizo UENO, Michihiko TOKUNAGA and Takeshi HARÁ (Vol. XXIII,

Table 2. (f)

We shall not, in the present state, touch

the problem of what a role the new friction

formula has in the model-ship correlation,

which would be examined in future compar-ing with other friction lines.

6. Conclusion

The paper gives a complete experimentally

derived formulation for the frictional resis-tance of smooth plane surfaces in turbulent

R, C1 R Cf R, C1 R, C1 2.140 3.633 2.360 3.557 4.929 3.460 4.681 2.564 2.591 3.641 2.485 3.450 5.327 3.373 5.897 2.864 3.310 3.630 2.887 3.619 5.787 3.374 6.917 2.890 3.615 3.616 3.239 3.687 6.261 3.350 7.970 2.838 4.016 3.611 3.797 3.658 6.554 3.368 7.612 2.868 4.539 3.266 4.292 3.552 7.015 3.322 8.390 2.799 5.145 3.319 2.655 3.690 7.290 3.233 8.627 2.856 5.550 3.469 3.565 3.726 7.823 3.267 9.895 2.689 5.750 3.446 4.391 3.603 8.139 3.245 10.530 2.693 6.015 3.478 5.672 3.509 8.956 3.092 10.740 2.661 5.409 3.368 6.125 3.410 8.274 3.185 11.300 2.712 Corrected Test 5.741 3.446 5.087 3.353 8.540 3.160 8.784 2.789 4.230 3.483 5.320 3.411 9.272 3.116 9.510 2.623 Results _2.388 3.347 4.736 3.479 9.958 3.088 10.200 2.674 2.916 3.659 4.485 3.412 9.583 3.104 11.700 2.666 3.381 3.562 5.804 3.501 10.130 3.108 12.200 2.684 3.679 3.497 6.349 3.411 10.400 3.059 12.990 2.639 4.636 3.313 6.978 3.325 10.580 3.051 12.510 2.734 5.164 3.344 7.853 3.327 4.090 3.495 13.500 2.740 5.602 3.301 7.453 3.430 3.841 3.427 13.530 2.771 5.832 3.426 3.786 3.552 4.825 3.494 13.220 2.615 6.293 3 378 4.309 3.561 4.631 3.484 13.960 2.639 7.101 3.352 4.802 3.517 2.487 3.473 14.010 2.625 6.491 3.390 5.250 3.433 2.948 3.521 14.510 2.580 7.254 3.288 6.846 3.306 5.268 3.289 14.940 2.643 8.035 3.250 7.664 3.333 5.507 3.285 14.950 2.675 7.918 3.184 7.185 3.303 6.031 3.337 15.070 2.662 8.689 3.127 7.224 3.471 7.412 3.269 15.510 2.623 8.911 3.171 8.570 3.164 10.810 3.054 17.060 2.591 9.127 3.165 8.922 3.197 16.370 2.517 10.010 3.110 9.216 3.161 15.860 2.482 9.863 3.110 8.798 3.176 16.690 2.529 9.511 3.159 8.389 3.232 16.820 2.599 10.090 3.096 9.137 3.182 17.260 2.611 10.680 3.105 9.625 3.121 18.660 2.483 10.450 3.142 10.050 3.100 17.820 2.541 10.280 3085 10.660 3.071 18.250 2.467 11.220 3.116 10.750 3.087 19.100 2.553 10.350 3.116 18.760 2.471 19.410 2.561 19.000 2.498

Values of R are given in units of 19.370 2.558

1 million. 17540 2.567

17.910 2.516 Values of Cj are given in units of 18.760 2.488

0.001. 18.590 2.485 19.190 2.528 Length (m.) 6.000 6.000 6.000 9.000 Draft (im) 0.200 0.170 0.140 0.200 Thickness (mm.) 30 30 30 30 Temperature (DC) 9.49.7 8.5-9.0 9.4 19.0-19.1 Test Number 16 17 18 19

flow, covering a range of Reynolds number

from 3.6x10 to 2.9x107. The new formula is as follows:

0.173/ (LogioR - 2)255

The new formula reveals a steeper slope

than any other hither-to derived friction line, and gives higher C1 values than the I. T. T. C. 1957 friction line in the range of smaller R,, than about 4 X loe, lower C1 values than

the G. Hughes friction line in the range of

larger R,, than about 5 X 10 and shows the

middle C1 values of the above two in the

range of R,, from about 4x108 to about 5x

10 which corresponds to the large model

test range. Length (m.) Draft (m.) Thickness (mm.) Temperature (CC) Table 2. (g)

We shall not, in the present state, touch

the problem of what a role the new friction

formula has in the model-ship correlation,

which would be examined in future compar-ing with the other friction lines.

Finally it should be noted that the present

experiments of flat plates were carried out

to the extent of R,1=2.9x 10 in the 76

meter-long Ship Mode Experimental Tank of Kyu-shu University, and Dr. G. Hughes's

experi-ments of pontoons were carried out to the

extent of R,, =2.445 X l0 in the 207 meter-long

No. 2 Tank of the National Physical Labo-ratory at Teddington. It may be said,

there-fore, that in the larger tanks than the above

might be able to be made plane

friction

9.000 9.000 12.000 12.000 12.000 0170 0.140 0.200 0.170 0.140 30 30 30 30 30 18.8 18.1-48.6 24.l-24.7 24.6-.24.8 24.8-.26.2 20 21 R,, Ci R C1 4.588 2.541 4.431 2.319 5.285 2.743 5.201 2.824 6.077 2.906 5.954 2.887 7.008 2.906 6.767 2.904 8.175 2.800 7.605 2.832 8.784 2.671 8.487 2.668 10.560 2.630 8.170 2.591 10.540 2.634 9.171 2.566 11.240 2.664 10.040 2.564 9759 2.633 11.160 2.623 9.202 2.703 11.940 2.567 12.130 2.641 13.290 2.537 13.370 2.627 15.660 2.549 12.630 2.630 14.830 2.517 14.020 2.593 15.490 2.482 14.460 2.535 16.130 2.513 14.900 2.580 16.850 2.471 13.170 2.548 16.700 2.512 14.500 2.558 17.700 2.415 15.480 2.514 17.930 2.416 16.750 2.468 18.260 2.464 15.830 2.482 17.370 2.511 17.340 2.540 19.240 2444 17.120 2.561 15.100 2.542 16.860 2.524 14810 2.536 16480 2.478 7.823 2.858 18.280 2.461 7.502 2.811 17.390 2.544 12.330 2.550 19'430 2.499 12.720 2.440 18.760 2.434 15.820 2.475 19.010 2.462 13.550 2.568 19.560 2.532 13.100 2.475 17.790 2.476 15.530 2.493 19.400 2.464 16.110 2.512 17.110 2.546 16.550 2.523

Note Values of R,, are given in units of 1 million. Values of C1 are given in units of 0.001.

22 23 24 R,, C1 R,, Cf R C1 6.917 2.782 7.520 2.936 10.210 2.427 8.771 2.898 5.753 2.988 7.607 2.509 10.130 2.498 8.322 2.901 6.914 2.568 11.540 2.562 9.473 2.860 10.930 2.583 13.750 2.631 10.200 2.828 9.232 3.293 12.100 3.035 10.680 2.710 8.473 2.819 8.063 3.109 11.550 2.571 11.040 2.509 11.000 2.625 13.360 2.342 12.560 2.528 12.960 2.515 13.870 2.363 13.920 2.369 10.800 2.619 14.750 2.522 14.450 2.298 9.332 2.404 16.040 2.535 16.510 2.290 15.290 2.317 16.600 2.470 15.780 2.295 16.380 2.465 17.370 2.544 14.910 2.354 18.280 2.455 18.570 2.364 14.100 2.645 15.740 2.436 19.580 2.375 19.090 2.337 17.450 2.414 20.740 2376 18.250 2.401 19.590 2.251 21.550 2.324 17.590 2.372 20.380 2.300 22.320 2.330 20.070 2.366 19.890 2.341 24.480 2.333 16.510 2.340 21.410 2.341 25.450 2.335 20.150 2.479 22780 2.414 23.900 2.307 25.090 2.290 22.420 2382 26.440 2.210 22.610 2.242 23.880 2.344 26.350 2.101 21.470 2.206 25.120 2.347 27.920 1.832 23.110 2.233 24.310 2.425 29.170 1.739 24.100 2.225 26.250 2.286 6.447 3.032 26.240 2.211 25.320 2.346 19.310 2.519 26.800 2.179 23.160 2.297 20.830 2.412 28.620 2.193 28.720 2.245 27.440 2.268 25.870 2.268 20.940 2.469 Test Number Corrected Test Results

206 Keizo UENO, Michihiko TOUN.&GA and Takeshi HA1 (Vol. XXIII,

experiments to the extent of larger R than

the above, for instance, R=about 1O, which corresponds to an actual ship range of

Rey-nolds number long-awaited and also that

plane friction experiments for such a large Reynolds number as an actual ship range

should be considered.

Acknowledgement

The authors wish to express their hearty ¿ .7 6 3 3 2 2 4 a .5. 4 /04,05 _s TEST No 3 LEN&TH OF PLATE. 5READT/-j OF PLATE. Fig. 13. (b)

thanks to many students2)3)4)s) who were

directly engaged in carrying out the towing

experiments of flat plates before cited, and

also to Assistants S. Uchino and K. Oda for their cooperation in carrying out the

experi-ments, calculations and drawings.

Appendix

Correction for the Artificial

Turbulence Stimulation Device

/8/o

0,239 ,,

i I I

2 3 .

TEST NO 4

LEW&TH OF "LA TE, /8/O a..

BREADTH OF PLATE. 0.309 n - Rn. I

III

6 7 8 9 /06 I.E Fig. 13. (a) LE 2 3 4 3_ 8 7 6 5. o 4

For an artificial turbulence stimulation de-vice in our towing experiments of flat plates

we used piano wires of

1 mm. diameter

around the plate surfaces at the stations of loo

mm. abaft the leading edges of the plates. In order to obtain the frictional resistance of fully turbulent smooth plane surface Rf

we must carry out the following two

correc-tions for the measured resistance of the

7

a

G

TEST NOS

LENCiTH OF PLATE. /8/Orn.

BkEAÛTN 0F PLATE. 037g,,,.

Fig. 13. (c)

TEST NO /4

LENOTH OF PLATE. BREADTH i2F PLATE.

R,,.

-3

/0_4'o

S 6 7 8 9 /0

Fig. 14. (a)

plate R',0. Namely, we must add to the value

of R',0 by some amount of allowance for laminar part of the boundary layer within the range of length from the leading edge to the position of the trip wire of the plate

and subtract from the value of R'3-0 by some

amount of allowance for two effects of both

the resistance increase due to the proper

resistance of the trip wire own and also the

2.O0O,.

0.090

increase of boundary layer thickness due to the existence of the trip wire. If it would

be assumed to be carried out the towing

experiments within the range of speeds from

0.4 rn/sec. to 2.5 m./sec. corresponding to the

range of local Reynolds number R from about 3.5x10' to about 2.2x105, which is

al-most smaller than the critical local Reynolds number of transition from laminar to

turbu-9 8 7 6 s 4 3 2 R7. -3 I I I I I '°4S 5 ¿ 7 8 9 /0' TEST NO IS LEN&TH OF PLATE. BREADTH OF PLATE. Fig. 14. (b) 2.000 77t O.2O . /5 2 3 4 5 6 7 TEST NO 6 LENST/4 CF PLATE BREADTH 0F PLATE. Fig. 15. (a) 3000 m.. 0090 "z.

208 Keizo UENO, Michihiko TOKUNAOA and Takeshi HARA (Vol. XXIII,

lent flow, the flow in boundary layer before the trip wire position would be laminar wi-thin the above speed range. Now, the value

of 4R10, the amount of total allowance in-cluding the above two corrections and the

value of Rf0, the frictional resistance of the

plate after those corrections should be

4R= (Rr&r R51) - (1) and therefore

RjoR'j'o+ 4R50. (2) K10 R1± R1e (R1 Rftx) +R, (3)

On the other hand, Rjo=Rj+Rje where all of the above symbols are explained

/0

-

9-

8-7 6 5 4 3 2 /J I

11111

1 I I I I

II

46/0 3 ¿ 7 8 9 /06 /5 2 3 4 5 ¿ 7 Fig. 15. (b)

q-8

2 6 TEST NO 8 LENGTH OF PLATE. BEADTN OF PLATE. 4000 .O9O a

I.

¡ j t t

11111

IS 2 3 4 5 6

789,'o7

Fig. 16. (a) FIÚ /'T (/2) -i-esT F60? LE/K6-TH 0F PLATE. .3000 ,t

8REAOTH OF PLATE. O.24Om.

7 ¿

S

4

.3-I 210 Keizo UENO, Michihiko TOKUN&GA and Takeshi H&s& (Vol. XXIII,

in the Section 2. Rewriting every term of in the Section 2, and also suffixing the very

the above two equations of (1) and (3) in signs of the above resistances, we get the form of non-dimensional factors such as _{4C = (C1 - C1) R4 R,, - CRXJR,,, (1)}

the frictional resistance coefficients indicated

4 9 8 7 6

s

3 2 0/06 TEST NO 9 LENGTH OF PLATE. BREADTH OF PLATE. ,e,1 I I I I /5 2 3 4 5- ¿ 7 8 ?io Fig. 16. (b) 4000 n?. 0.24-0 »v TEST NO IO LENGTH OF PLATE. BREAVTI-I OF PLATE. Fig. 17. (a) 5.000 in_-0.090 m. I i I I

III!]

J /5 2 3 4 5 6 7 8 /07 /5 8 7 6 5 4 .3 2

C'=C1+C,..

(3)'

The values of (C1C7S) which are f

unc-tians of R, calculated by using two formulae of both H. Blasius's laminar friction line and K. E. Schoenherr's turbulent friction line are

little different from those calculated by using

two formulae of both H. Blasiuss laminar

friction line and G. Hughes's turbulent

fric-tion line, and the approximate mean value

8 7 ç 4 2 4 3 2 /0 z t R,. f I I /.s 2 3 Fig. 17. (b)

of the above two is 0.00315 within the above

range of R.

In the right-hand side of the equation (3)'

every term is dependent of Reynolds number, but independent of the trip wire effect except

the last term, and the variation of trip wire diameter d affects directly on the value of C in the last term and accordingly on the

I I f I I I 4- 5 6 7 8 7 /07 I I I I

III

4 5

6 78 9Q7

Fig. 18. (a) 7EST N /6 LEBLITH OF PLATE 6.000

ONAFT OF PLATE O20 ,,,

8

7

6 TEST MO II

LENETH OF PLATE. OOO 17

212 Keizo UENO, Michihiko TOKUNAGA and Takeshi HARA (Vol. XXIII,

value of C'10 in the left-hand side. The

tow-ing experiments of the fiat plate of 1.787m.

lengthS) were once carried out in the Ship

Model Experimental Tank of Kyushu

Univer-sity in order to obtain the turbulence

stimu-/0 8 7 6 s 4. 3 2 9 8 7 6 s 4 3 'o /06 _____Ra I I I I I I I

II tu

/5 2 3 4 5 6 7 8 9 /07 Fig. 18. (b) /5 2 3 oo TEST NO 18 LEI6&T/-1 OF PLATE. DRAFT OF PLATE. Fig. 18. (c)

lation effect by the trip wires of various

diameters and various positions. Among

these experiments, we could apply to the

present case the experimental results of trip

wires of various diameters for the definite

TEST NOI? LENGTH OF PLATE PRAFT OF PLATE 6.000 s. 0. /40 io. 6.000 m.. O. ¡70m.. I I I I I I 4 5

6? 89/o?

/5

position, that is, for x=0.100 m., in which the values of C'jo were plotted in base Rey-nolds number R,, for four kinds of trip wire

diameters d such as 0.3 mm., 0.6 mm., 0.9 mm.

and 1.5 mm., and the mean lines through the spots were drawn for each definite value of

R ¿ 5 4 J TEST NO /9 LEN2TH OF PLATE. DRAFT OF PLATE o Ro. 9000 ,. 0.200 m. o J 4 5- 6 7 89/07 Fig. 19. (a) Fig. 19. (b)

d respectively.

The cross values of

C'5-0 read from these mean lines were plotted for

various definite values of R,, in base the value

of d, and it might be considered that the

values of C'.,0 of the mean lines through these

spots at d=0 would approximately give the

I I I 15- 2 2 /02 /06 9.000 , 0.070 n TEST 0/020 LENC-TH 0f PLATE. DRAFT OF PLATE.

214 Keizo UENO, Michihiko TOKUNAGA and Takeshi HARA

values of C',0 at C=0 in the equation (3)',

that is, the values of C'1-0 eliminated the trip

wire effects for various definite values of R

and the differences between the values of

C' for d=d and the values of C',0 for d=0 would give the values of CCRIR,, for definite

8 7 6 7 G G G /0 -_,ioò 4 3 6 7 8 :.-1"?. Fig.

values of both d and R,,. Thus

of CR,/R,, would be able to be

any definite value of d in base t

R.

In the present case of d=1 i

0.100 m., we obtained approximat value of C,R7IR,,=0.00012 for ti

9 io 19. (c) Fig. 20. (a) TE-ST W022 LENCrTH 0F PLATE. DRAFT 0F PLATE. /2.000 m 0.200 1?

i

(Vol. XXIII, the values plotted for he values of nm. and x= ly the mean

ie range of

4-'/0 5 7 8 9 /07 TEST NO 2/ LENDTH OF PLATE q000 DRAFT OF PLATE 0 /40 0 @0 ₀ o 'h o 0 s 4 7 'h 00 _o

R from 1.30x106 to 5x106 corresponding to the value of CR/R=000012 approximately

the range of R from 7.3x104 lo 2.8x10

corresponds to the value of C,=0.00215 since

from the above experimental result& , while the value of R/R,=x/L=0.1O0/1.787 in the

-2 'Q T 8 7 6 5-4 3 2 I I

i

I I i S 6 7 8 ' io? (j o o -3 "05r/06 6 7 5 7 TEST NO 23 LENGTh OF PLATE DRAFT OF PLATE R7 o Fig. 20. (b) TEST NO 24 LENGTH OF PLATE ORAFI OF PLATE Fig. 20. (e) 7 /2 000 0/70 ì I I 2 3 /2 000 QL 0./SO 7L o 's

present case.

Thus, substituting the value of (Cft,CfI)

=0.00315 and also the value of C=0.00215

in the right-hand side of the equation (1)',

we obtained

= (0.00315 - 0.00215) RJR = 0.00lx/L.

Accordingly, we get

4R,0 = 4C). xpILV2= 0.00lpxlV2.

References

G. Hughes: "Friction and Form Resistance in Turbulent Flow, and Proposed Formulation for Use in Model and Ship Correlation", T. R. I.

N. A. 1954.

T. Ochiai and M. Tanaka: "Some Experiments on Frictional Resistance of Flat Plates", Gra-duation Thesis, Department of Naval Architec-ture, Faculty of Engineering, Kyushu University

(1952).

A. Itoo, T. Itoo and M. Hamasaki: "Some Ex-periments on Scale Effect of Model Resistance Tests. Appendix, Experiments on Frictional Re-sistance of Flat Plates", Graduation Thesis,

Department of Naval Architecture, Faculty of Engineering, Kyushu University (1953). A. Ushio: "Some Experiments on Frictional Resistance of Flat Plates (Cases of 6 m., 9 m. and 12 m. Length)", Research Thesis, Advanced Course,Department of Naval Architecture, Facul-ty of Engineering, Kyushu UniversiFacul-ty (1954). K. Taniguchi and K. Tamura: "On the Blockage Effect", Report 307, Experimental Tank (Naga-saki) Laboratory, Mitsubishi Shipbuilding and

Engineering Co., Ltd., Aug. 10, 1958.

Y. Hiraga: "Experimental Investigations on the

Frictional Resistance of Planks and Ship-models",

Journal of the Society of Naval Architects of

Japan, 1934.

K. Ueno: "On the Ship ModeL Experimental Tank of the Kyushu University", Technical Report of Kyushu University, Vol. XV, No. 1, 1940.

T. Nomura and R. Mikami: "Frictional Resis-rance Experiments of a Flat Plate on Turbulence Stimulation Effect of Trip Wires", Graduation Thesis, Department of Naval Architecture, Facul-ty of Engineering, Kyoshu UniversiFacul-ty (1953). 216 Keizo UENO, Michihiko TovNAGA and Takeshi HA (Vol. XXIII,

'7 8 7 /06 8 SUMMAP 0F TEST RESULTS

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