Study on scale effect of propulsive performance by use of geosims of a tanker

(1)

October 1966

MITSUBISHI HEAVY INDUSTRIES. LTD.

-....-- -___,,i,_. .,.-.-

'---Lc.

ACi

T, a

5 '(

A

MITSUBISHI TECHNICAL BULLETIN No. 39

MITSUBISHI TECHNICAL BULLETIN MTB 010039

Study

on Scale Effect of Propulsive

(2)

Study on Scale Effect of Propulsive Performance

by Use of Geosirns of a Tanker

Kaname Taniguchi*

Abstract

On three geosim tanker medels, 4.2, 7 and 10m in length, resistance test, self-propulsion test, propeller open-test and wake survey test were carried out. Analyzed results on these models are compared with the standardization trial data of the corresponding actual ship.

One of the most important items to be made clear to improve the model-ship correlation seems to be the scale effect on propeller open-characteristics.

1.

Introduction

The method to predict the propulsive performance of an actual ship from the tank test data obtained 011 her model, i.e. the so-called model-ship correlation method has been the most important subject for the tank people concerned in the practical application. Thus a lot of papers dealing with this subject statistically have been published and we are now in the situation to be able to predict an actual ship performance within a practical tolerance, say 1/4 knot for constant power, from the model test. To improve further the present state of accuracy of prediction, however, it is much more impor-tant to study the propulsion factors themselves than to accumulate statistically the correlation data. This is the authofs opinion and the purpose of the present investi-gation is to contribute to this direction of study.

In this study various tests were carried out on three geosim tanker models, 4.2. 7 and 10m in length each-resistance tests, self-propulsion tests, propeller

open-water tests of the corresponding geosim propeller

models and wake survey. They were repeated several times in various water temperature during two years.

Table 1. Particulars of models used and experimental tank

Dr. Eng.. Manager, Nagasaki Technical Institute, Technical Headquarters

Based on the test results, investigations were made il:tD the resistance components and scale effect on self-pro-pulsion factors and on propeller open-water character-istics, and the standardization trial data of the corre-sponding actual ship were analyzed using these data.

2.

Test Scheme and the Models Used

A standardization trial of a 46000 DWT tanker was run by our Nagasaki Shipyard on the measured mile

course. Off Miye" and the trial results are reported in

ref. (1). Her three geosim ship models were made of wood with all appendages and the propeller models were made of aluminium alloy. The ship models were painted with polyurethan paint and their off-sets were checked using a model shaping-machine and confirmed to be in coincidence with the design values within tolerable error. Besides these three wooden models, a 7m paraf-fin-wax model was also made by our routine practice for comparative test.

The particulars of these models and the experimental tank are shown in Table 1. The method of experiment and analysis is omitted in this report, since the details are described in ref. (2).

M. No.Material Scale ni

B mm

Skin area Blockage ratio Lj/b in

F.L. eq. (2) Remarks FL. inni B.C. nm F.L. in B.C. en F.L. zn2 B.C. mn2 F.L. B.C. B.C.

1270 Wood 1/50.714 4.2 602.71 223.82 115.51 4.2854 4. 1347 3.737 2.787 0.00618 0.00316 0.7037 0.6789 Small tank

used 1382 'i 1/30.429 7 1004.5 373.03 192.51 7.1424 6.8911 10.381 7.743 0.00474 0.00243 0.5714 0.5513 Large tank used 1483 'i 1/21.3 10 1435.02 532.91 275.02 10.2033 9.8446 21.186 15.802 0.00968 0.00496 0.8163 0.7876 " 14.82 Paraffin 1/30.429 7 1004.5 373.03 192.51 7.1424 6.8911 10.381 7.743 0.00474 0.00243 0.5714 0.5513 « Wax

P. No. Material Scale Dia

mm

Pitch

nm

Pitch

ratio Bossratio Ae/Ad (t/c)o.7 Z

1280 Al-Alloy 1/50.714 130.14 95.63 0.7348 0.1818 0.5600 0.07371 5

1281 'P 1/30.429 216.90 159.38 a ', mi p, , 1282 " 1/21.3 309.86 227.70 " i' 'r i' 'i

Experi-mental

tank Length_ni Breadth_m Water depth ni Mean sect. area of tank water in' Small 120 6.1 3.65 21.65 Large 165 12.5 6.50 78.35 Small+ ₂₈₅ Large

-

(3)

The Mitsubishi Experimental Tank (Nagasaki) has

two kinds of sections, i.e., a bigger tank (165 x 12.5 x 6.5ml is connected longitudinally with a smaller one (120x6.l X

365m). We can, therefore, check the blockage effect very accurately by towing the same one model through-out the both tanks. Our formula for correcting blockage effect has been thus deduced and by this correction we can test a relatively very large model in our tank with-out any fear of excessive blockage effect. The reason of adoption of 10m model is based on such circumstances.

The first test run was commenced in the summer of

1961. Then the tests were repeated several times for

each model in the various water temperatures. The

tests were conducted for full load condition as well as ballast condition and the final test was run in April

1963.

3.

Resistance Tests

The particulars of resistance tests performed are

shown in Table 2. The residual resistance coefficients calculated by routine Froude's method from these test results are shown in Figs. l'-4. In these figures. no blockage correction is made and Prandtl-Schlichting's formula is used for frictional resistance, in accordance with our then routine practice. The mean lines of these figures are compared altogether in Fig. 5. from which we can see remarkable discrepancies between the four geosim models as well as between the case of different water temperature for the same model.

Nowadays it is considered more reasonable to assume that the total resistance can be split into viscous and

Table 2. Particulars of resistance tests performed

U) Full Load Condition (Sa/5=6.340)

(2) Ballast Condition (2% Aft-trim)(Sa/V5*=7.548)

a Abandoned due to questionable results

0.020 o u 0.015 0.010 0.005 0.020 0.015 0.010 0.005 Load

rond, M ker TentNo. Date ofrapt. Temp.("C) 4 ( k

Full load O T,-738 T,e-741 T5s- 742 1961.10. 4 1961.10. 9 1961.10.10 24.3 23.9 23.5 _452.5 T,s-947 1963. 3. Ballast eond. 2% A tnm T,-739 Ta-740 i9C 10 5 ' 6 24.5 24.4 224.4 T,,-948 1963. 3. L I _Mark m) (rn°2)S ¡WL Bid CB 'L _diept.Slap 224 3.737 4.285 2.893 47.3 60500 Ballast rond. 2% A trim o 116 2.787 4.135 5.218

-

30000

s

Loa1 Mark

'T4(kg

Foil load - Ta-1522 Te-1527 1961.7.29 1961.8. 5 21.2 21.6 Te-1575 1961.9.21 23.5 Ballast rond. 3% A tr,m 5 Te-1523 Te-1528 Te-1584 1961.7.31 1961 8. 6 1961 9.20 22.2 21.6 24.0 1039 Mark _(ms) LW Bid C.B.WL 373 50.391 7.142 2.963 47.31 60500 Ballast rond. 2% Atrim O 193 7.743 6.891 5.218

-

30000

M. No. To-No. Date of expt.

Temp.

wer,

"C m2/s 1270 Tas-739 1961.10. 5 24.5

>6

740 1961.10. 6 24.4 0.905 948 1963. 3.19 11.8 1.242 948 1963. 3.22 12.8 1.208 1382 TRL-1523 1961. 7.31 22.2 0.953 1528 1961. 8. 6 21.6 0.966 1576* 1961. 9.25 24.3 0.907 1584 1961. 9.30 24.0 0.913 1482 Tat-1521 1961. 7.28 21.1 0.978 1526 1961. 8. 4 21.9 0.959 1483 TRL-2018-1 1963. 4.12 11.5 1.252 2018-2 1963. 4.14 11.4 1.256

M. No. Te-No. Date of expt.

Temp. of water, "C e m2/s 1270 Tes-738 1961.10. 4 24.3 _0.907x106 (4.2 m, 741 1961.10. 9 23.9 0.916 wooden) ₇₄₂ _1961.10.10 _23.5 _0.924 947 1963. 3.18 11.8 1.242 947 1963. 3.20 12.0 1.235 1382 T-1522 1961. 7.29 21.2 0.975 (7m. wooden 1527 1961. 8. 5 21.6 0.966 1575 1961. 9.21 23.5 0.924 1482 Tt-1520 1961. 7.27 21.0 0.980 7m. paraffin; 1525 1961. 8. 3 22.1 0.955 1483 TaL- 1587 1961.10. 6 24.1 0.911 10m, wooden 1588 1961.10. 9 23.5 0.924 2016 1963. 3. 9 11.5 1.252 2017 1963. 3.11 11.5 1.252 310 0.12 0.14 0. 6 0.18 0.20 22 FOWL = V/L1L

Fig. 1. Residual resistance coeff. curves (M. 1270)

0.10 0.12 0.14 0.16 0.11 0.20 22

FSWL= v/,/gLwL

(4)

u 0.020 0015 0010 n 0.005 0020 0.015 - 0.010 u 0.885 0.0 15 0.010 a 0005 u 0.0 15 a 0.0I0 0.00 5

Mark M. No. Remarks 1270 42m Model I Wooden) - - 1382 70m Model ( Wooden) - 1402 : 70m Model I araRhiii 1403 10m Model Wooden) ç2 nn'' .-.-1eo1523&1528 oo" M.1270 %400s'r....__: - -M. 1483 M. 1482 Tn, 52101526 Tno738,741&42 M.1270 leo2OI6&2 Te,152261527 I lso947 8.1482 T,, 152081525 13 82 17 . I;

/

Loan! tond. Tent No. Date of expl.

'°r

Full load even keel Te-1520 1961.7.27 21.0 Te-1525 1961.0. 3 22.1 2095 Ballast rond. 2% A trim '1e-152l 1901.7.28 21.1 Te-1526 1901.8. 4 21.9 1039

Load rond _(nm, _(m'i)s i O'L _B/sd cew _do0ot.Sh,p

Ful! load

373 10.381 7.142 2.693 47.31 60500 Ballant ronod.

193 7.743 6.891 5.218 - 30800

Mark

Dof

'1io I

Full load r en hes.! 0 Te-1587 Te-1588 6 6 19 1.10. 9 24.1 23.5 6188 're-°816 Tes-2017 9 1903. _ll 11.5 BallanO tond. 2% A trim C Tes-2018 1963. 3.12 11.5 3029 Load _{Mark nm,} rn'i B,d CBL Pt. 533 21.186 10.243 2.693 47.31 60500 Ballast rond. 2% A trim 275 15.802 0.845 5.218 3808) FOOL V;KL,,i.

Fig. 5. Comparisons of residual resistance curves for

Geosim Models

c'=R/

v°S: Wave-making resistance coefficient

based on S

cfo=O.066 (Iog,0R,,-2.03)2 : Hughes' frictional resistance coefficient in two-dimensional

flow

k : Form factor.

In the case of full shipform such as tanker, the value of k is O.45-O.5 . Since the frictional resistance by Prandtl-Schlichting's formula is approximately 1.12 Cfo in the range of Reynolds number considered in the present study. the difference Cjo (l+k)-l.l2cfo is includ-ed in the residual resistance component. which is, there-fore, affected by a temperature and scale effect. This

seems to be the reason of the discrepancies revealed in

Fig. 5. To verify this point, the following analysis was

made.

In order to remove the blockage effect, the measured velocity was corrected by the following formula11:

.iv/v=l.l m(L/b) (2)

where.

iv: Velocity correction due to the blockage effect m=A/Ar: Blockage ratio (ratio of sectional area of model to that of experimental tank) L: Length of model

b: Breadth of experimental tank

After making the blockage correction, corrected c,'0 were plotted on the basis of corrected F,, as shown in Figs. 6 & 7. From these figures e,, was read at certain round Froude numbers and plotted on the basis of c ,, as

shown in Figs. 8 & 9. From Figs. 8 & 9, we can see that the relation (U holds good and that k and c, can,

therefore, be analyzed as a function of Froude number respectively. Namely (l+k) is determined by the slope of co-cf, curves and c,, by the ordinate at cj0=O in the figures. Thus analyzed (l+k) and e,0 are plotted in Fig. lo for both load conditions. These analyzed c.-curves are compared in Figs. 11 & 12 with the measured results of e deduced from the following relation:

e,=--a-- c'=

s

{c,-cfo(l+k)} (3)

V3

where k is taken from Fig. 10.

From Figs. 11 & 12 we can see that the analyzed c-curves fit the measured points reasonably. Therefore, we can conclude that the relation (1) holds good and the

2.10 0.12 0.14 0.16 0.18 0.28 0.22 FIOWL= v/gLwL

0.10 012 0.14 0.16 0.15 0.20 0.22

Fao,. =

wave-making resistance than to follow the traditional Freude's concept that the total resistance can be divid-ed into a flat plate frictional resistance and a residuary resistance. Thus, we may put:

c,=c0 +c,'= cjo(l + k)+e.' (1)

where

v2S : Total resistance coefficient based on S

v°S: on S

Viscous resistance coefficient based

0.10 0.14 0.16 018 0.20 0.22

'M,1382 M.1483 1,, 1575

(5)

4

TOO

l"ig. 6. Plot of total 'resistance coefficients (Full load

cond.) (Blockage effect corrected)

X Q5 0 142 I 1941,10,10 23.5 r'941 1063. 5.18 + r947 2 rooi. 5.20 20 0_0 o . ç) X

.

rk 220 4 Orrr Moan NO 238 961,10. 4 24.3 O 741 1541.10. 9 23.9 M. 482 t 7m Oodel Paralltnt No Date 520 961 1.27 1525 961. 8 3 Dol, Torro TOrrID 21,0 22.! .13821Irr 004011 Mark To No. J Dale temp 1522 1961. 7. 29 21.2

-

o 11s'3f Ti50T'8. 4 31.8' - r iSTO 'T296! '9 o'fl'20.s -aol a 9.10.. 0.0 15 0.005 0- 0.005

Fig. 10. Analyzed C & (1+K)

0.010

0.00 5

Fig. li.

Plot of wave-making resistance

(Full load cond.)

0.10 0.12 0.14 0.16 018 0.20 0.22

Fro WL VfLi7o

coefficient

thus analyzed c4 involves no scale effect in contrast with the residual resistance coefficient which is affected by scale effect as mentioned above. Also from Fig. 10, we can learn that the value of form factor is not con-stant for a speed range to be considered in practice. Although it is constant at low Froude number, it

shows a remarkable decrease at higher Froude number

p,

p'

4Ij'

1 :y'r

-Ill

Mono 'Tn-No. I DOlo Tolere C

1581 '961. lO. 6_ _?&.5 -0 20 1588 2016 960, tO. 9 1963. 5. 9 23,5 00,5 -r- 2017 0963. 5. tI 105 IBo0oot 0004 M. 02721 42m Model)

r

0 _o _ -oTTI-I . 739 1961.10. N 740 ' I9MJjQÇ ' '' . 945T 1963, 9 19 . __J 963. 2,22 t

:

M. 1482 I 2m Model Porollirol

::°.:

. 1576 t961.9Z5J a -. X -. O M 1443112m Modell Mark 0-Pt. D le Temp C 201M 0963 3,22I 11,5 314) 114 o 2018 t 1963. _7 000 'OT 23.9 7

''-,..I'.

.,o,toOT,s Il_0 120 - -. _1T26

I

UIl1Ø

,reiL

'\

II

U

_4

o tM.1402 ' IØ)M.l382

.4

483

.4g

1

010 0.12 0.14 0.16 0.10 0.20 0.22 Fnwt'' v/JgLwL

I-ic. 7. Plot of total resistance coefficients (Ballast cond.) (Blockage effect corrected)

2.0 2.2 24 2.6 28 30 3.2 3.4 3.6

C10 X10

Fig. 8. Plot e1 on the basis of Cfo (Full load cond.)

20 2.2 24 2.6 28 30 32 34 3.6

Ci0 x100

Fig. 9. Plot of et on the basis ofCfo (Ballast cond.)

OID DIO 014 016 618 020 0.22 30 U O 0.12 0.14 0.16 0.10 0.20 0.22 FOWL = X a ç) .25 3_Q 4 5.5 5.0 '.1 _4,5 (.1 3.5 3.0 2.5 0015 Dol Q 0.005 o 5.5 5.0 2 4.5 X Q COO a 3.5 3. 25

(6)

s ç) 0.015 0,010 a II 0.005 0.015 :0.010 ç) 0.0 05 ç) 0.015 0.0 10 0.005 - 0.010 - 0.005

Table 3. Particulars of propeller open-tests performed

Used for analyzing self-propulsion resuits

0.35 030 005 0.25 h 0.04 0.20 0.03-' 0.15 0.02- 0.10 I-. O 5

Tos-No. Date of expt. R.P.S. R,6 1/D

1036' 1961. 9.30 24.5 13.0' 1280 1037 « e 34.0 3.522 (130.14mm) 1252' 1963. 2.21 11.2 13.0 0.963 1 1253 " e 34.0 2.518 o 1020' 1961. 7.26 21.5 10.0 2.684 e 1281 1021 e o 17.0 4.561 o (226.90 mm) 1026' 1961. 9.19 23.6 10.0 2.819 e 1027 e " 16.0 4.507 'i 1038' 1961.10. 2 23.9 8.0 4.631 0.806 1282 1039 e ,, 5 4.922 « (309.86mm) 1254' 1963. 3. 2 11.5 8.5 3.587 0.775 1255 " " 9.5 4.017 e .

___,

I_

UI./?.í'..,

#

'1!

i

ID 130.I4,,4I 0.1818 P 05,63,, '1o' 7.371% A,/Ad 0.7348 8) o,s600 no.,) Ô se,5, 854e M S 5 . " e, 020 ' -D 236SOn,, dID O 1818 Stco,,O) A,Ì4d B5Ôt _174 O5600 7, 7 tod, 5 ep ---Oe 1038.1d39_{,, 0254} . Too 1255

U'o

I

!ALiUll

'Wl D ,309.90,, L D O 0. 7.371% M-6 518 P 227. 70m )í)

POonst.!j 0,7348 81o,eoeohoo . ' . i

Ap/Ad 0.5600 No ot6odn 3

0 1 02 0.3 0.4 0.5 0.6 0.8

J Vp/IlL)

Fig. 13. Open-characteristics of P. No. 1280

O I 0.2 0.3 0.4 0.5 06 0.7 00

J Vp/nD

0 0.1 0.2 0.3 0.4 0.5 06 0J 08

J= Vp/flD

0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24

Fnw: v/jgLwL

Fig. 12. Plot of wave-making resistance coefficient

(Ballast cond.)

(F>0.l5). The decrease of k-value at higher speed may be considered due to the fore trim caused by running. Another point to be noticed in Fig. 10 is that c at low Froude number in ballast condition is not zero but finite

(0.0004). The non-zero c, at very low Froude number is theoretically reasonable for a shipform with remarkable blunt stem (bulbous bow) and this fact raises a question about the validity of routine Hughes' method which assumes a zero C at very low Froude number in order to analyze k. The another Hughes' method of analyzing k which utilizes the minimum point of c-F, curve, was also tried to analyze the present experiments. The re-sults. however, was not satisfactory as seen from the fact that c at very low Froude number is not zero in ballast condition, while the Hughes' method assumes the

relation c0ccF,,,P.

From the above-mentioned discussion the following conclusions may be derived.

(I) _{The traditional Froude's method is not reasonable}

especially for full shipforms and the relation

1

should be used as a new substitute. When a resist-ance test is analyzed by the relation (1), no appreciable scale effect in wave-making resistance can be seen, provided a model of moderate size and an appropriate value of k are used.

On a shipform with an extremely blunt stem, there might be a case that a wave-making resistance coef-ficient is not zero even at very low Froude number. In such a case, the usual method of determining k in low Froude number range assuming c)=0. has to be

improved.

Although the value of k can be taken constant in low Froude number range, it is not valid to assume so in higher Froude number range (F5>0.l5 irs the present case of study). To improve the model-ship correlation, this point should be taken into

considera-tion.

4.

Propeller Open-Tests

The propeller open-tests were performed as shown in Table 3 on the three geosim propeller models which had been made in scale in accordance with the 4.2, 7 and 10m ship models respe2tively in order to carry out the self-propulsion tests.

The test results are shown in Figs. 13-45. The

char-0.0! - 0.00 0.3 0.3 0.05 0.2 - 3.20 0.04 rs OIS f-, OEI0 0.03 -s-0.02 0,0! - 0.05 0.7 0.6 0.5 0.4 '-''t 0.3 0.2 0.! 0.35 0.30 0.25 rs 0.20 0.04 rs 0.03 15 0.02 - 0.10 0.05 001 07 0.6 0.5 0.4 0.3 0.2 07 0.6 0.5 0,4 0.3 0.2 0.!

(7)

6 0.15 0.25 0.20 0.10 0.05 0.030 0.0 15 eolo 0.7 0.6 0.5 04 2 3 Relc X l0'

Fig. 17. Plot of Ka for Three Geosim Propellers

2 3

ReO X 1O"

Fig. 18. Plot of ep for Three Geosion Propellers acteristic values, K0, K0 and e4 are read at J-values of 0.4, 0.5 and 0.6 on the mean lines of open-characteristic curves and plotted on the basis of R074 in Figs.

1618,

where

Rek=(Go.?/14.'1v2+(0.7ltflD)2 : Reynolds number by

Kempf's definition.

From Figs. 16-48 it is seen that the propeller charac-teristics vary reasonably with Reynolds number except those for the smallest propeller (P. 1280). The charac-teristics of P. 1281 and P. 1282 coincide with each other within 1% error approximately and the similarity of these propellers can be considered satisfactory. (Since

these propellers were made carefully by use of the

Kempf and Remmer's point drilling machine, it can be thought that the finishing tolerance is within ±0.03mm

approximately.)

The value of K, becomes constant

approximately above R,744x10 and that of K4 shows gradual decrease with the increase of R074. This is in accordance with the well established trend by many researchers.

The test results of P. 1280 (the smallest), however, show a remarkable scatter and from this it may be con-cluded that the reliability of self-propulsion test on the smallest 4.2 m-model is considered insufficient, even if the resistance characteristics of ship model is reliable.

5.

Self-Propulsion Tests

The self-propulsion tests were carried out repeatedly on each model as shown in Table 4. The self-propulsion factors, w,,,, t and er were analyzed for each measuring run by the usual method of thrust-identity by use of the propeller characteristics tested in the nearest occasion to the self-propulsion test (Table 3). Thus analyzed self-propulsion factors are plotted for each model in Figs. 19--26. in which the test results obtained within a week or so are marked by the same symbo], but those obtained at interval of six weeks or more are distin-guished by a different symbol.

Table 4. Particulars of self -pro pvlsion tests performed

SFC5 not stated are for ship point of self-propulsion (Cf8=O.49O/(1ogl,RnJ)" os) ----,.-c---*J=0.4

_r-&.

-

J 0.5

-. sb-.-

-J-0,6 Massi p.t D I 0284 130.14 O 1281J 21690 0282 J 309,86 J =0.4 ./ 0.5

--J=O.6 + -

'.

_e Mark P.90. D + 1280 130.14 O 1281 226.90 2282 309.86

Cond.S M. No. Ts-No. Date of exr,t. Temp._C Dynamo-_meter SFC

1270 (4.2 rn) T50- 181 182 183 185 1961.10. 9 'r 1961.10, 10 1963. 3.18 23.9 r, 23.5 11.8 lSD. No. 3 r lSD. No. 2 SSD. IV Zero (Wooden) 187 1963. 3.20 12.0 'r u O o 188 r, ,r r Zero 1382 (7m) T55.-1259 1263 1961. 7.29 1961. 8. 5 21.2 21.6 lSD. No. 2 'r (Wooden) 1280 1961. 9.21 23.5 'i a5 1280 r, r, i, Zero 1482 (7m) Ts5.-1257 1961. 7.27 21.0 r' (Paraffin) 1261 1961. 8. 3 22. 1 I' Ts5.-1283 1961.10. 6 24.1 r' 1284 1961.10. 9 23.5 'r 1483 (10m) (Wooden) 1284 1470 'r 1963. 3.11 r, 11.5 'r 'r Zero 1472 1963. 3.13 11.5 'r 1473 r, r, Zero T55- 178 1961.10. 5 24.5 lSD. No. 2 179 1961.10. 6 24.4 lSD. No. 3 1270 180 184 'r 1963. 3.16 r, 11.8 r, SSD. IV Zero 186 1963. 3.19 11.8

Î

189 1963. 3.22 12.8 'r 190 r r r, T01-1260 1961. 7.31 22.2 lSD. No. 2 1264 1961. 8. 6 21.6 r' 1382 1282 1961. 9.30 24.0 .5 a 1282 'r r, r, Zero TSL-1258 1961. 7.28 21.1 r' 1482 1262 1961. 8. 4 21.9 'i 1483 T55.-1471 1474 1963. 3.12 1963. 3.14 11.5 11.4 r, 1475 ,r ,r ,r Zero 5 280 130.14 o 1281 216.90 2282'309.86 : + J=0.4

Fig. 16. Plot of K4 for Three Geosint Propellers

2 3 4

Rk X 10'1

0.025

'e

(8)

0) LO 0.9 0.6 0.5 1.1 1.0 09 0.6 0 0.5 0) 0.4 0.3 0.1 1.1 04 0.3 0.2 0.1 Fnw,,= v/gLwt

Fig. 19. Plot of self-propulsion factors

(M. 1270, Full load cond.)

1.1 1.0 0.9 0.6 0.5 0.4 0.3 0.1 1.0 f 0.5 1.1 09 0.6 0.4 0.3 0.2 0.1

...-....

--

--...

010 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 F,WL= V/JgLwL

F,1.= v/gLItL

(M. 1382, Ballast Gond.)

7

Ta-No. Date of pxpt. Temp. "C Mark

T,.-1283 1861.10. 6 24.1 0

T-i284 1961.10. 9 23.5 0

T,-l47O 1963. 3.11 11.5

T-1472 1963. 3.13 11.5 S

Te-No. Date of expt. Temp. "C Mark

Tm-181 1961.10. 9 23.9 0

Ts-183 0961.10.10 23.5 C'

Tm-185 1963. 3.18 11.8 5

T-187 1963. 3.20 12.0 S

Uoo UflUU

T,-No. Date of capt. Temp. "C Mark

Tm-178 1961.10. 5 24.5 0

T-179 1961.10. 6 24.4 ç)

T,,-1$4 1963. 3.16 11.8 5

T-i86 1963. 3.19 11.8 5

Ts-i89 1963. 3.22 02.8 S

Te-No. Date of expt. Temp. "C Mark

Ts,-1259 1961. 7.29 21.2 0

T.-1263 1961. 8. 5 21.6 ₀

T-i280 1961. 9.21 23.5

Te-No. Date of expt. Temp. "C

Tek-1257 1961. 7.27 21.0

Ta-1261 1961. 8. 3 22.1

l'e Ne. Date of capt. Temp. "C

Tell58 1961. 7.28 21.1 Io.'1252 1961. 8. 4 21.9 Li 1.0 1.3 1.2 0) 1.1 09 0) 1.0 0.6 0.9 0 0.5 0.8 0.4 0.6 0.3 . O0.5 0.2 0.4 0.1 _0.3 0.2 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.10 0.18 0.19 0.20 0.21 0.22 0.23 Fflu-L v/igL- 01 010 0.11 012 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.10 0.1 0.12 0.13 0.14 0.15 0. 6 0.17 0.18 0.19 0.20 0.21 0.22 0.23 FnWL= v/JgL'L

Fig. 23. Plot of self-propulsion factores

(M. 1270, Ballast cond.)

g:.. .Î

0.2 _0.2

010 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.23 6.23

FnWL V/.fJiij

(9)

1.! LO 09 0.6 0.5 04 0.3 0.2 0.1

. 1 Standard deviation of self-propulsion factors, etc.

It will be noted that the scatter of plotting of self-propulsion factors shown in

Figs. 19-26 is

rather remarkable. In these figures, the standard deviation band is drawn so that a half of the total points lies in-side of this band and another half lies outin-side of the band, excluding a few exceptional points. The mean half breadth of this band may be considered approxi-mately as the standard deviation of the plotting and it is read from the figures and summarized in Table 5 with those for k,, k, and k0 (non-dimensional thrust, torque and revolutions in self-propulsion tests

respec-tively). From this table the standard deviation of the

self-propulsion factors and others may be taken as

follows:

4er/er0.005, 4wm/(1_Wo,)0.008, 4t/(1 - t)0.0l0

4k,/k,=0.0l1. jk5/k5=0.012, .4k0/k0=0.005.

The standard deviations for the

smallest model (M. 1270) are generally bigger than those for the larger models as expected. But it is also to be noted that the standard deviations for the largest model (M. 1183) are not improved so much as expected from the size. The cause of this fact seems to be not only due to the insuf-ficient accuracy of measurement. but also due to the complex nature of the velocity field surrounding a stern part of full shipform. Therefore. the effort to improve

8 1.1 1.0 0.9 0.6 0.5 O u 0.4 0.3 0.2 0.1

...--..

010 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 FOwL= v/JgLv,.

(M. 1488, Ballast cond.)

such situation is considered necessary for the improve-ment of model-ship correlation.

5.2 Scale effect on the self-propulsion factors Although the scatter of plotting of self-propulsion factors is not small, the mean lines can be drawn with considerable confidence as shown in Figs. 19--26 and they are compared altogether in Figs. 27 & 28. From

these figures the following facts can be seen:

As for e, no appreciable scale effect can be noted except the case of ballast condition of M. 1270 and the mean lines for each model are fairly coincident in the speed range commonly used.

The mean lines of w5, show a regular trend of de-crease with the inde-crease of R,, except those of M. 1482

(7 rn-model made of paraffin wax).

As foi' t, no definite trend can be found but a some-what similar trend to w,,,, may be thought.

In order to make these points clearer, the values of self-propulsion factors at P, =0.15 and 0.20 are selected as representatives and plotted on the basis of length of ship model as shown in Figs. 29 & 30. From these figures the following conclusions can be drawn:

It may be considered that there is practically no scale effect on e,.

There is a definite scale effect on w.,, and upon this point some discussion will be made later.

T,-No. Dote of eapt. Temp. 'c Mark

T,,.-1260 1961. 7.31 22.2 0

T,-3264 1961. 8. 6 21.6 0

T,,-1292 1961. 9.30 24.0 S

T,-No. Date of Capt. Temp. c

T,,-1471 1963. 3.12 11.5

T,,-1474 1963. 3.14 31.4

M. No. 4er. (4e,,/e0) 4',,,,, \

/ 4t

4t. -')

No.8 of

tests M. No. 4e0, (4e0/e0)

4w,,, \ 4w / 4t ' No.S of_tests 4Wm. (i-wm) m' (

i-W)

4t,

-)

1270 0.010 (0.010) 0.005 (0.010) 0.010 (0.013) 2 1270 0.012 (0.012) 0.006 (0.013) 0.007 (0.009) 2 1270 0.005 (0.010) 2 1270 0.007 (0.007) 3 1382 0.003 (0.003 0.005 ('0.009) 0.010 (0.013) 3 1382 0.006 (0.006) 0.004 '0.008) 0.010 (0.013) 3 1482 0.004 (0.004) 0.002 0.004) 0.007 (0.009) 2 1482 0.003 (0.003) 0.002 (0.004) 0.005 (0.006) 2 1483 0.004 (0.004) 0.004 (0.008) 0.007 (0.009) 4 1483 0.003 (0.003) 0.005 (0.009) 0.005 (0.006) 2 No.S of No.8 of M. No. 4k, (40'1,,) 4k5, (4k5/k5) 4k,,, (41c,,/k,,)

tests M. No. 4k,. (4k,/k,) 4k5. (4k5Ik) 4k,,. (4k,,/k,,) tests

1270 0.0003 (0.013) 0.0004 (0.016) 0.05 (0.009) 4 1270 0.0004 (0.013) 0.0006 (0.019) 0.03 (0.005) 5

1382 0.0002 cO.009) 0.0002 (0.009) 0.02 (0.003) 3 1382 0.0003 (0.011) 0.0003 (0.011) 0.03 (0.005) 3

0482 0.0003 (0.015) 0.0003 (0.014) 0.03 0.004) 2 1482 0.0002 (0.007) 0.0002 (0 007) 0.02 (0.004) 2

1483 0.0002 (0.010) 0.00.2 (0.006) 0.03 (0.004) 4 1483 0.0003 (0.011) 0.0002 (0.007) 0.02 (0.003) 2 Table 5. Standard deviation of plotting of self-pro pulsion factors, etc.

(1) Full load cond. WLO15O2O) (2) Ballast cond. (2% Aft-tr'm) (FnwLi'O.2O)

N.B. k0=2,nQ[-v.V*, k,,-' (n/v,,,)V+

0,10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23

Fnwj V/.'gLwL

(10)

.0 0 0.5 0.4 0.3 LO E 0.5 .1 0.9 0.6 0.2 0.1

4

0.10 0.11

Fig. 27. Comparisons of self-propulsion factors among

4 Geosim Models (Full load cond.)

1.1 09 0.6 04 0.3 0.2 0.1 In 105,187 T 581.183 -(s 1470:1472 0.12 0.13 0.14 0.15 0.16 0.17 Foivi. = v/..ÍgLwL 0.18 0.19 0.20 0.21 0.22 0.23 Ia 184,186 189 0.50 0.15 0.12 0.13 0.14 0.15 0.16 0.17 0. 8 0.19 0.20 0.21 0.22 0.23 F,oI'L= V/fL'

Fig. 28. Comparisons of self-propulsion factors among 4 Geosim Models (Ballast cond.

(3)

The value of

t decreases with increasing scale.

However, it seems to be approximately constant for the models bigger than 7m.

The last conclusion is to be noted, since some of the theoretical studies suggest the increase of t with in-creasing scale

To discuss the scale effect on w,0, w,s's of each model at F5 =0.15 and 0.20 are plotted on the basis of e, in Fig.

31. In this figure a theoretical relation between w and e, is also plotted, which has been deduced according to

ref. (2) using Wm, of M. 1382 and the analyzed value of e,

for the actual ship (see the next section). The scale effect on w seems to be somewhat bigger than that estimated by the theoretical calculation and the true value of w5 is supposed to lie between the theoretical calculation and the analyzed value by method H.

5.3 Effect of propeller loading on self-propulsion factors

It is reported that the self-propulsion factors show a little variation when the propeller loading is consider-ably changed77. To check this, the self-propulsion tests were run at the model point as well as the ship point of self-propulsion as shown in Table 4. From the analysis made on these two kinds of tests, the following

conclu-0.0 0. E 1. 1.1-1. 0.5 0.4 1.0 50.9 0.6 0.5 0.4

-

0.3 0.2 0.1 06 0.5 loo s s 0.5 0.4 0.6 0.5 '0. 0.3-0.2 0.2 0 1382 M. 0.20 04 j ,oFflWL=r0,15 i ,_nwL=015 42m 7m 10m 1.0 15 log Lj'p

Plot of self-propulsion factors of ship length (Full load cond.)

05

0.20

0.3

2 3 4

Cs X IO

Fig. 31. Plot of w on the basis of Cv

06 0.5 0.4 0.5 2 04

sions may be derived:

As for er, the effect of propeller loading can be

neglected except the case of M. 1270 (4.2 m-model(, on

which the scatter of measured points is so large that the definite conclusion can not be derived.

In the case of w55, a slight effect of loading on w,0 can be seen, i.e. the values of W,0 decrease a little with the increase of loading.

Although it is difficult to find out the effect of load-ing in the case of t because of considerable scatter

03

Mark M. No. Remarks Mark M. No. Remarks

1270 _(Wooden)42m 1482 _(Paraffin)7m 7m (Wooden) . 10m (Wooden D ° FOWL= 0.20 g_iZ Fnw= 0.15 Tos 178,179

'

.2 °'-T,, 104,186,189

Mark M. No. Remarks Mark M. No. Remarks

1270 _(Wooden)42m _(Paraffin',7m - - - 1382 _(Wooden')7m 1483 _(Wooden'10m -1 0- norL= 0.20

.,-a-

fFnwL=o.15 0.6 0.5 0.20

0.4----0---FOwL' 0.15 0.3 -02 M. 01 M.1382 FOWL=020

4820

. Fiow0.15 42m 7m 10m 213m 5,5' Ballast cood. f

Full load cand.

M M Na JnsI Method 5I-,...' MetIrod 0 05 1.0 15 20 25 log, o L,'p

Fig. 30. Plot of self-propulsion factors on the basis of ship length (Ballast cond.)

01 O Fig. 29. 213m 20 2.5 on the basis

(11)

lo

of measured points, the effect of loading seems to be of the same order or less than that of w;.

Wake Survey

In Figs. 32 & 33 the measured distributions of the axial

component of wake on three geosim models are shown. Although the contour curves of the wake fraction are very complicated, similar tendency will be noted in the wake pattern. Therefore the wake distribution was analyzed by Fourier series and the scale effect on each

component was investigated. The fundamental term

(corresponding to the volumetric mearn shows the simi-lar trend to cj on the basis of Reynolds number, while the higher terms show much complicated trend in con-trast to the simple conclusion described in ref. (5). The

problem of scale effect on the distribution of wake is very important not only to an improvement of model-ship correlation, but also to avoid propeller induced vibrations and to prevent the local cavitation on

propel-ler blades.

Model-Ship Correlation Analysis

As mentioned in section 2. the corresponding actual ship, Everest-maru' was built by our Nagasaki Shipyard of M.H.I. and her standardization trial was conducted on October 21, 1959 on the measured mile course, "Off Miye' near Nagasaki in accordance with the research program of the Shipbuilding Research Association of

Table 6. Model-ship correlation analysis

N.B. Method I refers to the first method (model propeller characteristics). Method II refers to the second method (ship propeller characteristics).

W'm=l-VpVm

)

V,= 16.6kn. 173kn, mean -. M.1270 n4.2m M.1382 (7 rn -M.1483 10m *w0,

1

-i

J,

_{Fig. 32.} _{Comparison of} _{Fig. 33.} _{Comparison of}

wake distributions wake distributions

(Full load cond.) (Ballast cond. 2 % Aft-trim) Japan. The standardization trial code is essentially the same as the ITTC 1963 Propulsion Trial Code7. Since the test results and the wind- and current-corrections are described in details in ref. (1), the corrected values

of ship speed, propeller rpm and SHP are taken as

starting point here.

In analyzing the model-ship correlation factors (wake

W',rn= I - Vp/ Ve,

Va=17.Okn. ISOke, mean M1270 42m M.1382 7

rn

-M1483 10m

-Load 3/4L. Nor. Mer. Remarks

V 14.76kn 17.12 17.61

Corrected for wind & tidal current

N 91.98rpm 110.90 115.10 SII? 9,034 16.183 18,138 Log RSsWL 9.21819 9.28258 9.29584 e0.998X106m2/s Lwi'217.21 m Fnwr 0.1645 0.1908 0.1963 Q 70,343m.kg 104.510 112,862 Qf=2,292 m.kg Q-Qf 68.051 m.kg 102.218 110,570 Ka 0.02217 0.02290 0.02300 e,. 1.004 1.008 1.009 Kqa-=Kq e,. 0.02226 0.02308 0.02321 Method I II I II I II J 0.5076 0.4780 0.4872 0.4572 0.4836 0.4536 1-0.21384 (JN/V8) 0.0549812N2K, R,/1.01 Fig. 6 Sa/P'e,i=6.397 Fig. 6 Kt 0.1648 0.1777 0.1738 0.1866 0.1750 0.1881 aa'8 0.325 0.364 0.326 0.367 0.325 0.366 T 76.70t 82.656 117.4 126.2 127.4 137.0 t 0.212 0.212 0.212 0.212 0.212 0.212 60.43t 65.14 92.46 99.45 100.4 108.0 R,0 59.84t 64.50 91.68 98.47 99.40 106.9 i06__vlvaf 4.5648kg 6.1477 6.4978 0.01311 0.01415 0.01490 0.1600 0.01530 0.01645 ce,, 0.00151 0.00401 0.00494 0.01150 0.01254 0.01089 0.01199 0.01036 0.01151 (Fai/Sa) (c,,,-ce,)103 1.798 1.860 1.701 1.875 1.620 1.800 c10X103 1.277 1.255 1.250 1+0 1.319 1.2726 1.256 103c10(1±k) 1.684 1.597 1.570 iO34c 0.114 0.176 0.104 0.278 0.050 0.230

(12)

correction factor and resistance correction) from these corrected values, two kinds of method were tried. The first is our routine method except one point that the form factor is taken as a function of F (Fig. l0( instead of being taken as constant. In this method the propeller open-characteristics to be used to analyze J and K are

for standard Reynolds number (R =4.5 x l0) as described in ref. (2(.

Since the scale effect on propeller open-characteristics is remarkable as shown in Figs. 16 & 17. the full scale characteristics might be considerably different from those estimated at the standard Reynolds number above mentioned. The second method of analysis is to use the estimated full scale characteristics of propeller which are calculated as described in appendix.

The model-ship correlation analysis made by use of two kinds of propeller characteristics is shown in Table 6. The analyzed ship wake fraction w is 0.325 for the first method, while it is 0.366 for the second method. These values of w are plotted in Fig. 31 as mentioned

before. Resistance correction 4cj is (0.5-.-1.0( X 10 for

the first method, while it is (l.8'-2.8) x 10 for the second method. The correlation factors obtained by the second method may be considered more rigorous than those by the first method, but the second contains an unestabli-shed procedure of extrapolating characteristics for full scale propeller. This situation has to be improved to make a model-ship correlation method more rigorous and the author inclines to the opinion that the scale effect on propeller open-characteristics is one of the most important items to be made clear in order to im-prove the present state of model-ship correlation method. 8. Conclusions

From the model tests carried out on the geosim models of a 46000 DWT tanker and her model-ship correlation analysis based on these test results, following conclu-sions are deduced:

The traditional Freude's method splitting the total resistance into residual and frictional components is not reasonable especially for full shipform and the Hughes' method should be used as a new substitute.

Although the value of form factor is constant in low Freude number range, it is not constant in higher

speed (F,>0.15(.

There can be the case in which the wave-making re-sistance coefficient at sufficiently low Froude number is not zero but finite, e.g. the case of shipform having a remarkable full ending such as the case of ballast condition of the model used in present study. In such a case the analysis of k and c presents a new prob-lem to be solved. Splitting of the total resistance into viscous and wave-making resistance components by use of the geosim models such as described in the present paper or the direct measurement of viscous

or wave-making resistance9 may be used to this

purpose.

The thrust coefficient of an open-propeller may be considered constant in the range of RCk higher than 4>< l0. while the torque coefficient shows a steady decrease with the increase of R7k and it suggests the necessity of proper correction for scale effect provid-ed the rigorous ship-characteristics of propeller are

desired.

Although the scatter of the analyzed results of

self-propulsion factors are considerable, it may be concluded that

(a) the scale effect on e7 may be considered

negli-gible.

(b( the scale effect on w7 is definite and it seems to

be slightly larger than that expected by the theo-retical calculation described in ref. (2),

(e) the thrust deduction coefficient decreases with

increasing scale, but it may be taken

approxi-mately constant for the model longer than 7m in

length.

(6 Fourier analysis of the axial component of wake

showed that the fundamental term corresponding to the volumetric mean of wake followed approximately the same trend as (J against R7 but that the higher terms underwent complicated variations so that no

simple rule could be found.

Two kinds of model-ship correlation analysis were tried. The first was by routine practice except that

the form factor was taken as a function of F. In

the second method of analysis the estimated char-acteristics for full scale propeller were used. The analyzed wake fraction and resistance correction by the second method are considerably larger than those obtained by the first method. Although the second method involves an unestablished procedure of ex-trapolating propeller characteristics for full scale. the analyzed correlation factors by the second method seem to be more rigorous.

Finally the author comes to the opinion that the scale effect on propeller characteristics and on wake

distribution are the most important items to be

pursued in order to improve the present state of

medel-ship correlation problem.

Acknowledgements

The author wishes to thank Mr. K. Ohira for his as-sistance to prepare figures and calculations and all the members of Mitsubishi Experimental Tank for their enthusiastic accomplishment of these comprehensive experiments extending more than two years.

Appendix

Estimation of open-characteristics of the full scale propeller

The characteristic values of propeller read atRCkx l0 = 1, 2, 3, 4 and 5 on the mean lines shown in Figs. 16-48 were analyzed by the method described in ref. (8) into the two-dimensional lift and drag coefficints of the equi-valent profile. The analyzed results are shown in Fig. 34, in which practically no scale effect is found in lift coefficient above R5=3x l0. whereas the drag coefficient shows a regular decrease with increasing R7k. And the drag coefficient curve has almost the same shape with one another and approximately it can be thought that the curve moves down as a whole parallel to the vertical

axis with increasing R7A.

The minimum value of drag coefficient for each R. shown in Fig. 34 is compared with the fiat plate resist-ance at corresponding R7k and shown in the following table:

* Prandtl-Schlichting's}

Taking the inaccuracy arising from the experimental error and the assumptions involved in the theoretical

calculation of analysis into consideration, it was

roughly assumed from the above table that the value of CDmin/2 ej of the full scale propeller is unity. Since the

11

RCk 1x105 2x105 3x105 4x105 5xlO

CDmin 0.01538 0.01298 0.01134 0.01039 0.00953

e1 0.00716 0.00615 0.00566 0.00534 0.00511*

(13)

465 IO Ox 10e- 2xlO-¿ Rk=lx 10

\

--- __b_r-_.

Assumed or actual propeller

r Rk-= 4.4x 1O°J 0.03 0.02 L) 0.01 n 3 -2 I 0 1 2 3 a (d

Fig. 34. Analyzed two-dimensional characteristics of the equivalent profile

values of R0k of the full scale propeller in the trial are

3.7. 4.4 and 4.6x 10° at the engine loading of 3/4. Normal

and MCR respectively, the representative value of Ra5 of the full scale propeller may be taken 4.4x10°. Then the frictional resistance coefficient at R5=4.4x 10° is 2.39 x 10° for smooth surface by Prandtl-Schlichting's

formula.

The roughness allowance was estimated +0.60 x 10° by

use of resistance coefficient diagram of sand roughened

platebaI. taking C0.7/k,5=l x 10°, (C0.7= 1.612m, k0=161e).

Then the CDmin of the full scale propeller may be esti-mated as follows:

CDrnin=2x(2.39+0.6Wx l0-°=O.00598

Hence the CD-curve of the full scale propeller was as-sumed as shown in Fig. 34, to have a minimum value of

0.00598 and the meaned shape of the analyzed CD-curves

at various R(ke shown in the figure. The CL-curve of the full scale propeller was taken the same as those for R=4 and 5x10° as shown also in Fig. 34.

Then the open-characteristics of the full scale pro-peller can be calculated by the reverse process of the analysis by use of these CL- and C0-curves. The results are shown in Fig. 35.

12 0.25 0g 0.04 0.20 0.03 4.01 0.35 0.30 0.15 0.10 0.05

-0

Reprinting or reproduction without written permission prohibited.

We would appreciate receiving technicol literature published by you.

U4 L

vi-4AUI

Rek No Ko'

-

Opeacharacteristics - Open-characterislico Rrk.4.5,x 4.4 ollniticant and fKi/j st actual br of modal 10° ditterunce I propel propeller io found tsr er -0.2 0.3 0.4 0.5 0.6 07 J

Fig. 35. Estionated open-characteristics for

the actual propeller (Rc5 4.4 X 10°) & those

for the model propeller (R5 4.5x10°)

References

I i I Investigations into the propulsive and steering performance of

super tankers. The 41st Research Committee, The Report of the Shipbuilding Research Association of Japan No. 31, Nov. 1960 2) K. Taniguchi: Model-ship correlation method in the Mitsubishi

Experimental Tank, Mitsubishi Technical Bulletin, MTB 01012R, Dec. 1963 and Journal of the Society of Naval Archi-tects of Japan. Vol. 113, June. 1963

K. Tamura & K. Taniguchi: On the blockage effect. Mitsubishi Experimental Tank Report No. 307. Aug. 1958

S.Tsakonas, W. Jacobs & J.Breslin: Wake fraction and thrust-deduction scale effects, Proceedings of the 9th ITTC. P. 293-298

5 ) J. P. Breslin: The influence of scale or size on ship model tests

involving propeller-generated vibration, Contribution to the

10th ITTC (1963, London)

D. I. Moor: The effect of EHP loading on propulsive perfor-mance in still water. Proceedings of the 10th ITTC. P. 66-69

ITTC 1963 Propulsion Trial Code, Proceedings of the 10th

ITTC, P. 80-86

8 1 K. Taniguchi: Studies on open-propeller (effect of blade

inter-ference) Journal of the Society of Naval Architects of West

Japan. No. 7, Dec. 1953

9) K. Taniguchi: Measurements of wave-making resistance. Lec-ture in Wave-making Resistance Symposium. Bulletin of Zosen

Kiokai, No. 436, Nov. 19611

(10) H. Schlichting: Boundary Layer Theory (1955. London)

ç) 0.02 0.4 0.3 co 0.2 0. 1.8 0,7 1.7 -1.6 0.6 1.5 1.4 0.5 1.3 -1.2 1.1 0.4 1.0 -a - 0.8 0.3 0.7 0.6 0.2 0.5 - 2.4 o t - 2.3 - 0.2 -0.1