Theoretical calculation of tandem propellers and their open water test series

4 DEC. 197

ARHIF

Shanghai Ship and Shipping

Research Institute

Ministry of Communications

Shanghai, China

Lab.

y.

Scheepsbouwkunie

Technische Hogeschool

Dell t

Research Report No. 78-1-3

Theoretical Calculation of Tandem

Propellers

and their Open Water Test

Series

Sun Qin

Gu Yunde

Theoretical Calculation of Tandem Propellers arid their Open Water Test Series

By Sun Qin, Gu Yun-de and Zheng Shu-zhen

S UARY

Th18 paper conaiata mainly of two parts, i.e. the sim-plified method of the theoretical calculation and the open water test series for tandem propellers. First, by using

lifting line theory and analysing the changes of Induced velocities between two free vortex sheets of each propeller,

the interference of flow at a certain radius induced by each propeller on the other roel1er cari be calculated. verag-Ing seperstely the interference velocities along radius of forward and aft propeller, but considering influence of the

contraction

f the foardpropeller's slIpstrea

on the

aft rropeller we may treat the forward propeller and aft ropeller separately as a so-called "equivalent" conven-ional propellers. Then, with the help of the conventional propeller's open water characteristic curves

_which

got from either theoretical calcula tion or test results the

hydro-dynamic perfoance o

tandem ro;el1er can be obtained. The method pro;osed by the authors shows the calculating

reolts agree

reasonably well

_{th the test results. Next,}

r

the influences o various rarmeters of tandem

propelle :s,

butfi

ropeLlers of tandem pro;ellers their pitch match, radial pitch distribution as well as diameter difference etc., on the performance have been investigated. :inally,

two sets of open water test series iamed CLB4-40-2 and CLB4-55-2 are given.

e have investigated tandem propellers for several years, p till floWs there are more than thirty ships of various kind equipped tandem propellers in our country.

It has been proved already that tandem propellers can de-crease the propeller induced vibration at the stern of

ships and increase the proulsive efficiency when

coni-paredto the conventional propeller with the equivalent

large blade disc area ratio.

..

TTTTn

The tandem propellers arrangement investigated in this paper means having two designating propellers located close-ly on a single shaft and locked together so they turn in the sanie diretion at the same rpii. Pig. 1.1 shows tandem pro-pellers running in cavitation tunnel.

2

For convenience the word tandem propellers is abbre-viated to "T. P." in the following.

Since the past decade, the horsepower of most types of ships have Increased. rapid1, consequently, brought about the propeller load of single screw ships

increas-ing. As a result, not only is lt necessaly to further Improve the propeller efficiency, but also to take the cavitation and propeller Induced vibration into

considera-tion. The scope of certain open water test serles with conventional propeller has been extended to account of large 'oladed area ratio and inultiblade propellers. In the meantime some people have paid attention to i'. i-. again,

therefore, several model tests with T, . have been

con-*

ducted in succession in certain countries, (i), (2), (3),

(4), (5), (6), (7), (8), (9) and most of them are

con-cerncd of large tankers. It is obvious that people are much interested in the solution of vibration at the stern

of some fullness ships by using T. P. Based on the results of our research the specific features of T. . can be stated

as fcllows

In the case of the unrestricted diameters and ving same large bic ded area ratios, the efficiency of T. P. is

eouivalent to or slightly

higher

than that of the conven-tional propeller. For certain ships having high loading propeller, such as bulk carriers, tankers etc., more

beni-fits in effici ;y can be obtained v

th T. r,

In the .a5e of the restrict d diameters, it is evident

*

Tunibers in brackets refer to references at the end of the paper.

that the eificl'ncy of T.P. is higher than that of the conven-tional propeller, the bigger the power coefficient and the smaller the speed coefficient , the more bene'it in

efficiei-cy cari be obtained.

3.

T.P. might

improve the propeller-induced vi'ortion at the

stern of ships. Now more than thirty ships including passen-ger-cargo vessels,

tugs,

surveying ships and inlard water,boats

etc., few of trnem are twin screw ships, have been equipped with T.?. in our country. In addition to various benefits in

efficiency, most of their propeller-induced

vibrations

at the

stern have been improved by T.. too, when

comred with the

originally conventional propeller, for instance a certain coastal

passenger-cargo ves8l of 1600 tone equipped T.. h9 reduced the

nmaximu vertical vibra tin accelara tjon to bcut cne-second of

ito

;revioua value as compared with the oriji'-olly conventional

propeller rnning at the same speed.

'e consider thot T.P. is valuable to be used for trawlers, coasters and boats sailing in the h.i1cw torretial inlard water

ways.

Por fullness

ship such 5 large t3ners sm bulk-carriers etc., it would be worthy to use T.?, too. In

certain

cont1on,

because T.?. canai ts of two separa ted propellers so each ha ve

rather sil

uladed area ratio and lighter weight, this is tue Special advantages to overcome the disadvantag es of perf;rirance

and productive technology caused by the iare bThded area or

ultra heavy weight of conventional propeller. It has been

emphasized already that T.?. is getting

pr:ticaily valuable

to be

applied to controlable pitch propell :s,because T.-. would allow

higher blade area ra tios and decrease concern over spa ce for blade esing. In this field we should like to point out that Lips Propeller 06. might be the one of the

earli-est advocator (io)

II.

TFFTIOAL CALCUIATION OP T.?.

Fr the point of view of the marine propeller circula-tion theory, in principle, it is likely rather simple to make a calculation of the flow induced by each propeller on the specific blade position of the other propeller for T. P.,

the flow naned interferential velocity. This calculating process is then repeated with the propeller order

recipro-cated. If the flow induced by one propeller on the other

j superimposed on the intake velocities (incld.g.axial and

tangential)of the water into the other latter propeller disc and vice verse, the effective wake of each pro;eller(direct

pro'blem)or the pitch correction (irverse problem) may be

de-teir.ed.

After several iterations, the final forward and aft propeller geometrical data (inverse problerü) or charac-teristic curve (direct problem) are determined. In fact, to form a complete calculating progranie of T. P. in accc ice

with above mentioned principle, the amount of calculations is too large for conventional computer's capacity to do.

Thj is the reason that not only must the self-induced flow pattern of each propeller 'ce calculated by using the lifting surface theory lik sed foi' convention L propeller calcu-lation, 'out al the interferentio velocities induced by each propeller on the other must be taken into

considera-tion using the lifting surface field peint method, what is

more, the whole calculation

i

an approach

of alternative process.

However, tne authors have proposed a simple theoretical method

here in which both te propellers of T.P., after taking account

of t''e interfere'.tial veiccities Induced by each propeller on the

other, are treated as a individual conventional propeller

operat-ing in the open Water respectively.

It is assumed that:

The vortex systems produced by forward

and aft propeller of

T.P. are Independent from each other.

Both the propellers of T.?. having the saine pro;;ller

dia-meter, boss diameter arid number

of blades, are either lihtly

or roderately loaded so that

the Betz conditions of mininum energy

loss are suitable for them.

It is well known that according to the propeller circulation

theory for a conventional propeller operating in operi water, the

axial Induced velccity Increases continuously from zero at the

point Infinitely far ahead of propeller to

at the propeller

2

disc and to Ua infinitely far behind the propeller.

The

tan-gertial induced

veloc'Ity remains zero ahead of the

ropeller

and IedIately rises to

2

at the propeller disc and to

just behInd the propeller.

?ig.2.l shows the velocity diagram.

at 'alade section of radius

r

for a conventional propeller.

VA

2 771? C

ig.2.l

The velocity diagran for conventional propeller.

?ig.2.2 show' the velocity diagram at ûlade s'tion of radius

r

for forward and aft propelle

f T.i.

271/1 r'

_271/li'

Fig.2.2 The velocity diagram for tandem propellers

Comparing with the conventionEl propeller, the induced

velocities of both propellers of T.P. at propeller disc

possess not only the self- induc ed velocities conjTonen ta U

2

5ndJt

_{but also the Interferential velociti es components 1Tai}

2 2

(axial direction) and liti.

(tar.ential direction) induced by

each propel

r on the other.

urthermore, the

elf-thduced

velocities of both propellers of .I. should be equivalent to

th3t caused by each propeller individually operatiri

_{in the}

velocity field which composed of the intake velocities and

the

interferential velocities induced by the other propeller

on

this both Including axial and tangential directions.

_{In other}

words, t:ie self-induced velocities of the ecjuivslent each

in-dividually actual ppeller must be produced under

considerii

the res:ltant of the int3k' velocities infinitely

_{far ahead of}

propeller (V1,2,

_N1,2).

'or the forvard proe1ler

Axial

_{\== VA}

-h

(2.1)

The subscript l2 refers to the forward and aft

propeller respectively.

According to the concept of induced velocities for an individual propeller, the interferential velocity of one propeller depends upon the self-induced velocity of

other propeller and the axial distance as well as angular spacing between two adjacent blades. In otherwords, the Interferential velocities can be represented as a function of the self-induced velocity, axial spacing ratio and

an-u_tu =

=

U_t'2

where g is a distance spacing factor, is a phase factor

which g depends upon the axial spacing ratio, fa depends upon

the angular

spacing.. On the basis of a succession of ring

vortexes whose otrenths vary with propeller rodius Tachrnindji derived t'e distonce factor for propeller with infinite number of blades (li). The values of g

derived by Tschrnindji are to be used in this paper, where g is a function of r -1iension i. radius X, the induced

advance coeffi

n,las well

5 axial :acing rtio L/D. It is given by

Tangential N1 N (2.2)

and for the aft ro;eller

Axial y_A2 y

-Ua2

(2.3) 2 Tangential N2 = N

Ut2

(2.4)

2 irr

guiar spacing at a specified radius r.

The value of phase factor fat is a function of the anZular

apCiflg O ,

namely, thei r relative loca tioi of cree vortex of the adjacent fore and aft blades. Che angular spacing de-finites the angle between the reference line of one

fore-blade to the reference line of adjacent aft blade. It is known from the circulation theory that the tangential

In-duced velocity reaches the maximum value LJ at the location

being free vortex and the min irnum value

J,,

In the middle

of two adjacent free vortexes., lie consider that the

opti-mum angular spacing will be realized if the aft propeller blade is just placed in the middle of two free vortexes shed

from tho two adjacent blades

of the foard propeller

(i),(7).

In this paper we assied that the distribution of the

tangen-tial velocity between both loca tion of it3 maximum and. mini-mum values is linearity, heoce

U1 l2Utriri

_=KU

and,

(2KIi)Ut)

furtheinore, when the free vortex sheets are treated as solid walls due to the free vortex sheet cöinciding with the rda tive streamline so that the flow between two ad-jacent free vortex sheets of forward propeller blades may

be loo1ed lihe running in open channel, therefore, from

U

the assunption, the relationship 2

-9-2wnr

' -

=

may be satisfied not anly at the locationS of the

free-vortex sheet òut also in the middle of them in the vicinity

O tie f3re--roPe1ler .sc, then

LJ1 U0 mn

it

(»<,J)

Ùa (2.11)

*

It is shown that beside a above mentioned tangential induced

velocity the xi1 inducd velocity betxeen

two adjacent free

vortex sheets of forward propeller blades is linear

distribu-tion too. The result coincides with tiat of lifting surface

field point method published in the U. S. A. recently (4).

Therefore, the phase fQ tor of the aft propeller may be

ex-pressed as

ftfa2K_/

For the forward propeller, in general,

there is relationship of 422K2 I but the influence of the

aft propeller on the forward one is sal1 that fa2 nay be also

simplified. In a word, after considering the trutal

interfe-rence between the fore-and-aft propellers the T. . may be

treated as two conventional propellers separatively, the in-take velocities in front of propeller disc at radius r of

T. . are expressed as follows:

or forward propeller

= v+

(2K2IXi-2)

2

N

For aft propeller

Va,

= V

t

LJt

N2 = N - (2K-l)

The values of the above expressions may be averaged along radial direction by the following form:

R

Jrß

F(r)rc[r

rr

Therefore, we can obta

Thr the forward propeller

VAl = VA [

(2K2VÚ

2) I2) N - lo -(2.12) (2.13) (2.14) (2.15) (2.16) (2.17)

(2. 20)

(2.21)

(2.22)

(2.23)

- li

For the aft proeller

(2.18)

N2 =

N [i

2K,I)H]

(2.19)

s usual, where

Kì2and

,2refer to

the values at

non-dimensional radius X

=

0.7

The integrals of the induced velocities may,be ¿iven by

Ua rr/rr

I

- o

r2VA -í U

rr/cRrir

HJ2r

_/)r

_i-xf

c'=

À2J17

i+À2

x A1

J

-nhl1

It is indicated that once the induced advance

coeffi-cient A and the Ideal efficiency

7.

of the forNard

pro-peller are determined, the interferentlil velocities of

the aft propeller could

be

obtained and vice verse.

For the reason, the ideal thrust coefficieit must

be calculat d by

¿K

_{ffl2D4 -}

2rU2r-)dr

_f2J34

After integral of aoove expression, we can obtain

K

/«2

(_ I)Ix2)(27);?(/ -)J

_(2.24)

when usin (2.24)

we assume

1'T = /.O2Kr

Owing to the contra cti on

f

t .:e fc rrd

propeller 's

slipstream and two propellers having ec1ual diameters, a

part of blade tip of the aft propeller is not

_influenced

by the slipstream s-oi from

the

_{foardpropeller.}

_he heavier the propeller load, _{ore the contraction of}

the fo.. 5rd propeller's slipstream will be, therefore,thls

contraction effect cust be taken into

_{cons deration in}

cal-ltng the averse disc value

of intake velo ities at the aft

propeller disc.

According to the cndi tion of flow

continuity at the disc

of both propellers we

con obtain the following

equation;

R2IT

_u EV E, 2 2

22_a2)J

R-(R 2'T

1r

j

SIVA 2 Va2

(/+)))rdedr

'here E1

₂

j

UflkfloWfl coefficient de to the

non-unifority of the axial induced velocity along the

dr-curferential direction, i.e. it reaches the maximum value at the lifting-line and the minimum in the middle f

them.

Because K is

represented for

circumferential average Value, therefore, carrying out the integral of (2.25) with respect

9 from O to27r , we may get expression (2.26)

J{VAKI

_r,

2

K2Ù-2)

Ua22

rr

o Va

=

_r. { VA +K2 2 + K1 _{+ )} 2

3 rar

.'oin, tarrying out the average of the above expression

along the radial direction, we

ay get expression (2.27)

I +K,

!I, +K2I2)J2=!+K212tKI,i,_)

_(2.27)

F()=)+

V42F()

_{+(!g,)K1r2(r)}

_(2.23)

VA

The mean values i,2' g1,2on the both sids of equation (2.27) exist sornewhere on the interv1s (X

_f)

and (

-)

according to theory of the mean, but, since

is small, it

might be considered that those mean values being X

O.7

on the

th sides of equation ae

ul respe'tively.

X

Fii=

_JRdr

12

-(2.25)

(2.26)

(2.29

R LJa2 R

rdr

/

_2r

r2

-j

r r

i X2 i r;ç R

(JûrJr

_

- "

2 r

=

fRrar

,2

Therefore '4i

(K,

v

L

K212r2Xi-)

2(1

VA2K2(2)

V, K,()(4T)

VAC%+2)

V4

Cn account of the contraction of the slipstream, the flow

velocities of the aft propeller which are equivalent to

the intakevelocitiein front of propeller disc of a

conven-tional propeller alone are of the

forE:

p-z

JrJr

±(2K -i)(-,)

_4rr

I = VA, 1 2 VA

_2

2 R-tR

Jrar

r,

2rrnr

--2

(2,-I)

)

=NÍI(2KH)H-21

2 2

rdr'

1.

r

=

[i

(2i) H)

here nelectin

second order ters s,allness

Now up to here, by uin fonrulS (2.15)

(2.17)

(2.33)

(2.34) (2.23) _{(2.21) (2.22) (2.23) (2.24) (2.32)}

and on 1as1s of two equivalent conventional propellers _with the diven open water characteristic curves, the calculations

bot direct and inverse roblers for T. i. which have the optimu

anu_

spa

cing. the same dianieters and tue

_same

number of bladeS of both proellers may be proceeded by mèans

o: itetive ret:od.

(2.30)

(2.31)

(2.32)

(2.33) (2.34) -

13

-t

I

i

't

The open water -s'" racteris tic curves of two conventional

prpe1lers equivalent to the forward and aft propeller of T.P.

separatively may be obtained either from the recuits of the

open water test conducted in 'ossir (e.g. the thrust and torque

coefficients may be expressed as polynomials with respect to

advance coefficient and pitch ratio by means of regression

ana lysis) or using circula tion theory cal cula ti on, herein we

have used Prof. Kerwin's method (12), however, the lifting

sur-face correction

is oarried out wjt

'r. 1organ's method (13).

A block diagramof the program Is

¿iven in Fig. 2.3.

(

STarE

)

\

read

\

CmpIg

\

C,e1(;c,'e,,t

\

read

\

Aei/o

\R/D,/ r

Jj_' =û

st

T. (=0.0

L .LL=O

LTÌJJ

(ampute

g,, T,,H

Q5 _no T2fnpute prc,oe ¿Lei' ,°,rf2 rn"ance

(SLI 6 rotA t,>ie)

Jar-ony/an

in t-r,00L2 LIOn

j,,

Ka1

(Sahroatiiie)

K, _,J4-,

(Sabroti')

LrrA1

,r t rpû

J2

. Kî2.K, (Subrûti, tine) Co/nptLt. K2, ? ¿2

(omftt

2, 12, Ji

14

-UeS Comp.4I

(

[e 4=

/-(2,'(,-i)H

Krr k',-, Xrì tA '1

ikAt4

Wri1 Agi/Ao, Ae2/Ao =JS 'io. .1 .1

ì/D

P2/D

J

Set 112

J,

2 ?i L= I Stop

00

05

03

02

Fig. 2.4 has shown that the

_{ca1cuting results of a}

_certain

se t of T

_{P. agre}

_{reasonably well w:th the}

_{tes t results.}

EXPERZ/1ENT

0/ 02 23 _{04 05 05}

07 08

J

OP

Fig. 2.4 comparison between c3lculstlon

_{and test results of}

y using the rnethod

_{mtioned above the}

perfoxance of

a great number of T.

_{P. sets}

_{was calculated by}

reans of an

ordinary corìputer at

_{our thtitute.}

_{tnder the guidance}

of

.w].ich,

_{everl sets of}

_{open water test}

series of T. P,

_with

better par5neters

_{arc chosen.}

III. o:E:; 7F

T2.

In additien to

_gener1

_'ters

r'on1y used to

a :nventiona1 pro;.

_there

_{re two ccmbining}

;arameters

for bath ft_ward

a

ft jropeiÌer

_{of T.}

.

i.e. t1e axial

- 15

o

-_----â---

Cûi'IPO' lED

-o.

I

o

_

N N /0 Tic 07

spacing ratio and the angular

spacing.

Some parameters for T. 1. are discussed as follows:

1.

xial spacing ratio L/

It means the ratio of the axial distance

between the

reference lines of the forward and the aft propellers

at

0.7 R radius to the diameter.

The influence of L/D on the

perforance of T. P. with 3 and 4 blades has been studied.

Fis. 3.1 shows the test results of T. P. 3-30

(The numbers

mean each propeller having number of blades

Z = 3, expanded

area ratio AE/Ao

0.3), having forward propeller pitch ratio

1.1, aft propeller pitch ratio

1.3

o. D D

06

J=o.6

-

16

-05 n

rig.

0/5 020 ₀₂₅ L/D

3.1

_{The influence of ax1j spacing ratio on the}

0,6

05

It is realized that'the efficiency cf T.., generllj

speak-ing, will be higher for i/i = 0.2 - 0.25, obvious.y. it

.ould

be reasonable to choose larger L/D for higher P/D.

2.

jigulsr spacing

O

It means the angle between the reference line of one

for-ward blade and the reference line of one adjacent aft blade,

which Is related to axial spacing ratio closely.

It is

general-ly reccgnized that the optimum angular spacin

is obtained when

the blades of the aft propeller can be arrayed in the middle

position between the two trailing vortex sheets shed fron tne

adjacent forward propeller blades (1), (7).

In view of the

point the authors have derived an optimum angular spacing

formula a

fol1ows

L/D

_x

₃₆₀₀

t'o

(3.1)

where L/D = axial Spacing ratio

-

pitch ratio of the forward ;ropeller

Z.

= nun:ber of blades

A typical test result of the influence of anular spacing on

the performance of T.F. is given in Fig.3.2, in which e = 22.6°

i

the optimum anular s;acin

colcl5ted from forula (3.1).

.2

e

in "ence of angular sccing

_{on the perform2nce of T.k.}

U

J 08

____..;,_______ =0.7

II

--FT

L -. LT 0.6 I (r /0" 20° 300 íO' 590 50" 70 84:.1" 990

6

Diameter ratio =

It means the ratio of the diameter of _{the forward propeller} to that of the aft propeller. _{The diameter ratio in our T. P.} tct series is unity, i.e. 'both propellers have the same diameters,

Radial pitch distribution

The pitch radial distributions of the forward and the aft propeller should be adapted to _{their flow fields in front of} propeller disc _{Since the reveal of the a ctu3l flow field} sur-vey behind the propeller disc has _{indicated that the velocity} distribution over the slipstream _{range of conventional propeller} is rather homogeneous (14), _{therefore, it is plausible to choose} the radially constant pitch distribution for aft propeller. In

consideration of the tendency towards radially constant pitch and large disc ratio for _{certain methodical model}

test series

pu1ished and in order to _{et more T. P. combination}

in our test series within a _{limited number of propellers}

the radially constant pitch distribution _{of the for'w9rd propeller}

is also chosen.

Pitch match of both propellers for T. P.

It is In essence how to distribute _{the power to}

forward and aft propellers _meet

certain requirements _{of design thnist} whIle operating at optimum _efficiency.

As we know, _until re-cently, there _{18 lack of paper dealing}

with marine T. P. test serles available _{for design application, because}

certain people could consider that a methodical

series of T. P. _involving

many paraméter _{Is no practical.}

But, as mentioned

above,on the basis of the

theoretical method the authors _{ad arried}

out to calulate

a large sets of T. P. with _{a view to lnvest}

-gate the influences of the inflvidual parameter leing

sys-tematically varied on t'e T.

i.5

performance.

The results

of analytical calculations have indicated how many pitch ratio

differences between both pr&pellers for £.P.re better in the

case of heavy or light load respectively.

For the

convenien-ce of running test and drawing up the methodical series

dia-grams,both sets of T. 1.

for open water test series

pro-posed here have constant pitch ratio differences between both

propellers of them and it equals 0.2, because the two sets

are suitable for moderate and heavy loaded peopellers .

The

geometric data for two sets of T. P.'s model test series

named CLB4-40-2, CLB4-55-2 are given in table 3.1.

Table 3.1

eometiic data of T.I.'s model propellers

The blade section and the outline of Wageningen B series

are adopted in our series, but a little differences from

those that our series all have a radially

constant pitch

and no rake.

Open water tests

_{re rn in our institute model basin}

íth instniment in type 04 on it

nda

propeller bo

-driven frcm dwnstrear and by

u

ng routine t-'t prc adure,

i.e. over a

range of opera tins condition

_{by var'1ng the}

speed of

vance an

remainin

constant ri.

Th

rsed

1g D

Diameter

(mm) Z

21ae

nuniber

Blade area

ratio of

Scre

t/D

Thickness

ratio

d/

BOS

diameter

ratio

Data for blade

sections and

outline

220 4

_O.-3

0.045

_0.182

B series

t"' h.

j

a' j

:' I .G Cri

t'1rìtH

'r.

i:,

L - p '

fj

fL'

of

rLcId.. :'r

)fl

te T. ?. 's prforiace, the o'er'. wter tet

with the T.

P.

No. ?OlBlT1C h.'v

}:een ri

fer chc1cthg test accurac:r with

different

t.

an

water tp-r'rtures.

It L s-.c-

in F'ig.

3.3

that the res'.dts in th

fie'ds Of

e2.3-3.4xlO5

are

hai-cally steady and of coincidence of the characteristic curves

reasonably well.

p.. £ '.4 :3 t t H -'1

---- ---f.

. /975.'0. 33./0 4!Ja4..5 o /974.2. ìj.,o J04.'2 72 7 28,O t.J0.&25. . /978.2. 2.0./C' I /877. .9. 3.4. 'o J3.s. )5 .

Ht

--- ...

---.. I L CI . :: C4 :5 c / C8 29

?jg.

.3

Repo: i

ent rOsui ts of T.

P.

T-llC.

t.;

T.

'.)' 5

)05

4-.O...-2,

ds of

.î.'

rhe results of

rpen wat'r test.

_{re exresed in the farm of}

non-dimensional coeff'icient,

namely:

VA

advance coefficient

J

flD

7-thrust coefficient

_T

-torque coefficient

The pitch correction

_{of T. ï.S model}

has been made according

to their actually

_{measuring data.}

_{The test results}

have been

made data processing

_{by means of}

regression analysis

_with

res-pect to two

_{variates through}

_{computer and expressed}

in polynomial

coefficient,i .e

thrust coefficient

_KT

torque coefficient

K

The plmomial

_{coefficients, the}

_{eon square deviations}

and the

ccrrelatjon coefficients are ¿iven in table

_3.2.

'ig. 3.4 shows the open water characteristic curves for T.P.

CLB4-'3-2.

TheJJ'

_{.e sign chart is}

calculated and given

_in

Fig. 3.5.

'1g. 3.

_{shows the oren water}

_{characteristic}

curves for T.P.

OLB4-55-2.

_ThCri

_{deoiri chart is}

calculated and given

In

Fig. 3.7

\fter comparing

_{the T.1.'a}

_{:erformnsr.ce .vith the}

ageningen

B series

_{4-3O aLd B5-105}

_{with the equivalent}

blade disc

_area

ratio

t

_conclusions

_{ay be drawn as follows;}

-21-Table 3.2 Polynomial coeffici ente, rnean square deviations

and correlation coefficients

-C L B 4

- 40 - 2

No.

X Y

i

0 0

_0.0800615185

_{-0.0000714555}

2 0 1

0.2526930696

_0.0000000000

3 2 0

_0.6014192596

_0.0719167679

4 1 1

_{-1.6932060320}

_{-0.0295512197}

5 0 2

_{-0.0430520459}

_0.0695116812

6 2 1

_1.3380236960

_0.0000000000

7 3 1

_{-0.2803367306}

_0.0000000000

8 2 2

_0.0000000000

_{-0.5113313226}

9 1 3

-0.0850891645

_0.0000000000

10 5 0

_{-0.2779032386}

_{-0.0024756567}

11 3 2

_0.0000000000

_0.6911005060

12

5 0

_0.1227554372

_0.0000000000

13 4 2

_0.0000000000

_{-0.2767168544}

14

i

_0.0000000000

_0.0012366813

15

4 3

_0.0376340270

_{-0.0120053868}

16

6 2

_{-0.0085102591}

_0.0115812191

17 6 3

-0.0059761304

_0.0050785269

Mean suare

deviations

0.0021600

_0.0005793

Coe1ation

coefficients

0999875()

Ta1e 3.2 Folynomial coefficients,

_{niean square deviations}

and correlation coefficients

23

-C L E 4 - 55

- 2

iTo.

_{X Y}

i

0 0

_{-0.0379680316}

0.0234384119

2 1 0

_0.3548465156

-0.0658l2236

3 o ₁

_{-0.1391804624}

0.0127281851

4 2 0

_0.2650951132

0.1149378970

5 1 1

_{-0.2986437862}

-0.0530862942

6 0 3

_{-0.0634071516}

0.0000000000

7 4 0

_{-0.0461357783}

0.0000000000

8 1 3

_-O.066054600

-0.0170861467

9 3 2

_0.1371004660

0.0000000000

10 5 1

_o.0000000000

_{-0.0115399028}

11

6 2

_o.0000000000

_0.0124720298

12

6 3

_0.0030000000

-0.0033510788/

:.:c

LU3I

deriations

0.0033054

0.0005231

Correlation

coefficients

0.9998271

0.9997983

't .1 N' I' L) ¿2 i.' '9 dg 08 07 01 as 4 03 02 'Ji o t'.

Fig. 3.4 Open water performance curves for T.P. CLB4-40-2

r

N I-'

/

/

it; -

. . . . ,.

---ir.L/

...

L

...

'i',

., . /

.

. . . I,.: I : -

--'-'.

..-

.

...

-.

Pig. T .5 ]ß- digrarn for T.. CLB-40-2

24

XC ' 83--SC 4.5-Ja -:

'4/t':

L-i 0X44 r r - .1 ',a) -S'' Sc - .5 .5

ij; ¿,Lr7

_i 36'

it

-;.-:

Erl

1

fH

'.1:

:

t'

8 JLL 04V

"

8 0.'

CLB4-4O-2

,

i-Kf

f44ii1T

:

1-::

u:_.::

:-

u

:.

'L:---

_t -L-+

:.

.-___'_'_1

.t:::::zf4

g .:.

tt1f

!

:-'- X z 4 4

--i

u -Ii

_:;--; -JsXIS 'Jß

-i-:---i:zp

_.ifl:!

.

-:: 0182 C/ ft?.-a/7

'i.i1

t

i

_t

;ft-

t%

-o:

t' q o .LL..::-:.2-::-t

',:I

¡

i-ti'

ï

L1:L

ii

t'

_illhIH'NI!

s

.

_..-1!Fr

U!

'î

-1II

- _/

/_-- A / . j,

.-.-

. - ?-- ?--?--

i '':i

f-L.

I

X i__i o a' 02 03 04 as 06 07 Q, 0.9 1.0 1.1 02 i) 4 '5 L' J

.16 C3

7.--H--- .-.-:;_ t 4-II

CLB4-55-2

Itk

-ft

as

oi l * Z 4 4 '

t

C435 0045 - ¿iQ 0/52 t. tt pv _' 0/ cg

ig. 3.6 Open water perforiance

_{curves for T.P. CLB4-55-2}

25

J

L

t'

In the case of tne unrestricted diameter, to compare with

the conventional prope1lr rit1

_{the eival3nt blade disc ares,}

ratio, forfd ,

the T.P.'s efficiency is slightly higher and

the optinum diameter is less.

In the case of the restricted diameter, to compere with the

conventional propeller with the equivalent blade disc ratio,

for

AJp

4

the T.P.'s efficiency is higher, for example, the

T.F.'8

efficiency increases about 4,

for D = 0.9

_{bigger the power}

oeff1cient B

_{and the more serious the restriction of}

_diameter,

the more the gain in the efficiency will be obtained.

Acknowledgments

The authors are grateful to Mr. Chal Yangye for his valuable

advice of this paper.

-R

F E R E N C E S

(i)

Jacques 3. 1-tadler, william

_{. Yongan and Kenneth A. Meyers}

dvanced Propeller iropi1sion for High-Powered

_Single-Screw

Ship6.

SNÀE 1964.

C.T. Davis and Richard Hecher

Open-water Perforance of Tandem Iropellers

AD - A056651

_{June 1973.}

Richard D. Kader

Cavitation and Open-Water ierfoance

_{of a Set of Tandem}

P rope li e rs

-

OO7478

_{November 1974.}

Stephen .i3, Denny

rocedure for the Desin of Tandem Fropellers

-

.O2'O27

Jurie 1973.

Marlin L. Miller

xperitenta1 Deterina tian of

_{Unsteady i'ropeller Forces}

Seventh Synposíum Naval

_{Hydrodynamics}

(5)

I.

_{k. Titoff and B.}

_{. Biskup}

Investibation into te Fossibilities

_{of Tandem Propeller}

.pp11cation with the

_{im of Decreasing the}

_Variable

Hy-drody.ric Loads Transmit

_{ted to a Propeller Shaft}

eleventh ITTC. 1966.

(7)

L. Sinclair ad

_.

_merscn

The iJeoin ar

_{eve1oent of}

_{ruj.eilers for 1ih-.oiered}

shipbldin

_{ercnt Vessels and Shipping}

_eco

?eb

i, li68.

(S) _{eitendorf, . ).}

xper1rente1le Untersunungen der von _{ropellern an der}

U3 _{enhaut erzeiten periodi}

_{scien Druckochwankunen}

Schiff und Hafen _{Heft l/'1970.}

(9)

_{Hans Hurs}

HSBPN, ein neuer Schiffspropeller

Schiff und Hafen _{Heft 8/1972.}

(io)

J.

_.

Van Aken and K. Taaseron

Comparison between the Open _{ater Efficiency}

_{and. Thrust}

of "Lips _{- Schede" Controllable}

- Pitch Propeller and Those of "Troost Propeller"

ISP _{1955 Nr. 5 Vol.2}

(ii)

J. Tachìindji

The Axial Velocity Field

_{of an Optimum}

Infinitely Bladed

Pro; e i le r

AD - 650547 _{January 1959.}

J. . Kerwin

Yachine Computation of larine Propeller _{Characteristics}

I3 _{Vol.6 no.60}

_{ust 1959.}

tm. B. Morgan, _{11pvic and}

tephen B. Denny Propeller Lifting -

Surface

_Corrections

SNÀJIE _1968.

vary Dickerson

Iniced Velocft.es

_{Forward and Aft of a Propeller}

- 650553 _{March 1959.,}

-wer coefficient f u?.5 .

0VA

J,Ïl S?eed coefficient NO1E LA

r

tber of revolution per minute and _second

Speed of advance of propeller, _knot , m/sec.

Power _{metric horse power}

D=2R :ropeller diameter , meter

Va

Axial induced velocity

Li _{Tangential induced velocity}

U0. Jx1al interference velocity

Tangential

interference velocity

Hydrodynamic pitch 3ngle of _{any propeller òlde}

section

ßAdvanced

angle of any

propeller blade section

r

adius

Hub radius

oss diameter ratio

D

r,

Nondnensjona1 radius

'xil interfere;ce _factor

antji interference fact

-Axial distance

factor

Induced advance coefficient

Ideal efficiency

1<

oldstein circulation

_reduction

_factor

VA

J=

Advance coefficient

T

Thru st co e f fic i e nt Q

KsTorque coefficient

propeller efficiency in open

2i7Ka

Ideal thrust coefficient

Y

Thrust

G _Torcue

5) ?resh

water

density

i

Slipatrears

contraction factor on

_forward

_propeller

i

Number of

blade

ce pitch

ratio

Re Reynolds

nurber

/,4

Blade area ratio

,4xy Regression coefficient

BXy _{Regression coefficient}

C L54-40-2

. tandem p'

celer s enes

wit"3blade

_{setin:. and outline,}

in which

_{nuibers incate number}

_{of blades,}

blade

are _io

of each propeller _{and pitc}.}

ratio differ- 'not

_{of two prollers}

in sequence water Kg. Kg. r Kg. sec. 2,/rn4