A theoretical scale effect study on the proplusion coefficient of a body of revolution

(1)

r

QV. 1q82

ARCH1

ij

SYMPOSIUM ON

"HYDRODYNAMICS OF SHIP AND OFFSHORE PROPULSION SYSTEMS"

HØVIK OUTSIDE OSLO, MARCH 20.-25., 1977

"A THEORETICAL SCALE EFFECT STUDY ON THE PROPUL-SION COEFFICIENT OF A BODY OF REVOLUTION"

By Gilbert Dyne,

Statens Skeppsprovningsanstalt, Gothenburg

SPONSOR: DET NORSKE VERITAS

Ref.: PAPER 5/3 - SESSION I

Lab.

v.

Scheepsbouwkurne

TecIinische Hogeschool

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STATENS SKEPPSPROVNINGSANSTALT

Contents

1. Introduction 2. Nomenclature

3. Description of calculation method

Calculation of, the boundary layer around the fore part of the bQdy

Ca.ldulation Of the viscous flow around the stern and the ideal flow outside boundaxy layer and wake

4. Comparison with experiment

5. Resistance and ±iominal wake fractions at Varying Reynolds number

6. Calculation of effective wake

7. Sürfäce hip -. equivalent bodies of revolution 8. Conclusions 9. Refetences BibIcth _{'ac) de} Afdeling Sc _{psbo_} en Shpvaainde Technjsche Deft DOCUMEATIE

1:

ç2-DATUM,

(4)

1. Introduct.ion

The drag of a body tdwed at constant speed through an unrestricted fluid is caused by forces of two kinds:

pressure forces normal to the hull an frictional fOrces tangential to the hull. BOth the.pressure drag coefficie C and the frictional drag coefficient CF caused. by these forces are influenced by the thickness

and the velocity distribution of the boundary layer

around the body. Thus the viscous drag coefficient C,

+ CF generally decreases with increasing Reynolds number RL being zero in an ideal fr.ictionless fluid at RflL = - d'Alembert's paradox.

The viscous drag C is larger than the drag C of a

V F0

flat plate with the same length, wetted surface and speed The difference is expressed in the form factor

CV.

(1)

C.

F0

k being caused by the viscous pressure drag and the change in friction due to the varying velocities around

the body. In general the form factor is assumed tp be independent of Reynolds number

The nominal wake w = 1 - where VA = advance velocity

at propeller plane, V = undisturbed velocity, cati

be spIit up into two components the displacement wake

Wd and the frictional wake Wf. The displacement wake is a measure of the flow retardation caused by the fullness of the body at the stern, hile the frictonai wake is

àaused by the loss of total head inside the boundary

(5)

STATENS SKEPPSPROVNINGSANSTALT 4

IPpo

Wf = 1

_VpV2

(2)

(3)

where

P = total head at the propeller plane P0 = free stream static pressure

Since the boundary layer thickness decreases with increas-ing Reynolds number in a similar way as the viscous

re-sistance, the frictional wake is less in full scale than in model scale. The variation of wd with RL is more

uncertain.

The problem of predicting the scale effect of form factor and nominal wake fraction for a body of revolution is

investigated in the present report with a streamline

curvature methOd recntly developed at SSPA [1]. With nominal wake distributions from these calculations as input data, the scale effect of the effective waké is then determined by the thrust identity method usin a

quasi-steady propeller method of analysis. Finally the posbiiity of predicting co±responding scale effects

for conventional ships by applying the method to equi-valent bodies of revolution is discussed.

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STATENS SKEPPSPROVNINGSANSTALT S

Nomenclature

a correction coefficient

a1 flow separation scale effect coefficient

Cf

local frictional coefficient based upon the velocity _Ue at the edge of the boundary layer local pressure coefficient; c P Po,2

P

4-pUo

P-p0

local total head coefficient; C

Pt _pUo

total frictional drag coefficient

total frictional drag coefficient of a flat plate

CFA frictiona1 drag coefficient of the stern CFF frictional drag coefficient of the fore part

of the body

CPA pressure drag

coefficient

of the stern

pressure drag coefficient of viscous origin

CT total drag coefficient

Cv viscous drag coefficient; C =

+ CFA + CPA

D propeller diameter

1.

àOnst'ans th the c-eqüation proposed by gj Webster & Huang

H1.2 boundary layer shape parameter; H12

0

k form factor; k = - 1;

- F0

L length of body

n coordinate normal the the streamlines

n velocity proiie exponent

static pressure

Po static pressure in undisturbed flow

(7)

q source strength radius

corrected radius

r radius of body contour

w

radius of body contour at x =

R drag of fore part of the body

RflL1 Reynolds number based upon L or 0;

R R L . R

nOJ'nL

' nO

s coordinate along the streamlines

S wetted surface of body

t thrust deduction

U velocity along streamlines

Ue velocity along streamlines at the edge of the

boundary layer

velocity in undisturbed flow (= speed of body);

Uo=.v

advance Velocity at the propeller disk

speed of body; V =

wàkë fraction; w =

displacement wake;

effective wake determixied by the thrust identity

method

coordinate along the axisof the body; see Fig 1 coordinate where calculations are started

coordinate for transition from lamThar to turbulent boundary layer flow

coôrdinate normal to the contour of the body

angle between velocity vectOr and x-axis

constant in the boundary layer prof.iie proposed by Coles

(8)

6 thickness of boundary layer

corrected thickness of boundary layer

thickness of inner part of boundary layer, see Fig 2

6* displacement thickness of boundary layer;

r±6

6*

= f

(l---)dr

0 momentum thickness of boundary layer;

r +6

0=

p fluid density

P0 longitudinal radius of curvature

Indices

model ship

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3. Description of the calculation method

a. alculation of the boundary layer along the fore

cart of the body

The pressure distribution along the fore end of the body of revolution is calculated by the ideal flow method of Hess & Smith [2], SSPA computer programme No 105. The

body is replaced by a surface distribution of ring sour-ces and sinks, satisfying the boundary condition that

the normal velocities at the naked hull must be zero.

Since the boundary layer at this part of the body is

very thin the. pressure at the edge of the boundary

layer is in the following assumed to be the same as the pressure at the hull of the body in ideal flow..

The development of the laminar boundary layer from the

Fig 1 nose, x = 0, tothe transition point, x = see Fig 1,

is calculated by a method proposed by Granville [3], SSPA computer programme No 191. The method gives the variation

of local frictional coefficient Cf and momentum. thickness O along the body profile as well as the location of the neutral instability point and the poin.t of self-induced

transition.

Starting from the known moentum thickness at the

tran,-sition point, x = the turbulent boundary layer is calculated by a modified version of. Head's enta.inment

method [4] up to the entrance of the run, x = xo, see

Fig 1, SSPA computer programme No. 102. The modification,

made by van Berlekom [51, means mainly that

o the streamline convergence and divergence occurring

in :axisyrnmetric flow is taken into account

o . the local frictional coefficient cf is calculated accOrding to 'Schoenherr [6] instead of

Ludwieg-Tillmann [7] to obtain more reliable results at

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Laminar flow; Granvilte (3)

Turbulent flow; Head (4]

TurbuLent flow; Dyne Eli

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STATENS SKEPPSPROVNINGS?NSTALT 9

Thus cf is calculated by the following fôrthula, derived by Webster & Huang [8]

from

Schoenherr's frictional line:

f(R0)

=

g (R0

where 0 and H.12 are the two-dimensional momentum

thick-ness and shape parameter respectively, see nomenclatue.

Since Head's entrainment integral method gives only the integrated boundary layer characteristics, the velocity

profile at x = x0, to be used as input data for the cal-culations at x >- x0, itust be dete-mined separately. In

the present version of the method the relation between

the total velocity U and the distance r _{- .r} from the

wall (r = hull radius at,x x0) is according to

.wo

Coles [9] assumed to be

where

Ue = total Velocity at the edge of the bOundary

laret

(S = boundary layer thickness

Approximating w with

=0019.10o.578f(R0)Hl2}

0.46 lnR nO

0.46 1nR0 -1

= -

0.256 + 0.0044 mR

nO

2.5.j4.

in

r-r

2.. 5 $

\j. r- r - 0

1)]

+ 1 (S R. nO

(6)

w((SWO)].

_,(7)

(8)

see [10], and inserting the. local frictional coefficient

(12)

parameter H12 as from Head's enttainment method.

b. Calculation of the viscous flow around the stern and .the ideal flów.outsideboundary layer and wake

The viscous flow around the stern is determined by. a

streamline curvature method [1], SSPA conpUter proganme

NO: 210. The flow field behind x = x0 is divided along Fig 2 the streamlines in four regions, see Fig 2, with

differ-ent assumptions of flow characteristics:

Region I - inner boundary layer

Ten streamlines., equally spaced in the range r. r

r + 6 , where 6 is about one-tenth of the boundary

w0 g

g

layer thickness at x = x0. .

Region II - outer boundary layer

Twenty streamlines equally spaced n the range r. +

+6 < r r + 6, where 6 is the thickness of the

boun-g . . .

dary layer at x =

x0.

Region. III - entrainment flow

Five streamlines distributed, between streamline 30 and the edge of the boundary layer at x = L.

Region IV - Ideal flow

Outside streamline .35 the flow ±s assumed. to be ideal.

The ideal flow outside the boundary layer and the wake

'is determined b' a singularity method 'simultaneously

with the calculations of the viscous flow. A flow

dia-Fig 3 gram of.. the tération process used is given ir Fig 3. The different operations can be described as follows:

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Region V Region III

Region

Region I

Edge of boundary Layer

Body contour

:team1ine

35

Carnljne 30

(14)

Read jflt)U.t data Calculate approx velOcities in flow field 'II ICalculate

stream-lines from the law Qf continuity $ Calculate source-sink distribution and determine pressure, along streamline 30 or 35 $ Calculate pressure gradient across

boundary layer and

deterthine press ure in, flow field

Calculate new

. Velocities in.

floW field

Correct boundary

layer -thickness,

add or correct flow

region iii and cal-culate new values of velocity and total head. alopg streamlines

Calculate total

dragC(x) at

various x-values

Fig 3 SSPA streamline curvature method.

Flow diagram

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STATENS SKEPPSPROVNINGSANSTALT ii

ir[r2

(i) -

r2

(i -

1) ]u cos a = constant

where

r(i) = radius of streamline No ± U = total velocity midway between

Nos ± and i - 1

angle between velocity vec.tor

Calculate source-sink distribution q1 and eterminë pressure along streamline 30 or 35.

In the ideal flow calculations the body and its boun-dary layer arid wake are represented by a polygonal sourôe-sink.distribution along the axis. The strength

of this dstribution is determined from the boundary

condition that

th

normal velocity is zero, along th body at x < x.0 and along streamline 30 or 3. at x x0. The. s.tatiá pressure

3o O.p3s

at these stream-lines is. then, determined, from the source--sink distri-bution in a conventional manner.

A comparison of the source-sink distribution of the

present nd the preceding iteration shows i.f the calculations have converged. ' .

o Calculate pressure gradient across boundary layer. and determine pressure in flow field.

-Neglecting the influence of normal Reynold stresses

El]

the transVerse. pressuré.gradient is' ptit

streamlines

and. x-axis

(9)

Calculate streamlines from the law of continuity.

Starting from the ontou.t of the body the shape of the

streamlines at x > x0 is derived from teJaw of

(16)

r353

£ dr

To obtain convergence the static pressure used in the following isa weighted mean of the pressures from

the' present and the preceding iterations.

Calculate new velocities in flow field.

In the first part of the calculation precedure the flowis assumed to be ideal also in region III, see

Fig 2. In region I. the velocity is calculated from a power law: (13)

'U = a(r-r)'..

, (14) STATENS SKEPPSP.ROVNINGSANSTALT 12 .2_= !L (10) Po.

where Po is the longitudi-nal radius of curvature of the streamlines obtained from

1 3ci

Po

-and s -and n are cOordinates along -and normal to the

streamlines. Since the longitudinal pressure gradient known from the preceding iteration the radial ptessure gradient can now, be calculated

- =

--cos a. + sin c' 3r' ri

(12)

the static pressure inside the boundary layer finally being

r.30

(17)

where the constant a and the exponent 1/ri are deter-mined from the three inmost streamlines in region II. The total head c is in region II assumed to be

con-pt

stnt along the different streamlines giving the

fol-lowing expression for the velocity

L

tgctrdx

-'

' ,

' (17)

9 ::

Cf()

-(18)

the local frictional co ficient Cf being eriyed from the momentum thickness 0 and the shape parameter

= I1cpt - cp (15)

This formula is also used fOr regions IIII in the

s'econd part of the calculation proceduie, the. varia-tion of theY total head in the flow field now being known, see. below.

Calculate' total drag C(x') at various x-values, correct velocity profiles and add or correct flOw region III. The drag coefficient CFF(x) of the fiore part of the body, is determined by the momentum theorem at

suc-cessive values of x from x = x.0 tox L:

RFF CFF(x)

_.fpUo2S

= _{1 [2} cos c(l-'---- ôos cx) -c Ir dr (16)

Sr

Uo Up

-w

Fig 4 see 'ig 4.

The pressure and the frictional drag coefificients of corresponditig afterbodies are integrated, along the hull:.

(18)

I

V..4 L..ILI.I

\

C, U ii;

,1

L.J.J.J.JJJ.J

I

\

4. Calculation of the drag of the fore part of

the

(19)

STATENS SKEPPSPROVNINGSANSTALT 1,4

(1+a(x)]6

(20)

while the local values of U/U0.and c are displaced

Outwards from r to

r

+ (l+a(x)](rfr)

The correction coefficient a(x) is chosen so that

Cv(x)

c,(x0)

(22)

The new values of r , U/U0 and c at streamlines 1-30

C. pt

or 1-35 are determined from the stieam function

H12 in the same way as in the ca1cu1.tions ahead of x

x0, see Chapter 2.a.

The total viscous resistance coefficient Cv(x)

iS

finally

Cv(X)

=

CFF(x)

+

CpA(x)

+ CFA(x) (19)

Cv(X)

= Cv(Xo) ?

The total resistance

_Cv(x)

varies, especially in the first

part of the ca1óulatios where the total head is assumed

to be copst.nt along the streamlines iti region II, with

the location xof the aft control surface,see Fig 4.

Thus if

_Cv(x)

Cv(xo) the total head assumption must be changed.

Correct boundary 1a.yr thickness, add o± correOt flow region III and calculate new values of velocity and total

head along streamlines.

To correct the total head assumption in a simple and re-liable way the thickness of the boundary layer at x

> X

(20)

r'

STATENS' SKEPPSPROVNINGSANSTALT 15

r

it U Oosctr dr

0 C C

Fig 5 at x0 and x as shown in Fig 5.

After the first part of the calculations, hëre the flow

in region III is assumed to be ideal, see Fig 2, five

new streamlines are added outside streamline 30. The dis-tance between these is chosen. sO that streamline 35 co-incides with the. edge 'o the boundary, layer a.t x L see Fig 2. The velocity and the tOt'al. head in this region are determined by a method similar to that illustrated in. Fig 5..

The correction method used has the, following

óharacteris-tiás:' .

the tOtal head is decreased along, the streamlines in such a way that the, total drag deternined by the momen-turn theory is independent of the, 'location of the aft

control surface . . ... .

the'flow in region III, being idealat.x x0 "is grad-ually entrained in the boundary layer 'for increasing'

x-valués .

Both the decrease in total head and the entrainment rate

depend inainlyuon the local: change in.'the frictional. coefficient C.-.

(21)

5. Determination of r, U/U0 and at the diffcrent streãml ines

(22)

r

_{w_l}

L .6 (0.9655

X)}l/2

_0.9333

(24)

4. Comparison, with eejmn,t

To test the reliability of the calculation method a

cornpari-son was made with experiments carried out by Patel, Nakayama & Damian [12] with a body of revolution The body was a

six-to-one prolate spheroid modified by a conical tail. ts contour is given by

= 0.43:33(1- ; 0.9333 1.000

The expériménts comprised tatic pressure measurements along

the body as well as velocity and pressure measuremer4ts of

anurnber of boundary'layer profiles along the stern. In the

latter case a transverse mechanism moved the measuring probes in the' 'y-dilection normal to the contour of the' body.

Patel et al [12,] used a tunnel wall pressure as reference

when calculating the non-dimensional pressure coefficients and c. Since this pressure apparently was lower than'

the free stteam pressure the total head coefficient c in the ideal flow outside the boundary layer was in [12]

c 1.06 instead of the maximum ideal value c = 1.00.

pt ' ' Pt

The experimental pressure coefficients given in this paper were' therefore corrected to give (cpt)max = 1.00. Te

cor-rection has been approved' by Dr Patel [131.

The experimental pressure distributions along the hull and along the edge of the boundary layer around the stern are

Fig 6 given in Fig 6 together with corresponding theoretical values derived by the present streamline curvature method

[1]. The calculations use the experimental nomentum thick-ness 0 and, the shape parameter H12 at x0/L as input valUe.

Also shown in Fig 6 is the'pressure distribution at the hull

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0.5 cp 0.1, 0.3 0.2 :01: - 0.1 0.8

PRESSURE DISTRIBUTION ALONG THE HULL

.1

I.

o

Experiment [12)

Ideal fLoW method t2)

Streamline curvature

method [1)

09 I

/

I. /

/

PRESSURE. DISTRIBUTION ALONG THE EDGE,

OF THE BOUNDARY LAYER

10

6. Exper.mental and

theoretical pressure

distributions arOund the stern

x/L

I

0.2 Cp 0.1

-0.1

(24)

STATENS. SKEPPSPROVNINGSANSTALT 17

As seen

o the static pressures at the hull calculated by the ideal flow method are lower than the corresponding experimen-tal values at 0.8 x/L 0.92 but considerably higher at x/L ' 0.92. The difference was..especially large at x/L = 1.0,. where the ideal pressure was c = 1.0, the experimental value being as low as c 0.17

o the static pressures at the hull calculated by the stream-line curvature method agreed well with the experimental values also at x 1.0

o the static pressures at the edge of the boundary layer

calculated by the streamline curvature method agreed

well with the.experimental values at O.8' x/L 0.9 but

less satisfactorily at x/L 0.9 . . .

A nunther of experimental. and theoretical boundary layer

Fig 7 profiles along the stern are shown in Fig 7, where the

velocity U/U.o is plotted against the coordinate y directed

normal to the hull. The agreement in result between theory

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1.0 U0 0.5

0

1.0 U U0

0

1.0 U A

0

0 Experiment

(12) Theory

(1]

0

I - I 0.01 0.02

0.03.

0.0.4 0.Q5

000

000 Experimental and theoretical velocity profi1es at

the stern

0.07

rIL

x/L0.99

x/L= 0.96

(26)

Drag and nominal wake fractions at varying Reynolds

numbers

The streamline curvature method gives good results, as shown in Chapter 4. Therefore it seems realistic to assume that reliable results also are obtainedwhen the method is used

to calculate the influence of Reynolds number on resistance, nominal wake and wake distribution of a stern propeller.

The àalculatioñs described below were performd in the

Rey-nolds number range l.26 106

RflL lO with the following

assumptions:

The transition from laminar to turbulent flow in the

boundary layer occurred in all cases at Xtr/L = 0.06.

The calculation of the flow around.the stern was started at x0/L = 0 7, where the input data obtained by Headts

entrainment method are tabulated below. Table 1

The thickness of the inner part of the boundary layer

was g/5 0.09 at x =

x0.

The Reynolds number variation of the frictional drag

coefficient CF of an equivalent flat plate was

e-termined by }ead's entrainment method (4], assuming

the transition to occur at x/L = 0.06.

The velocity distributions at xo/L 0.7 calculated accord-Fig 8 ing to eq ( 7) from H12 and Cf are given in Fig 8 As seen

the form of the velocity profile at

nL l0 deviates

slightly from the other profiles. The reason is that the low

H12-value of the method used gives abnormally large

boun-dary layer thickness in spite of the displacement and

(27)

1 oo uoo 0 0O 6000 9000 L000 9030 S000. 700'O wo:o Z0O0. 1000 0. .s.0 90 LO 80 60 on -n

0I

(28)

Table 2

this anomality can have influenced the result at RnL. =' iO slightly,, see below.

The result of the, drag calculation is summarized in the fol-lowing table.

The calculations give two interestin.g results:

o The pressure drag is extremely low. - all cases.

less than 2% of C

F0.

o The form factor k increases with increasing Reynolds

number, i!e k is larger in full scale than in model 'scale

The.pr.essure drag is built up by

o positive pressures at the. bow and.at 'the. stern acting

backwards on the fore and forwards on the aft part of

the body

o negative pressures around the shoulders acting forwards

on the fore 'and backwards on the aft part of the body

In ideal flow the pressures on the fore and aft parts Of

the body cancel each other and the p.res'sure drag IS.': zero.

In real flow the boundary layer along the fore part of the body is so thin that the pressure hëré is almost the same

as in ideal flow.. The pressure drag can therefore be

obtained by integrating (cpjd -c').r along the. afterbody:

(r ) . . . .

wmax

C = .f

.

(c.d-c)rdr

(25)

where c. . and c are the pressures in ideal and' real flow

pid. 'p ' : . .

respectively..

In comparison with ideal flow conditions the static 'pressure c at the hull in real flow is

(29)

0.2

0.1

-0.1

07

9. Pressure variation along the stern at different

Reynolds numbers 0.4

cp

(30)

Fig 10

o considerably lower around the stern at x ' 0.93 where the radius r is relatively small

w

somewhat higher around the aft.shoulder at Ô.7 x 0.93 where the radius r is relatively large

Fig 9 see Fig 9. Thus, the low pressure drag given in Table

2 above depends upon the contribution from the lower pressure at .the stern being almost counterbalanced by the higher

pressure around the aft shoulder.

The pressure drag is so small that its influence upon the

form factor k is almost negligible. The variation of k with Reynolds nufliber must therefore mainly be due to differences in frictional drag.

The form factor concept means that the viscous drag of the body of revolution is compared with the frictional drag of an equivalent fiat plate with the same lengti and the same wetted surface as the body. For this plate the boundary layer shape parameter H12 and the local frictional coeffi-cientcf decrease evenly ata slow rate along the plate, see Fig 10. For a body of revolution the slight decrease in H12, which occurs at the major part of the body changes into a rapid increase along the stern, see Fig 10. In the

model case (RflL = 1.26 10) this increase is so large that

the

frictional coefficient cf approaches cf = 0 at x = 1.0,

I ethe boundary layer is quite plose to separation at the

end of the body. Also in the full scale case (RflL = l0)

there are changes in H12and Cf along the stern. These

changes are, however, see Fig 10 much srnalierthàninthernodei

case and there is not risk of flow separations. A

compari-son of the two Cf curves shows that the frictional forces

on the stern are e1ative1.y speaking larger for the full scale. body than for the model. Thus the relation

(31)

H12

5 3.

C. 10

SHAPE. PARAMETER

1112

Equivólont fLat p(c'te

I'

1

Body of 'revoUtibn

RrL

1.26 1

LOCAL. FRICTIONAL COEFFICIENT

flL

i.26..;6

--..1

03

0.4

O.

0.6

07

10. Variation of shape parametr

}Ii 2 and fri.ctiona.l

coefficient

Cf

along the body of revolution and the equivalent flat plate at different Reynolds

numbers 0.9

x/L

1.0 0.8 .1

09. x/L 10

0.3 0.4 0.5 0.6 0.7 0.8

(32)

(cf)R _1.26 _io

(cf)R ₌

being between 2.9 and 2.4 at 0.2 x/L' 0.7 decreases to 1.7 at x/L = 0.9 and to 0.8 at x/L = 0.95. It can therefore

be concluded that the increase ih form factor with Reynolds

nuinber depends upon the contributions from the frictional

forces.at the stern to the viscous drag coefficient being

larger for the full scale body than for the model.

The form factor is primarily a measürè of the influence of

the form of the body upon the viscous drag. Unfortunately,

however, also the locatiOn of the transition from 1amin.r

to' turbulent boundary layer flow can sometimes influence

its magnItude. To avoid this the calculations described are

carried out assuming this transition to occur atx/ 0.06,

both at the body and at the equivalent, flat plate. As a re-sult the C -values used are lower than, corresponding values

F0

calculated by Schoeñherr's formula, which assumes the tran-sition to take place at the. leading edqe, of the plate. It is

believed that the form factor would be the same as given in Table 1 also if the boundary layer in the 'a1ciUations were assumed turbulent along the entire length of the. body.

The calculations also give the velocity and pressure.

distri-butions around the stern From these results the local nomi-nal wake. (w(r) at x/L = 0.98 is derived from

w(r)

.Uo.

in thea same way as when the velocity is measur.ed. by a Pr'andtl

tube. Corresponding displacement wake figures Wd are

ca.cu-lated with eq ( 3 ). . ., .

(26)

The variation of w(r) with the viscous' drag coefficient C Fig 11, at various radii is shown in Fig 11. The wake figures at

(33)

1. 2

3;

r/L

0.015 0.020 0.025 0!030 0.035 0.040 0.045 0.050

io

io8 1.26.106

Reynotds numbers

11. _{Variation of local wake with viscous drag coefficient}

cv

0.9 w 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. I fLow

00

(34)

Fig 12

=WM

where a = 0.116 1O b = 0.94 c = 1.46 ' io STATENS SKEPPSPROVNINGSANSTALT 22

character of the curves at C, = 1.7 depends partly upon the

slightly anomalous, velocity profile assumed at x0/L= 0.7, see Fig 8. The wake scale effect, i e the change of w(r)

with C, is very large at small radii, but decrèase with

increasing radius. For radii located outside the boundary layer in the model case the scale effect on local wake is

quite small, see r/L = 0.05 in Fig 10.

The scale effect on total nominal wake w for different pro-peller diameters is illustrated in Fig 12. The figure also

gives the variatiOn of displadertent wake with viscous drag

coefficient C. As seen the scale effect in w is less for

larger than for smaller propellers. The scale effect on dis-placement wake is small

With fifteen calculated w_C-pairs (five viscous coefficients and three diameters) as input data a linear regression

na-lysis gives the following relation betwee w, C and D/L:

= 0.202 + 0.19 i0 - 0.94 - 1.46 s iO (27)

The maximum deviation between the input and the calculated w-values is w = 0 017 and the standard deviation w = 0 010

If D/L is eliminated from eq (27) and the model wake WM de-termined at CVM is introduced, the full scale wake w at C can be approximated with

(c wa)(CC

(35)

0.7

w

Wd

0-- D/L0..06

0.08

--O--

0.10 0.6 0 Ideal

flow,

.1 I .1 I 10 io8 1O7 1.26106

Reynolds numbers

12. Variation of total and displacement wake with viscous

drag coefficient CV and propei1erdiaeter-ship1enth

ratio D/L 3

C".'

34

5

Total

wake w

Displacement

wake Wd

(36)

STATENS .SKEPPSPROVNINGSANSTALT 23

6. Calculation of effective wake

When calculating the effective wake the body was assumed to be fitted with a stern propeller of given geometry located

at x/L 0.98. Three propellers with different diameters

were investigated. They belong to SSPA standard propeller

family and have the following main dimensions:.

diameter D/L = 0.06, 0.08 and 0.10 pitch ratio P/D '= 0.65.

nuithér of blades z = .5 blade area ratio AD/AO = 0.60

The. hydrodynamiô characteristics of these propellers wre calculated by a quasisteady propeller analysis method based upon Goldstein's K-method, SSPA computer programmes 29 and

107,. see [16). The calculations were performed in three steps as follows:

The design advance ratio of the propeller was determined so that the calculated thrust coefficient, agreed wth the corresponding experimental value obtained from open water tests. The blade profiles were at thi loading

assumed to have shock-free entrance.

The thrust and torque values at off design advance ratios in uniform flow were derived usIng hydrodyamic

quantities calculated at the design load.1 see [16]. The

theoretical and experimental open water characteristics

agreed well.

The thrust and torque valueS in non-uniform floW were

calculated in a similar way using the axial velociti.

distribution at x/L 0.98 determined by the streamline curvature method. The propeller load was assunie to be

(37)

Table 3

STATENS SKEPPSPROVNINGSANSTALT .24

The effective wake WT and the relative rotative efficiency

were, finally derived by the thrust identity method using the thrust and torque coefficients calculated in behind con-dition, see. point 3 above, and the calculated open water

characteristics, see point 2. The result is shown in Table

3 below.

As seen the effective wake WT agrees very well wjth the nominal wake w, the influence of Reynolds number being al-most the same in the two cases. The reason for the good agreement is probably the assumption that the propeller

has. no influence upon the approaching flow.

The relative rotative efficiency riR calculated by the pre-sent method is in all cases = 1.00. .

(38)

Granville exemplified his method by _{predicting the full scale}

drag of the ferry Lucy Ashton [19] _{and a.fuil-shaped merchant}

STATENS SKEPPSPROVNINGSANSTALT 2.5

7. Surface. ships- equiv&lent bodies of revolution

The streamline curvature method can, as shown above, be used to calculate the scale effect on form factor and wake of a

body of revolution. The, question is now: is it possible to

determine corresponding scale effects on a suzface ship by applying the method to a body of revolutiOn equivalent to

the ship?

Granville [17] recently proposed a modification of Froude's

method for determining full scale drag of surf ace ships

from towed models where this idea was used. His prediction

method is summarized as follows:

The coefficient of total drag _{CTM is.determined in a}

towing tank.

The coefficient of viscous drag C of a non-separating

equivalent body of revolution is calculated by a method

proposed by Granville [18].. The method is adapted to

give a form factor kG which is independent of Reynolds

number:

'C

C..

1 + k

_{= CM =}

F0 F0

(29)

Other viscous drag components caused by wave-breaking; flow..separat.ions behind,blunt sterns or appendices,

bilge vortex separations and so on are assumed to be the sane in full scale as in model tests.

The coefficient of total drag _{CTS of a ship with smooth}

hull surface then becoittes: . . . ..

(39)

z

0 z

u) C-) -J

0

13.. Body contours of two equivalent bodies Of revolution compared to the body of revolution tested by Patel

(40)

Fig 13

Fig 14 Table 4

To obtain a comparison with these predictions the viscous

drag C was also derived by the streamline curvature method

-''L.'J'

--'

. - .-...

-using the cross-section equivalent radii, see Fig 13, both for the pressure and the boundary layer calculations. The

resulting form factors are given in Table 4 and in Fig 14.

Also in these cases the pressure drag C' was less than 2%

of C . is seen in Fig 14 the variation in form factor k F,,0..

with Reynolds number is similar for the two equivalent bodies of revolution as for Patel's body of revolution at

1.26 1.06

< RflL '< 108. The. relatively large change in k

between RflL = 1P8 and RflL for this body may be due to the anomality of t'he 'Input velocity profiles illustra-ted in Fig 8.

As mentioned Granvifle's method gives results which fall

in be.tween those of .Eroude's method (kG = 0 in eq (30))

and those of Hughes' form factor method (k '= k in.

G. exp

eq (30)),. If the form factor, as found in the present in-vestigation, is higher for' the ship than for the model,

see.Table 4, the calculated full scale drag CTS approaches

'the Froude method value.

Analyses of trial test results show, however, that

predic-tions made by Hughes' form factor method are .rtuch more

re-liable than those obtained by Froude's.method, especially

for large full-shaped ships '[22). If the finding that the

ship, Moor-Model .684 [20]. The cross-section area equivalont

radii were used to calculate the pressure distribution along the body and the wetted perimeter equivalent radii to

calcu-late the boundary layer. The method .gave full scale drags,

which fell in between those of Froude's original method with no form factor and those of the form factor method proposed

(41)

Form factor, k

0.3 0.2 0.1 S 1..0- 108 10

ReynoLds number, RnL

(42)

form factor increases with Reynolds number is true also for conventional ships a modification of the prediction method can be necessary.. Such a modified prediction method

can, as a suggestion, have the following features:

1. The coefficient of total drag CTM is determined in the towing tank

2.. The, experimental form factOr k is derived by the method proposed. by Prohaska [23] and the wave drag (including also the wave-breaking drag) is both in model and full scale put:

= C

- (l+k)CFM

The form factOrs kDM and kDS of non-separating equivalent model and full scale bodies of revolution arecalculated

by the streamline curvature method.,

Residual drag caused by flow separations of different kinds amounting to

(kkDM)CFM

(32)

in the model case is in full scale put:

a1(k

_{- kDM)CFM}

(.3 3)

a1 being a flow separation scale effect coefficIent vary-ing in the range 0 a1 1.0.

S. The coefficient of total drag CTS of' ship with smooth hull surface finally becomes:

(43)

Table 5

STATES SKEPPSPROVNINGSANSTALT 28

The choice of the coe icient a1 is very important fOr the result. Thus if

o a1= 1.0 the flow separation drag coefficient is assumed

to be the same for the ship as for the model. This may be true for very blunt sterns and appendages, but the trial test analyses indicate, as mentioned, that this assumption gives too high C8-values for more

conven-tional ships.

o a1= 0,0 no flow separations are assumed, to Occur oP the

ship. Though the calculations described in the present paper show that the risk of flow separation is much lower for the ship than for the model, this assumption probably

gives too low C8_va1ues for most ships.

An attempt to test the proposed prediction method and at

the same time determine the coefficient a1 empirically will

-be made ih the near futurewith the aidof .SSPA's extensive

tria] test material.

The calculations also gave the nominal wake for propellers with different diameter-length ratios located atx/L = 0.98.

The result is given in Table 5 together with the full scale wake w approximated by eq (28).

As seen the maximum difference in between theory and

eq (28) is 1w = 0.03.

The wàké prediction formula, eq (28) was also tested agai.nst

the wake scale effect (wM_wS) analysedfromSSPA's trial test

material. As expected the for1awhici is basd upi

theory for bodies of revolution gives too high.wke scale

effects when applied to conventional ships. If, however,

the formula is corrected to give the same mean value of

(44)

(1.46 SwMO.116)(CvMCVS)lO

= W w = 0.686

b.94 + 1.46 ' iO3

The standard deviation in w is

iAw = 0.049 (36)

This deviation is slightly lower than that obtained with

Sasajima's formula [241:

C

iw =w - (t+0.04) - (w

M M

VM

used in the cooperative work of the Performance Committee

of.the ITTC [25].

(35)

(45)

8. Summary

The,, flow around the stern of a body of revolution is

cal-culatéd by a new streamline curvature met1od capable of

solving the problem of a thick axisyrnmetric boundary layer with crosswise pressure gradients. A comparison with

experi-ments carried out by Patel shows that the method predicts the variation of velocity and static pressure around the stern very satisfactorily if the experimental boundary layer parameters 0, H12 and Cf at the aft shoulder xo/L = Q.7 are

used as. input vaiues..

The method is then used to determine the variation of the different drag components and the nominal wake of a stern

propeller by Reynold.s number. The input' values at xo/L =

0.7 are now obtained from a modified version of Head's entrainment theory. Of course the realibility of the

re-sults depends upon the assumptions made. The the expressions for

the entrainment rate 'at x < x0

the frictionai coefficient, eq (, 4 )

the boundary layer profile at x xo, eq (7)

as well as the choice of thetransition point x

_{=Xtr and}

the momentum thickness at x = Xtr influence both the vis-cous drag and 'the wake. To decrease this influence upon the

form factor, the drag of the equivalent flat plate is

cal-culated with Head's' entrainment theory assuming the'

transi-tion to take place at the same locatransi-tion as on the bocy of revolution.

With reservations for the uncertainties mentioned, the _main

(46)

STATENS. SKEPPSPROVNINGSANSTALT 31

The pressure drag is small - in all cases less than

2% of CF. The reason for this is that the positive

con-tribution to from the pressure at the stern is

al-most counterbalanced by a negative contribution from the pressure around the aft shoulder.

The form factor k increases with increasing Reynolds number, i e k i larger for the full scale body than for

the model. The reason for this somewhat controversial result is that the decrease in local friction along the stern is much more rapid at low than at high Reynolds numbers where the risk of flow separation is small. :The

frictional drag coefficient of the stern is therefore

comparatively largerfor the full scale body than for

the model.

The scale effect of lOcal wake, i e the change in wake with Reynolds number is large for small radii but small

for large radii. The total nominal wake w decreases with

decreasing viscous drag, coefficient C (increasing

Rey-nolds number). Though the relationship not completely

linear it is possible to approximate w with a linear

function of C and the propeller diameter-length ratio D/L without too large errors.

Starting from the calculated nominal wake distribution at the propeller plane, the effective wake is determined by

the thrust identity method in combination with a quasi-steady propeller analysis method. Th.e propeller is asstmed

to have no influence upon the approaching flow. The

agree-inent between the effective and the total nominal wake is

found to be good and the relative rotative efficiency s

at all Reynolds numbers = 1.00.

The streamline curvature method is, finalily used to

(47)

the ferry Lucy Ashton and a full shaped tanker, Moor-Model

684. Also in these cases the pressure drag is found to

be small and the form factor is larger for the ship thn

for the model.

In a suggested modification of Granville's method, used to:

predict the totaidrag of a ship from model test results1

the drag is divided into .the following components:

o the viscous drag of a non-separating equivalent body of

revolution o the wave drag

the flow separation drag

The possibilities of using SSPA's large trial test material to determine how the separation drag changes from model to

full scale will be studied in a near future.

It ispossible to convert the relationship between w,

and D/L mentioned above to the following wake scale effect

formula

-. (c WM - a) (CVM _{- Cvs)}

b + c ICVM

where the constants a, b and c are given. This formula gives reliable results also when applied to the calcuiate6 model and full scale wake values Of the equivalent bodies of evo-lütion.

A comparison with the wake scale effect analysed from SSPA's

trial test material shows that the formula giVés low

full scale wake figures for conventional ships. If, however,

it is corrected to give the same mean value of (wMws) as the analysis it predicts the full scale wake with sigitly (38)

(48)

smaller standard deviation than Sasajima's method used in

the cooperative work of the Performance Cornrnitte of the

(49)

STATENS SKEPPSPROVNINGSrNSTALT 34

References

DYNE, G. A Streamline Curvature Calculation Method for the Flow around a Body of Revolution. To be

published.

HESS, J L and SMITH, A M 0. Calculation of Potential Flows about Arbitrary Bodies. Progress in the

AerO-nautical Sciences, Vol 8, New York, 1966.

:3. GRANVILLE., P S. The Calculation of the Viscous Drag of Bodies of Revolution. DTMB. report 849, 1953.

HEAD, M R. Entrainment in the Turbulent Boundary Layer. A.R.C. R & M 3152, '1958.

van BERLEKOM1 W B. Calculation of Boundary Layer and

Drag for Axisyrnmetric Bodies Using Head's Entrainment Theory. SSPA report 2023-2, 1976

SCHOENHERR, E. Resistance of. Flat Surfaces Moving through a Fluid. Trans SNAME 1932.

LUDWIEG, H and TILLMANN, W. Untersuchüngen Uber die Wandschubspannung in turbulenten Reibungsschichten. Ing Archiv, Vol 19, 1949. See also NACA TM2475, .1959.

WEBSTER, W C and HUANG, T T Study of the Boundary

Layer on Ship Forms. Journal of Ship Research 14,

No 3, 1.970.

9.' _{COLES, D. The Law of the Wake in the Turbulent}

(50)

10. HINZE, J 0. Turbulence. McGraw-Hill, 1959

ii. DYNE, G. Förslag till undersokning av skaleffekter

p ett fartygs meström.. SSPA PM B].70-1, i967

.12. PATEL, V C, NAKAYAMA, A and DAMIAN, R. An

Experi-mental Study of the Thick Turbulent Boundary Layer Near the Tail. of a Body of Revolution. Iowa

Inti-tüte of Hydraulic Research Report.No l42 1973.

PATEL, V C. Private communication, 1974.

SASAJIMA, H and TANAKA, I. On the Estimation of Wake of Ships. Proc of the Eleventh International

Towing Tank Conference, Tokyo, 1966.

BRRD, R and AUCHER, M. The Prediction of Ship Performances in Calm Water. Proc of the Twelfth

ITTC, Rome, 1969.

16.: JOHNSSON, C-A. On Theoretical Predictions of Characteristics and Cavitation Prqperties of Propellers. Pubi No 64 of the Swedish State

Shipbuilding Experimental Tank, .1968.

GRANVILLE, P S.A Modified Froude Method for, Determining Full-Scale Resistance of Surface

Ships from Towed Models. Journal of Ship Research,

Dcember 1974.

GRANVILLE, P S. The Calculation of the ViscOus

Drag Of Bodies of Revolution. DTMB Report 849, 1953.

CONN, J F C et al. B$RA Resistance Experiments _on

the Lucy AshtOn. Part II The Ship Model correlation

(51)

MOOR, D I

l960

HUGHES., G. Friction and Form Resistance in Turbulent

Flow, and a Proposed Formulation for Use in Model and

Ship Correlation. Trans INA, 1954.

LINDGREN, H and JOHNSSON, C-A. The Correlation of

Ship Power and Revolutions with Model Test Results. Proc of the Ninth ITTC, Paris, 1960.

PROHASKZ, C W. A Simple Method for the Evaluation

of the Form Factor and the Low Speed Wave Resistance. Proc of the Eleventh ITTC, Tokyo, 1966.

SASAJIMA, H and TANAKA, I. On the Estimation of Wake

of Ships.. Proc of the Eleventh ITTC, Tokyo, 1966

Report of the Performance Committee. Proc of the

(52)

Listof figures

Definition figure The flow field

SSPA streamline curvature method. Flow diagram

Calculation of the. drag Of the fore part of the body

Determination of ri U/Uo and c at the different

streamlines

Experimental and theoretical pressure distributions

around the stern

Experimental and theoretical velocity profiles at

the stern

Input velocity profiles at x =

PressUre variation along the stern at different

Reynolds numbers

:Variation of shape parameter H12 and frictional coefficient Cf along the

body

of revolution and

the equivalent flat plate at different Reynolds

ni.nbers

Variation of local wake with viscous drag coefficient

cv

Variation of total and displacement wake with viscous

drag áoefficient C and propeller diameter-ship length ratio D/L

13.. Body contours of twO equivalent bodies of revolution

compared to the

body of

revolution tested

by

Patel 14. Variation of form factor with. Reynolds numbet

(53)

Table 2 _{Result of drag calculation.}

Comparison of calculated. effective and nominal Table 3 _{wake fractions}

RL

(OIL) H12 Cf 10 1.26' 106 1.340 1.452 3.36 1O7 0.922 1.356 2.43 108 0.6.54 1.280 . 1.79 l0 0.494 .1.196 1.41 R nL 1.26 106 l0 108 l0 i0 4.37 3.02 2.16 1.67 i0 b.06 0.04 0.03 0.02 c io3 F0 . 3.88 2.62 1.83 1.34

(C -C

)10 V F0 0.49 0,40 0.33 0.33 k 0.13 0.15 0.18 0.2.5 RflL

D/LO.06D/L = 0.08

D/L= 0.10.

WT W WT w WT. W 1.26 .10 0.66 0.6.5 0.50 0.50 0.36 0.37 0.51 0.50 0.37 0.37 0.26 0.27 108 . . 0.41 0.40 0.30 0.29 0.21 0.22 0.33 0.32 0.24 0.2.5 0.18 0.19 STATENS SKEPPSPROVNINGSANSTALT 38

(54)

ornina1 wake fractions for the equivalent Table 5 bodies of revolution L V Form factOr Experiment Theory (rn). rn/s Hughes k Granville kG Dyne kD

Lucy Ashton Itode1 6.1 2.5 0.08 0.04.3 0.09

Ship

58177 -

0043

013

Moor-Model 684 Model 4.9 1.5 0.26 0.185 0.16 Ship 122.0 7.7 - 0.185 0.25 D/L _WM theory eq (28) Lucy Ashton 0.06 0.24 0.17 0.19 0.08 0.15 0.11. 0.13 0.10 0.10 0.08 0.10 Moor-Mbdèl 684 0.06

0.59

0.40 0.37 0.08. 0.45 0.30 0.30

040

0.34 0.23 0.23 STATENS SKEPPSPROVNINGSANSTALT 39

Form factors for the equivalent bodies of Table 4 revolution

(55)

.STTENS SKEPPSPROVNINGSANSTALT

List of tables

Input data

Result of drag calculation

ComparisOn of calculated effective and nominal

wake fractions

Form factors for the equivalent bodies of

revolution

Nominal wake fractions for the equiyaient

(56)

POSTADRESS: Box 24001, S-400 22 GOTEBORG, SWEDEN

TELEFON: Växel 031/200130, TELEX 20863

TELEGRAMADRESS: SKEPPSPROV

GD/BK

Broder

Ref Symposium on "Hydrodynamics of Ship and Offshore Propulsion Systems"

Jag bifogar tv& kopior av manuskriptet till mitt föredrag "A Theo-retical Scale Effect Study on the Propulsion Coefficients of a Body of Revolution" samt originalen till figurerna. Jag yore tack-sam ow jag fick tillbaka dessa original nilr Ni anvilnt dew.

I

G5teborg 1976-09-29

övering T S$ntvedt

Head of Section for Propellers and Hull Forms

Det norske Veritas P 0 Box 300

N-1322 HOVIK NORGE

(57)

1 ga h1sningar

Gilbert Dyne

Eu

SThTENS SKEPPSPROVNINGSANSTALT

-POSTADRESS: Box 24001, 5-400 22 GOTEBORG, SWEDEN

TELEFON: VäxeI 031/200130, TELEX 20863 P

TELEGRAMADRESS: SKEPPSPROV

GD/BK

Göteborg 1976-10-05

övering T S$ritvedt

Head of Section for Propeller and Hull Forms

Det norske Veritas P 0 Box 300

N-1322 HOVIK NORGE

Broder

Ref Symposium on "Hydrodynamics of Ship and Offshore Propulsion Systems"

I det manuskript jag överslinde 1976-09-29 var fig 5 ofu11stndig. Jag bifogar därför original och 2 kopior av korrigerad figur.

(58)

Dune.: F

(59)

5.

Determination of

U/U0 and

the different streamlines

.Th

at.

(60)

5.

Determination of

U/U0 and c

at

(61)

Turbulent flow; Head [4]

Laminar flow ; Granvillle (33

TurbuLent flow; Dyne El]

x

3

(62)

Region IV

Region

Region I

Region

Edge of boundary Layer

/

(63)

Read input data Calculate source-sink distribution q1

and deterine

pressure along streamline 30 or 35. Calculate pressure gradient acrdss boundari layer and determine pressure in flow field $ Calcu'iätê new velocities in flow field

Fig 3 SSPI streamline curvature method Flow diagram Correct boundary

layer t1hidkness, add or correct flow region III. and cal-culate new values

o velocity and

total head along streamlines Calculate total

drag C(x) at

various x-values Slop

DLne

Ft no

I

Calculate approx velOcities in flow field $ 1 Calculate stream-lines from the law of continuity

(64)

Li

Control surface

Pt U oc CD 1 LO

(65)

5.

Determination of ri U/U0 and

at

the different streamlines

(66)

0.5 cp 0.4 0.3 0.2 0.1 - 0.1

-0.1

PRESSURE DISTRIBUTION ALONG THE HULL

.1

I I

Experiment [12]

---- Ideal fLow method

12]

-

Streamline curvature

method [1.]

/

.10

08

PRESSURE DISTRIBUTION ALONG. THE EDGE.

OF THE BOUNDARY LAYER

.

'-'- 0 9

DneF

x/L

1.1 0

08 09'

:10

x/L

1.1 0.2 cp 0.1.

(67)

x/L

1.00

0.95-

0.90-0;85 1.0 A U 0.5 0.01 1.0 -U

uo

A 0.02

0

Experiment (12) Theory

(1]

0.03

004

0.05 '0.06

00

0 _x/L0.99

0

0 _x/L:0.96

0 0

0 _{0 x/L=0.93}

y/L

0.01 0.02

yIL

0.07

r/L

y/L

0.03

y/L

(68)

t0.0 1 -1L00 0100 6000 9000 1.000 9000 S000 ,000 £00.0 0OO L00O E0 s0 90 CO 60 on

(69)

-0.1

07

.8

vne: t-i

:' 0.4 cp 0.3 0.2 0.1

(70)

H12 2

SHAPE PARAMETER H12

I

log

-0.3 0.4 0.5 0.6 0.7 0.8 0.9

x/L

1.0

LOCAL FRICTIONAL COEFFICIENT Cf

03

04

05

0.6 0.7 0.8 0.9

x/L

1.0

Cf1O

4

3

Body of revoLution

Equivalent flat pLate

(71)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 O Ideal f Low 3 C... I____

io

o8 iO 1.26106 ReynoLds number

(72)

0.5 0.4 0.3 0 0 1 2 3 Ideal

c103

flow

I I I I io9 1.26106

Reynolds number

00

4

tat

3ke w

placement ke Wd 5

LI1). ric

ii-0--

D/L:0.06

a--.

0.08 0.10 £ To > WI =

-J

p - -

--[w

0.7 w Wd 0.6 0.2 0.1

(73)

LUCY ASHTON

MOOR MODEL 684

(74)

Form factor, k

.0.3-106 _. _io7 io8

ReynoLds number, R

Vne

0.2 0.1