Design of propellers in ducts of given shapes Part I

.00l, Deft

Norwegian Ship Model Experiment Tank

The Technical University of Norway

TECHNISCHE UNIVERSITET Laboratorium voor Scheepshydromechanica Archief Mekelweg 2, 2628 CD De!ft Toi: O15-786873-FacO15.78835

Design of Propellers in Ducts of given Shapes

Part I

by

IKnut Minsaas

Norwegian Ship Model Experiment Tank Publication No. 115. January 1972.

DOCUMEN TA TI

lic

CONTENTS

Page

list of Symbols 2

2 Abstract 8

3 Calculation of Velocities on the Duct due to the Propeller 8

L Propeller-Forces and Distributions o Circulation _il

5 Profile-Characteristics 15

6 Cavitation-Safety 19

7 Wall-Effect 21

8 The Thrust of the Duct 23

Y Singularity Distributions on the Meanline of the Duct

2'-10 Pressure Distribution on the Surface of the Duct 29

1. LIST OF SYMBOLS.

Velocities and Circulation.

V speed of ship

V (x) l-w(x) V speed of advance

a

V axial velocity component

Vr radial velocity component

Vrq radial velocity induced by the sink and source distribution

Vd

radial velocity induced by the duct circulation

Vrp = radial velocity induced by the propeller

V axial velocity induced by the sink and source distribution

xq

Vd

axial velocity induced by the duct circulation

axial velocity induced by the propeller

Vd total axial velocity Induced by the duct through the

propeller disc

V = axial velocity induced by the duct circulation through

the propeller disc

= axial velocity through the propeller disc induced by the sinks and the sources on the meanlin of the duct

a velocity on the propeller surface due to angle of attack

V xp

V

o

J

velocity on the propeller surface due to camber

velocity on the propeller surface due to thickness

n revolutions per second

w

2n

angular propeller velocity

V(x)

resultant inflow velocity to each blade section

UT(x) circumferential mean velocity induced by the propeller

in the slipstream

U:(x) z axial mean velocity induced by the propeller in the

final wake

*

UT(x) z circumferential velocity induced by the propeller in

the final wake

*

Ua(x) z axial velocity induced by the propeller in the

final wake

UT(x) z circumferential velocity induced by the vortex-images

in the final wake *

AUa(X) z axial velocity induced by the vortex-images in the

final wake

w(x) z local wake fraction

F(x) z bound circulation of each section of one blade

G(x) F(x) z nondimensional circulation of one blade

- 7IDV

y(x) z circulation on the meanline of the duct

q(x) z sink-source strength on the meanline of the duct

V a

z _{z advance coefficient}

Forces and Pressures. T = propeller thrust Td duct thrust T = total thrust o Q propeller torque T

K p propeller thrust coefficient

p

2L

pn D

T

K total thrust coefficient

o _{pn D}

2'4

Q _{torque coefficient}

q _{pn D}

25

CL lift coefficient of the blade sections

CD = drag coefficient of the blade sections and the

duct profile

CD/CL drag-lift--ratio

ideal lift coefficient

drag coefficient due to angle of attack

friction coefficient of the suction side

C friction coefficient of the pressure side

fp

T shearing force

V

Po = static pressure at the actual propeller section with

a

o

C p

Dimensions, ratioes, factors and angles.

L length of the duct

D propeller diameter

R propeller radius

s propeller tip clearance

Rd R

+5

p 1(x) = chord length rh hub radius r propeller radius x r/R

f z camber of the blade section

g

t = max thickness of the blade section

p1 pressure in front of the propeller (fig. 7)

p2 pressure behind the propeller (fig. 7)

p3 pressure in the final wake

e vapor pressure of the water

cavitation number based on V_co

Rd

2 - x or p

k(x)

correction in angle of attack due to thickness

k(x)

correction in camber due to lifting surface effects

k(x)

correction in ideal angle of attack due to lifting

surface effects

ratio between total lift and lift by camber

ratio between liftsiope for viscous and ideal flow

ratio between angle of attack for 3 and 2 dimensional

2 a m1 m a flow

m2 ratio between zero angle of attack for viscous

and for ideal flow

R Reynolds number

n

a.

_u

ideal angle of attack for C_Li 1.0

a02 zero angle of attack in 2 dimensional ideal flow

a resultant angle of attack

3m

a03 zero angle of attack in 3 dimensional ideal flow

hydrodynamic pitch angle used in the calculation of induction factors

6i1 hydrodynamic pitch angle at the propeller

P propeller pitch

ACKNOWLEDGEMENT.

To SivilingeniØr Olav H. Slaattelid who has solved the problems involved in the adaption of the numercial methods best fit to satisfy the requirements of the theory, and who has done all the programming work.

ABSTRACT.

This report describes a theory for the design of propellers

in axial symetric ducts of given shapes. The theory is mainly

based on principles from the papers (l)-(1) and is used in the

first stage of the design. The propeller is calculated by

lifting-line theory with correction factors for lifting surface

effects. It tries to take into account the influence of

propeller-tip clearance. A computer program has been developed which allows

ducted propellers to be designed in the same way as conventional wake-adapted propellers, but the program includes the

inter-ference between duct and propeller. The work was financed by

the Royal Norwegian Council for Scientific and Industrial Research.

CALCULATION OF VELOCITIES ON THE DUCT DUE TO THE PROPELLER.

In the calculation of propeller induced time averaged velocities on the duct it is assumed that the propellers have an infinite

number of blades. The radial circulation distribution is equal

to the distribution used for finite number of blades. Using the

symbols shown on fig. 7 and applying Bernoulli's equation to the

flow relative to the propeller blades which are rotating with the angular velocity 2iîn, the relative angular velocity decreases

from 2irn to 2rn _- T/r and hence the increase of pressure is;

dT(x)

2ïrrp[2rnr - 0.5 UT(x)]UT(x)dr

where

UT(x) circumferential induced velocity immediately

behind the propeller.

The propeller thrust is also:

U (x)

dT (x)

2rp[V(x) +

a

]Uxdr

- (p - p3)2r'dr'

p

The equation of continuity gives:

U (x)

[y (x) _{+ Va(x)} + a

_lrdr

_[vAx

+ U(x)]r'dr'

La

where

Vd(x) axial duct induced velocity through the propeller disc ,

r propeller radius

rt radius equivalent to r in the final wake ,

U (x) a

axial propeller induced velocity through the propeller disc.

From the equations given above:

U (x) V (x)+V (x)+ a

U(x)

a d 2 V (x) + a _UT(x) [i

o3

V (x)±U (x) a 2 a a 2lrnr_0.5UT(x) U (x)_a pUT(x)(2nr_0.SUT(x))

The pressure gradient in the wake is balanced by a centrifugal force on the fluid and is given by:

2

and

d(p0-p3)

_UT2(x)

dr' P

The condition for constancy of angular momentum gives:

UT(x)r UT(x)r

hence

d(p0-p3) UT (x)r UT (x)

dr' 3

r' r

from which p-p3 is calculated by integration.

U

(x)

is solved as a function of

a

(x)

T 2irr

by assuming p-p3 0 in the first approximation. Ua(X)

determines the strength of the coaxial vortex tubes shed from the

propeller. The calculation ot tlie strength ot the vortex cylinders is illustrated in fig. 1. The cylinders consist of uniform distributions of vortex rings placed along the

streamlines as indicated. Axial and radial velocities induced by these rings are obtained from

2(---l) dV (y'r) 1

[Kk_1

r )E(k)] x

2r'

/()2(r+1)2

r

r

()2(r1)2

_r

_r

r

- d

r'

[Kk-l+

2 dV (y'r) r

2r'

rÁ)2(r+l)2

r r r r r r

k2 r and y x'-x 2

r r

K(k) and E(k) are the complete elliptical integrals

E(k) L

/1k2.2

K(k) - J da o /1_k2sin2

v and V are obtained by integrations from x i to x

and from rh to R (fig. 2). The integration in the axial

direction is restricted to a finite number of duct lengths dependent on required accuracy and the adequate computertime.

The total thrust of the system is:

R

U (x)

T 2rprV (x)+ a v (x)]U

xdr-p0-p32r'dr'

o _J

La

2 d a

rh

Hence the duct thrust is:

R

J 2prU (x)V (x)dra d. rh

L4 PROPELLER FORCES AND DISTRIBUTIONS OF CIRCULATION.

where x-x h x , or x' l-x l_Xh l_Xh d, e, p constants,

Xh hub diameter ratio,

by:

d/ ,e'

r(x) px 1-x

p[a _{sinix_xhfl+b sin(2u(x_xh)]}

where p, a and b are constants. It is further possible to use

the distribution

2 n

k(a +

01

x-4-a X + a x

2 n

which is illustrated in fig. 3b. An other distribution is:

r

k/1_z2(1+)

where k is a constant. This function is shown in fig. 4. The

distance between hub and propeller-tip is divided in two parts and the equation is used separately to each side of the dividing

point as shown in fig. 8. By using different "a" values for the

two parts and by moving the dividing point,most of the pitch

distributions in question are covered. The z values are:

Hub-Dividing point. X-XI-, z

X -X

_ph

, (fig. 3a)

Dividing point-Propeller tip.

x-x

and the torque:

p

[1+ S ]xdx

tgí31

in these equations tg81 is:

* * U (x) lU (x) U (xrnl) V (x)+V (x)+ a + a + a a d 2 2 2 1 U =

O for r(x1)0

a

The propellers are designed by ordinary lifting-line theory and

induction factors. The selfinduced velocities at x are:

U a -

f

.dGdx

(x)-

i V o

adxx-x

o Xh * U (x) lU (x) U V (x)±V (x)± a + a a d Applying the K p Kutta Joukowsky T 2 X

law, the propeller-thrust

U(x

i *

)+X[i_tgj

T is: pn2D

23

R F(x)[wr h 2 - Q -

2L

23

pn D 8w R Xh tg131

U(x) lU(x)

+ 2

where

i axial induction factor,

a

--tangential induction factor.

The induction factors are calculated as in [ 6], ana are functions

of the hydrodynamical pitch angle of the vortices. One possibility

is to use the angle at the propeller given by:

tg il U AU U (x=l) V (x)+V (x)+ a a 2 a d U 2 tg3

2nr_UT+UT

anotiLcr is to use the angle in the final wake defined as:

k

*

V (x)+U +TJ

+ U (xl)

a a a a

In this theory the angle is given by: V (x)+k(U +U + U (xl))

tg1

_2nr-k(U+U)

a

where k is a function of propeller loading and

5. PROFILE CHARACTERISTICS.

It is assumed that is a linear function of the effective

angle of attack. The liftsiope, ideal angle of attack and the

angle of zero lift are corrected for lifting-surface effects and

viscous flow. Further the lift is corrected for the influence

of profile thickness which alters the angle of attack. To obtain

the geometrical pitch of the different propeller sections the

following angle must be added to 3.'

57.3 k +573(a-l)C .+57.3k t(x)

ma

a C m1 2 03 2îm1 Li t D 3m Li 2î where

ma +ka.

a 02 a i2 a +a. 02 i2 ( 57. 3 a a. 02 2 ll)CLi total lift - lift by camber

kt correction in the angle of attack due to thickness

effect ,

t/D max blade thickness ratio

m zero angle of attack(viscous flow)

2 - zero angle of attack(ideal flow)

zero angle of attack(3 dimensional flow)

ma _{zero angle of attack(2 dimensional flow)}

C . ideal lift coefficient

a.

ideal angle of attack for C .

il Li

m1 ratio between lift slope for viscous and

ideal flow

k correction in the ideal angle of attack due to

lifting-surface effects

a

ma

03 a 02

he following standard values are used:

m1 1.00

m 1.00

a

m2 1.05 NACA a 0.8 (modified)

ide camber of the different sectiones are obtained from:

f

ck CLi(x)

where

0.06790

for NACA a 0.8 mean lines

c 0.06651 for NACA a = 0.8 (mod) mean lines

o 0.05515 for NACA a 1.0 mean lines

kc correction in camber due to lifting-surface effects. The

corrections for lifting-surface effects are calculated separately using lifting line and lifting-surface theory as developed for

conventional propellers. At the moment the correction factors

do not include the effect of the duct. This effect is taken into

consideration by using half empirical corrections. The drag

calculated separately. In the latter case:

where

C C +C

D p a

[l+2+60jCf5+Cf

C friction coefficient of the suction side

fs

0fp friction coefficient of the pressure side,

t/l maximum blade thickness fraction of the section,

chord length

C (a-l)C .sin[S73k

Li] (a-1)C

a Li 2iîm

where mt , k and a are defined above

= a function of the nose radius of the section (0.25 - 0.75)

The friction on the pressure and the suction sides can either

be preset or calculated. In the latter case radial flow along

the blades due to friction and centrifugal forces will cause an increase in drag compared to the drag in 2 dimensional flow.

According to Schiicting the tangential shearing force of a rotati

disc having laminar flow is given by:

TV(0) _{0.616 B(s)}

pV2

-

n°

IR(0)

For a flat plate:

0.332 B 2

PV

/R (0) /R (e)

n n

In the first equation

V0r

R (&) n

ihere (x) (z distance from the leading edge to the actual

point). If there is no pressure gradient in the flow direction the flow is supposed to start as the flow along a flat plate

and then approach the flow on a rotating disc at O . The

shearing force along the propeller sections can then be obtained

from: T

(e)

y

____

(0.6l6+

0.007572

pv2

IR(o)

(0.2+0)2.35 n giving: and 2 P V T (0) y 0.616 2 - /R (o) n T (0) y 0.332 /R (e) n for e 'r for e O

The friction coefficient is obtained by an integration from O

O to e. If the R of the sections are defined as

n

R

)xJjA2<)2

nx

model tests in open water seem to indicate that the suction side of

the propeller remains laminar up to Rflk

2-310

whereas the

critical R of the pressure side is: R

6-8l0.

For R higher

n nk n

than the critical number, but lower than Rn 3'lO , the friction

if

Cf

A(R

Q.L.55 _nk

(1ogR)258

R

where R critical Reynolds number.

nk

R

2i0

310

510e

io

3l0

nk

A 700 1050 1700 3300 8700

In turbulent flow and with rough surface: 0. 572[l.085 log(R)-0.51 f -

_(logR)3

.L55

R >3l0

Cf > (lo R )4.58 and If not: QL.55 (logR )2.58 n

The equivalent sandroughness iK is

(fig. 6)

where 1K' is mechanical roughness and T is a function of the

machining and. finishing of the propeller surface, (2-5). Typical

values for Cfs+Cfp are shor, in fig. 6.

2

CAVITATION SAFETY.

Due to the axial velocity vd induced from the duct through the propeller-disc the static pressure has to be corrected by

versa for accelerating ducts. This pressure correction is

obtained by assuming a propeller with an infinite number of blades

and by neglecting tangential propeller-induced velocities. From

fig. 7 and Bernoulli's equation:

U (x) 2 pl+p/2[Va(x)+Vd(x)+a2 I

p+p/2V2(x)

and and where U (x) 2 a p p/2[V (x)+U (x)]2 o

La

a :ssuming that: dT . 271r(p2-p1)dr pl pi-p2 Apd ₂ we get: Vd(x) U (x) V (x) pV 2 d a (x)v (x) [l+2

(x)2V

(x)l a a a

This equation is identical to the equation given in [io] . The

cavitation safety is defined as:

V 2 'V

F ()

_nl

G o

p(x)_e_pd

V U U 2Trnr----+ 2 2 cose.

2

1 X

and V

f ()

chordwise surface velocity.

p

nl

G

The chordwise distribution of the velocity is:

V V

(_)2

a)2

V

VVV

where + suction side,

- pressure side,

velocity due to angle of attack, = velocity due to camber,

y0 velocity due to thickness.

lv

(CL_CLj)fl()

00

V

(f3()-l)051 +1

f, , f2( and f3() are specific functions for tie actual

meanlines and thickness distributions in 2 dimensional flow. In fig. 21b pressure distributions calculated by the method

indicated above are compared to values measured on a propeller model at SNT [131

WALL LFFLCT.

11 velocities induced on and by the duct are regarded as

uncnrrecl-during the calculation of the hydrodynamical propeller pitch,

the result obtained will not be correct. The expression for the

pitch must also include the effect of finite blade number. This

effect is determined by prolonging the inner surface of the duct

from the following edge and down to infinity. The boundary

condition is satisfied on this cylinder downstream where the free vortices shed from the propeller blades are mirrored by using

vortex images. The vortex images will induce axial and tangential

velocities at the propeller. These velocities will together with

the "smeared out" velocities and the selfinduced velocities of the propeller (taking finite number of blades into account) give the

final pitch. The vortex images are placed outside the "tunnel"

at a given distance. To each free vortex shed at the radius

61

r with the pitch P and the strength -- there is corresponding

vortex with the strength - at the none dimensional radius.

,z f(R

p

(fig. 8)

The vortex image is given the same pitch P as the free vortex. The velocity induced by the vortex images on the lifting lines of the propeller is:

1 i

r.dGi

-(x )

z -

I 2 o L1 J dx Xh where:

i _{z induction factor given by z,}

and(-x.

2 and have been calculated for different circulation

Rd Rd2

distributions using the functions

z 2 -x, and

z

p

The choice of the function seems to have minor influence on the

result. For a given distribution of circulation the wall effect

will reduce the pitch at the propeller tip. The pitch reduction

will increase by decreasing propeller-tip-clearance as can be seen

from fig. 9, 10, 11 and 12. The velocities for the 15-biaded

I

identical to the case Rd/R = z The latter ttrepresentedfl

by z _{15, Rd/Rp} . If the propeller circulation is finite

at the propeller tip and s O the tip vortex of the propeller is

neutralized by the vortex image as indicated in fig. 15a. The

figure shows the down-wash for a 2 dimensional foil close to a

wall.

8. THE THRUST OF THE DUCT.

By using the radial induced velocities of the propeller on the meanline of the duct and the duct circulation it is possible

to calculate the axial force on the duct. This can be done by

applying the Kutta-Joukowsky law. Hence the ideal thrust of the

duct is: L Kdi _{2 =} J R(x)y(x)Vrp R

()dx

pnD

nD

o

where R(x) the radius of the meanline of the duct. The momentum

theory gives: i

U ()V ()xdx

Kdi Td 2n2D2

aR dR

pn D Xh

Several calculations indicate that the thrust obtained from these expressions are identical within reasonable numerical accuracy. The ideal thrust must be corrected due to the drag of the duct. A drag calculation based on the mean pressures and the thickness

of the boundary layer is at the moment aimless due to the uneven flow on the inside of the duct causd by the finite blade number. In addition the drag is a relatively small component compared to

the total thrust at least for heavy loaded systems. Thus the drag

where

C0

2R LV

2

m xm

R mean radius of the mean line of the duct

m

V axial mean velocity along the mean line

xm

total drag of the duct ,

t max thickness of the duct

L total length of the duct

and C are calculated as functions of the R as for two

-f5 fp n

dimensional laminar and turbuJent flow.

LV

R xm

n \)

The total duct thrust is:

IKd

_-

₂₄

1K._di

_2L

pnD

pnD

SINGULARITY DISTRIBUTIONS ON TEE MEANLINE OF THE DUCT.

distribution of circulation along the mean line of the duct is

assumed. If the shape of the meanline and the propeller thrust with its radial distribution are given, the velocities due to the

duct must, together with the propeller induced velocities, satisfy the boundary condition on the mean-line:

whe re

= radial induced velocity by the propeller

Vrd radial velocity induced by the duct circulation

Vro radial velocity from a body of revolution (hub)

V(x)

axial velocity from a body of revolution (hub)

Vxd axial velocity induced by the duct

axial velocity induced by the propeller

Vxq axial velocity induced by the sinks and the sources

representing the thickness of the duct

Vrq radial velocity induced by the sinks and the sources

representing the thickness of the duct

A circulation distribution as indicated in fig. 13 is assumed.

by calculating the velocities and satisfying the boundary condition in as many points as there are unknown columns, a system of linear

equations is obtained from which the circulation may be solved.

In the computer program a number of 30 columns along the mean line

is used. If the entrance to the leading edge of the duct is not "shock free", a "direct" calculation of the selfinduced velocities

on the duct will be incorrect. This is illustrated in fig. 13 and

table I, II, III. The down wash along the chord of a flat plate

at an angle of incidence in 2 dimensional flow has been calculated by a "direct" method and by the method used in the computer program

where a transformation is used. _{The transformation is based on}

the following reasoning: // If the function f(x) has a logaritmic

infinity at x = O the integral

J

f

f(x)dx

o

can be solved by:

n J /x F(x)d/ o V +V +V +V dr rp rd ro rq dx - V +V +V +V +V (R x xd xp xq A d (fig. 2)

n

where n is an integer greater than unity and / f(x) is not

singular at x O. The circulation of the duct is solved from the

velocity equations by placing the first control point at

the second at 2- , the third at 3- and the last at

Table II shows the down-wash along a flat plate having parabolic circulation distribution calculated by the above mentioned method.

X dx

V'

r V r V r (theoretical) X 1 4.8900 -4.8258 .0642 1.0004701 -1.0000000 0.0000000 77.5395256 2 4.4050 -3.9041 .5008 .9976770 -.9968000 .0016000 - .9995535 3 4.0144 -3.3540 .6604 .9880813 -.9872000 .0064000 - .9995513 '-t 3.7091 -2.9663 .7428 .9720886 -.9712000 .0144000 - .9995475 5 3.4614 -2.6684 .7930 .9496990 - .9488000 .0256000 - .9995423 6 3.2536 -2.4268 .8268 .9209128 - .9200000 .0400000 - .9995352 7 3.0747 -2.2236 .8512 .8857302 - .8848000 .0576000 - .9995263 8 2.9174 -2.0479 .8695 .84141516 - .8432000 .0784000 - .9995154 9 2.7768 -1.8930 .8838 7961775 - .7952000 .1024000 - .9995021 10 2.61493 -1.7540 .8953 .7418088 .71408000 .1296000 - .9994862 11 2.5324 -1.6278 .9047 .6810461 -.6800000 .1600000 - .9994671 12 2.4242 -1.5117 .9125 .6138907 -.6128000 .1936000 .999LtLt43 13 2.3231 -1.4039 .9192 5403442 -.5392000 .2304000 - .9994170 14 2.2279 -1.3031 .9249 .4604086 -.14592000 .2704000 -.9993842 15 2.1377 -1.2079 .9298 .3740869 -.3728000 .3136000 -.9993443 16 2.0517 -1.1175 .9341 .2813830 -.2800000 .3600000 -.9992953 17 1.9691 -1.0312 .9380 .1823030 - .1808000 .4096000 -.9992341 18 1.8894 -.9481 .9413 .0768559 -.0752000 .4624000 -.9991562 19 1.8121 -.8677 .9444 -.0349439 .0368000 .5184000 -.99905'43 20 1.7366 -.7895 .9471 - .1530723 .1552000 .5776000 -.9989161 21 1.6625 -.7130 .9495 -.2774851 .2800000 .6400000 -.9987193 22 1.5894 -.6376 .9517 -.4080922 .4112000 .7056000 - .9984182 23 1.5166 -.5629 .9537 -.51446796 .5488000 .7744000 - .9979049 24 1.4438 -.4882 .9556 -.6865903 .6928000 .8464000 - .9968489 25 1.3702 -.4130 .9572 - .8305622 .8432000 .9216000 - .9936168 26 1.2950 -.3364 .9586 - .8903161 1.0000000 1.0000000 -.9455958 27 1.2170 -.2572 .9597 28 1.13'l -.1737 .9605 29 1.0425 -.0829 .9597 30 .9298 0.0000 .9298 table I table II table III

and

The thickness of the duct is represented by a distribution of

sinks and sources with the strength q(x) along the meanline. This

distribution induces a vertical velocity through AD (fig. 114),

equal to:

y q(x)

n

The strength of the sinks and the sources is obtained from the equation of continuity: dV (x)

V(x)h(x)+qdx

[y (x)± dx] [h(x)+dh( dx] t dx dx «hich gives: dV (x) dV (x) dh(x) t t dh(x) q(x) 2V (x) +2 h(x)+2 t dx dx dx dx

dx,

or approximated: q(x)

_V(x)+t(

x)

dV(x)

dx dx V (x)_t

J(V

_rp+ _rq+ _rd+ _ro

)+[V +

_x + + + (R ) xd xp xq A d

t the thickness of the profile

The calculation starts without including the thickness effect

(without V_rq and V_xq). Then q, V and V are calculated. V

rq xq rq

and Vxq are included in the equations and y(x) is solved. The loop

is repeated 14 times. The induced velocities from the sinks and the

sources are obtained from

and E(k) 2--C-j--l) dV q 1 ÍK(k)-Fl- r r 1E(k)l rq 2rr'

2-A7 q r uy -xq 2irr

/()2(r1)2 [()2+(rl)2]

r r

Lr

r r r'

()2+(_l)2

2 r r r r where

k2=

r

aridyx'-x

()

2r

2

IK(k) and E(k) are the complete elliptical integrals

iî/2 (k) I dx o /l_k2sin2 IT/2 E(k) J /1_k2sin2a _dx

Jhe circulation on the meanline of the duct will induce an axial

velocity through the propeller disc. This velocity vd(x) is

obtained by summing up the contributions from each of the columns

showed in fig. 13. A typical v(x) distribution is shown in fig.

i 5b.

10. PRESSURE DISTRIBUTION ON TEE SURFACE OF THE DUCT.

The pressure on the surface of the duct is given by the tangential

velocities on this surface. Compared to the pressure at infinity

*2

2

-V

or dimensionless: C P Va where

V tangential velocity on the surface of the duct

The tangential velocity on the meanline is:

v(x)

+V +V +V )2+(V + + +V )2 np rd ro rq x xd xp xq

The velocities ori the inside and on the outside of the meanline

are:

v(x) = v(x)

₂

where

((X) circulation on the meanline

+ inside of the meanline .,

- outside of the meanline

The tangential velocity on the surface of the duct is obtained

by using the Riegels-factor. hence

* _v(x) V

/

+ 2 dx Il. EXAMPLES.

in fig. 25. The propeller is operating in uniform wake and is

placed in the middle of the duct. The radial circulation

distribution is:

0.60/ 2

f px vi-x

which is illustrated in fig. 3a. The propeller is designed for:

J 0.2880 D 300 p

K 0.l90L , n 21

p

For

this condition the duct thrust is:

1K

= 0.073.

p

An example showing input and output is given on the pages 65 - 7+.

The figures l5b, 16,

17,

18, 19 and 20 shows velocities induced

on and by the duct. Further the distribution of sinks, sources

and cifculatiOn on the meanline is shon . Fig. 20 shows the

selfiriduced velocities of the propeller and fig. 19 the pressure

distribution on the duct. From fig. 19 it is clear that the duct

thrust is mainly ttproducedt on the inside of the duct forward of the

propeller. An example showing the influence of k and d, (in 1

pxd/_x2

_{on the pitch distribution is given in fig. 2la. There}

is an influence from tip-clearance on the characteristics of the

propeller and the duct. For given advance coefficient propeller

thrust and propeller circulation the duct thrust is decreasing

with increasing tip-clearance. The axial velocity induced by the

duct will decrease together with the propeller torque. The final

result is a reduction in efficiency. This is illustrated in fig. 23.

Fig. 22 shows the influence of tip-clearance on pitch distribution

and axial velocity through the propeller disc. Fig. 23 shows that

the influence on the efficiency from tip clearance is minor for the

most actual values of R /R (1.01-1.02). Further calculations show

d p

that the influence of tip-clearance is also a function of the type

of propeller circulation. The duct thrust is for a given advance

where calculated 'd values are compared to values from model tests

with a bladed c.p. propeller in duct 19A. Model tests with the

ducts shown in fig. 25 indicates that the thrust of 19A is essen-tially less than the thrust of the other duct for given advance

ratios and propeller thrust coefficients. Calculations give

similar results. The pressure distributions on the ducts for

J z O.35 and identical propeller load distributions are given

in fig. 28. The figure shows that a higher value is necessary

for 19A than for the other duct to give equal duct thrusts.

Fig. 27 shows a comparison between calculated and measured pressure

distributions on the l9A duct for different loads. As well

calculations as measurements indicate that the marked pressure -peak shown on fig. 26 does not appear on 19A due to the softer nose of this duct.

REFERENCES.

B.D. Ccx: "Vortex Ring Solutions of Axisyrnetric Propeller Flow

Problems". M.I.T. Department of Naval Architecture and Marine Engineering Report No. 68-73.

H.E. Dickmann: "Grundlagen zur Theorie ringförmiger Tragflügel

(frei umströmte Düsen). Ing.-Arch. l90.

D. IKüchemann, J. Weber: "Aerodynamics of Propulsion", New York, 1953.

W.B. Morgan: "Theory of Annular Airfoil and Ducted Propeller", th Symposium on Naval Hydrodynamics, Washington 1962.

H.W. Lerbs: "Moderately Loaded Propellers with a Finite Number

of Blades and an Arbitrary Distribution of Circulation". Trans.

SNAME, Vol. 60, 1952.

L. }Kobylinsky: "The Calculation of Nozzle Propeller Systems Based on the Theory of Thin Airfoils with Arbitrary Circulation

Distribution". Intern. Shipbuild. Progr. 8, 1961.

H.E. Dickmann, J. Weissinger: "Beitrag zur Theorie optimaler

Düsenschrauben (IKortdüsen)", Jahrbuch STG, Bd. 9, 1955.

K. Wiedemer: "Ein Beitrag zur Theorie der Düsenschrauben

(IKortdüsen)". Mitteilung aus dem Institut für Angewandte

A.J. Tachmindji: "The Axial Velocity Field of an Optimum

Infinitely Bladed Propeller". David Taylor Model Basin Report

129'4, January 1959.

G. Dyne: "A Method for the Design of Ducted Propellers in a

Uniform Flow". Statens Skeppsprovningsanstalt. Publication

nr. 62, 1967.

L. Boliheimer: "Lin Beitrag zur Theorie der Düsenpropeller". Schiffstecknik Heft 76, April 1968 (15 Band).

W.T. Durand: "Aerodynamic Theory", Volume IV Airplane Propellers

by H. Glauert. Dover Publications, Inc. New York.

0. hiby: "Three-Dimensional Effects in Propeller Theory", Norwegian Ship Model Experiment Tank Publication No. 105, May

1970.

1)

B.S. Gulbrandsen: "Utvikling av metode for analyse av

propell-forsØk ved lave Reynoldstall med full eher delvis 1aminr-strØmning". _{Hovedoppgave ved SMT 1970.}

15) J.D. van Manen and M.W.C. Oosterveld: "Analysis of Ducted

Propeller Design". Trans. SNAME Vol 7, 1966.

15) _{J.D. van Manen: "Effect of Radial Load Distribution on the}

Performance of Shrouded Propellers", Publication No 209 of the N.S.M.B.

J.D. van Manen: tRecent Research on Propellers in Nozzles", mt. Shipb. Progress No. 35, 1957, Vol. .

M.W.C. Oosterveld: "Wake Adapted Ducted Propellers", Publication No. 35, Netherlands Ship Model Basin Wageningen, Netherlands.

SINGULARITY - DISTRIBUTIONS

SOURCES

VORTEX RINGS PROPELLE R

DISK

AR S

4J \J \

¿R

w w w

AR AR

AR

AR SINKS

FIG 1.

5= PROPELLERS

TIP 1Sr

tg

im A R,

ARI

jARi

AR7

'AR,

VORTEX' RINGS

Va(X) # kU(x)

2nr -k

( for

k see page 1')

df'

dr

X

y

R

Vp#Vrd+'rq

Vro

Vxp+VxdtVxgVAI'Rd)

(p (p

IL

Ir

_lii:.

II

oo

"#

co II V-C. co C')

L-FIG. 3a

CIRCULATiON DISTRIBUTIONS

"o Q.

FIG

3h.

Q Q Q t." U) Q Q Q L II

L

CIRCULATION

DISTRIBUTIONS

FIG. 4

BLADE

AREA RATIO

0,432

Rd/Rp

1,01

DUCT 19A

(LID - 0.5)

Î

77

MODEL TESTS

Fl5

o

CALCULATED O

0,030

0,100 Kd

0,050

o

INFLUENCE

_{FROM K ON THE PITCH}

3

0,350

DIAMETER (MODEL)

245 MM

(CALCULATED)

300 MM

RPS

(MODEL)

30

K0 RPS (CALCULATED)

21

0,035

NUMBER OF BLADES

4 0.8 0,9 1,0

P/D (0.7)

030

K0 Kp

0,25

0.20

0,15

7 5 to 2 3 6 70' 2 3

Equivalent sandroughneSs

Measured roughness

6Z

2 ¿

FIG6

IÌI

=

ìiiJiÌIIÍII

iiiUIIIiii!ii

IIIiIui!!!.iU!iA!!

jïiIip!!

IIiI!!!!I!!!

IIii!!-

-imii

i

*II

_{III iuíïir i}

I

II

Iii _- I

II

Iii _-

II

II! --

Il

L

I I III

- t .d

id,o

9

VA po

PROPELLER

CAVITATION

P, P2 i

VA+Vd+--.u16

PROPELLER DISC

P2 -1-Pl g Po Po

Pd= PÍP2 -t-P2

4

FIG. 7

Po

I CONTINUOUS VORTEX' DISTRIBUTION

_______

Z= -1

x=1

AXiAL VELOCITY INDUCED BYL

Lc ONTÍNU OUS VORTEX DIS TRIBU TION

zo

DIVIDING'

-t!Q!'I

INNER SURFACE OF THE DUCT

or

/ ox

ICONCENTZE9

VORTEX'

!t4QEJ

CONTINUOUS VORTEX D14STRIBUTION 2Ttr -

or

- OX

VORTEX IMAGES

W W W

s

\or

s

Ia

OX

I

or

ox8

VC LU a 2 FIG. O Z= -t-1 OUND R TEX) Xh XL, PROPELL ER DISK

G) '8 .7 6 .5

tW

Ua«

VVA

.4 .3

i

o

C4,76571

_30,32O

z = 5

a#,r

R:j/Rp=co

Rd/Rp =1,0005

Rd/Pp =10209

r-px°'52

V/_x21 o

i

2 3 .4 .5

6

7 .9 1,0

FIG. lo.

"9 Lo

/

8

/

/

q

y

INFLUENCE FROM THE VORTEXIMAGES

FIG. 11

o) t-' C) Lfl II N I' N B

q

C-('o

X

L!)

4r 4

t1J

FIG. 12

U) N IC, Q. 'I

L-7//II

Q:

Method used in Program

n =m J I___

-

dX = 2i

> f(x)

dX dX _mJ FIG. 13

ï

BOUND CIRULA 1/ON

OF THE FOIL

j"

FIG. 14

DOWN WASH

FROM VORTEX IMAGES AND

FREE

VORTICES

dxdx

FIG. 150

VORTEX IMAGES

FREE VORTICES

,

t

I

t

_t

,

t

y

t

t

1.0 0,8 0.6 0.4 0.3 0.0

AXIAL VELOCITIES INDUCED THROUGH

THE

PROPELLER DISC.

Vd va

v'i

DUCT 19 A

=

0,288

0,073

Kp

0.1904

EXP

0,600

Va Va e 4 4 _e 0.3 0.4 0.5 0.6

01

0.8 0.9 1.0

A= tIR

FIG 15b

6.0

-y-Va 5.0 4,0 2ß 1,0

.lß

SINGULARITY DISTRIBUTION ON THE

MEANLINE OF THE DUCT

X/L

FIG. 16

79&

J

L/i= 0,5

= O,20

9

\

I

s73

seo

\

\

\

LE

\

\

FE

N

-

- -

q

-p

02

s

04

Q6

08

1.0

1,2 V Va

'p

48 04 04 42,

co

42

- 0,4

PROPELLER AND SEL FINDUCED AXIAL

VELOCITIES ON THE MEANLINE OF THE DUCT

Vxp + Vxd. Vxq

/

-

DUCT

J

=028ã0

19A

= 0,1904

Kd

=0,0730

EXP

=

0.600

\

/

\

\

_1Vxp

\

VxdVxq

-N N

ççJ

A A A A A A

40

0,4 0,6 0,8 1,0

X/L

FIG. 17

04

0,2 010

-02

-0,4 -Oß

-09

- 1,0

PROPELLER AND SELFINDUCED RADIAL

VELOCITIES ON THE MEANL/NE OF THE DUCT

4 I

/

/

I

/

LE

/

/

/

F L

/

/

\VrpVrdVrq

I

/

/

.. Vrp

y

DUCT

19A =

0,2&0

=

0,1904

Kd =

0,0730

EXP

0,600

A A A A A 0,0 0,2 0,4

₄₆

_{0,8 1,0}

x/L

FIG. 16

0,8 V Va 0,

PRESSUREDISTRIBUTION ON

THE DUCT

19A L/D=Oa

J

=

0.2500

=

0,1904

-

00 730

QQ _{Q2 0,8 1,0}

X/L

FIG

19

cp

27

OUTSIDE

-20

-40

INSIDE

-iqo

u:/VA 1,3 1,2 1, 1 1.0 Q9 0.8 0.7 0.6 0.5 0,4

u/1:4

0.4 Q3 0.2 0.1

SELFINDUCED

VELOCITIES

BY

THE PROPELLER

0.6

0.7

X=r/R

0.6 0.7

X= nR

FIG. 20

0.3 0,4 0.5 0,8 0.9 1.0 0,3 0,4 0,5 0,8 0,9 1,0

p/p0.?

1,0 0,9 0,8 0,7 0.6 1,0 0,9 0,8 0. 7 0, 6

INFLUENCE FROM THE ciRcuLAr/ON AND K

ON THE

p/rcH

+v

'-s

/

K_=0,5 1,0 19A

LID

0.5 J =

0,2ã80

/+//

s K =

0,1904

Kd =

0,0730

EXP- KONSTANT 0.55

Rd/Rp

1,0 1

7.

EXPb0.65

//

19A

LID = 0,5

J

= 0,2080

/ExP=a55

= 0,1904

,

_/

Kd

= 0,0730

K

= KONSTANT 0,75

Rd/Pp

= 1O1 0,3 0,4 0,5 0,6

0,7

0,8 0.9 Q95 1,0

X= rf R

P/p07

0,3 0,4 0,5 0,6 0,7 0,8

0,9 0,95

¿0

X =r/R

FIG. 21a

-0,4 Cp

-0,2

0,0

-0,2 QQ

-Q6

Cp -0,4 - 0,2 0,0 Q2 0.0

-0.8

Cp

-0.6

-0,4

I

0,0

I

42

0,4 0,0

CHORD WISE PRESSURE

DISTRIBUTIONS

I

0,4 0,6 0,8

_x

1,0

RADIUS 0,7R

Q2 04 0,6

=

RADIUS 0.9R

0,4 _{0,5 0,8 X 1,0}

I MEASURED

- CALCULATED

08 X 1.0

RADIUS 06R

FIG. 21 b

110 0,9 0,8 q, 7 0.. 6

.L01

1,02 1,05

/

79A

=0.26'CO

L/D=0,5

Kp =0.1904

/

/

101

RdIRp

1.02 -105

I----

19A LID 0,5

J

0,2880

= 0.1904 0,3

0,4

0,5

0,6 0,7 0,8 0.9 1,0

X=rIR

0,2 0,3 0.4 0,6 0.7

08

0,9 1,0

Xr/R

_{FIG. 22}

Vd va 1,0 0,9 0,8

0,7

0,6

0,5

0,4 0.3

0,4

KdK

1OKQ

0,35

O 30

INFLUENCE FROM TIP-CLEARANCE

19A

L/D=0.50

.7 = Q2&«O

= 0,1904

7

_O =

KP#Kd

_2,r

\

_N

N

N

N

N

O

K.,,/K

NU

N

P

N

N

N

K; .10 101 1,02 103 1,04

1:05

1,06

d

/R1,

FIG. 23

THE

THRUST OF THE DUCT

/

j =0.2.38

=0.288

j

0,345

CALCULATED O

K 1,0 0.10 0

0,05

FIG. 24

DIAMETER (MODEL)

(CALCULATED)

245 MM

300 MM

Kd

RPC

_(MODEL)

30 i O

(CALCULATED)

27

NUMBER OF BLADES

4

BLADE

AREA RATIO

0,524

19A

LID =

0.500

DUCTS USED FOR COMPAR/SION OF

PRESSURE DISTRIBUTIONS

L/D= 0,5

19A

LID = 0,5

-FIG. 25

4

-3

-2

i

#1 0.1 u.b

X/L

,

_/

/

_/

K

j =a345o

=0,18800

/

_I

\

fi

i

I

/

_/

\

's

/NHDE

.1

I

'4 '4

J

/

_I

I

\\

I _I

0US/DE

-6

C

_-5

- 14 Cp

12

10

8

6

o

2

10

ci

Cp

6

4

2

o

2

0

7

6

5

4

Cp

3

2

1

o #1

0,1 X/L

0,2 0

01

0,2 XJ/ MEA SURED 0,3

14

Cp

12

10

6

o

+2

-to

-o

Cp

-6

4

2

O

+2

0,3 0

_{0,1 X/L 0,2}

7

5

4

Cp

3

O

01

02

X/L

CALCULATED

J= 0,230

CALCULA TED

J

=0,208

F/G.27

0,3 ULA - 0.3 TED

46

0,3

J

3"

2

1

0,3

0,1 X/L

0,2 0,3

0,1 X/L 0,2

O +1 -

e

-s

STRESS CO5T : REL.RAD. 0.280 0.300 0.400 0.500 0.600 0.700 0.803 0.9no 0.950 EDGE 1H. 0.005 0.005 o.o0 0,002 0.001 0.001 0.000 0.000 0.000 L.FORF 3.026 0.028 0.036 0.043 0.049 0.052 0.052 0.046 0,038 L.AFTFR 0.026 0.028 0.036 0.043 0.049 0.052 0.052 0.046 0.038

W.DISTR. 1.00

1.000 1.000 1.000 1.030 1.003 1.003 1.000 1,000 THICKNESS AT O,3R 0.011 M THICKNESS AT O.6flR 0,007 M 3 0 F3ERG DIPLOMOPEG.

1)YSEpROFIL: NACA 20 (MOD)

L130.5

THIS CALCULATION 15 BASED

ON

THE FOLLO1NG SET

F ¡N PUT NUMBER OF BL ADES : 4 DATA DIAMETER PRoPELLER: 0.300 i PERMTS.STRESS 800.0 KP/CM2 HUB D1AM[TFP

0.O4

' DEsIRrr POWER 6,0 DESIRED THRUST

71.3

KP SPEC.GRAV.PRUPILLL. 7.850 KP/DM3 SHIP SPEED 3.53 KNOTS ROUGHNESS 10.00 MY PROPELLFTP SPEED 1260.00 RPM ROUGHNESS COREC 3,500 TAYLOR WAKE : c.000 1H (KIJf5S BLADET j: 0.001 M PROPELLER DrPTH : 0.350 M CAV.SAFETY AT 0.8R: 30,OC % BLADE AREA PATIO

0,520

GD2B SJRK.FKSP. : 0.603 CLASSIFICATION PEUIREMENT DNV MAX,POWFR : 6.0 BHK rlAx.PPOF'.SPEEI): 1260.00 RPM RAKE

0.00

DEGRØ

AO=0 INNLEST AkE FOIDELING

A1'4

STANDARD 5TRKULASJONSFOt?LNj(EKSP))

P21

INNLEST SN! TT1EIc,()ER

A32

VFR!TAS KRAV fOP VPIBAR PROPELL

M4=3 TYI"KELSE3FORDELIN SE [3rSKRIVELSE

A51

MTNSAA STIGN. KORREKSJON A6=j RUHETS KORREKSJPN EITER MINSAAS

A71

STANDARD + YR0DYNAM1Sv UTSKRIFT A8c1 THRUST DJMENsJoERFNCE FOR RETTA!

A92

INDIJKSJONS FAKTOREP

A101 UTSKRIFT AV KAv/TRYKK FflRÛELING Al 11 IF'!NLEcTNG AV

i

Aj21 INNLESING

V NYE STANDAU VERnIER

A131

vFGCErFEKT: KAPA2*ROF-X A1'40 VRQ FR IKKE MEl) I FES1, AV KILDESLUKFORI)ELINGEN

PAELLOH RESULTATEP FEA PROCEI)UPE ÇIRKULASJON El

CTS 6.O&O0 O 591 '$

2.85007

7 2291 's 5.98783 X STR FX1 eri KR 0.9500 0.4082 C.3507 11.4545

O.587

O.9fl00 0.5402 0.464n 12.1950 0.7575 O.8rf2O O.hóOa C.59' 13.8166 0.9073

0,700

Q.'H0P

C.R7P

15.6529 0.9602 0.6000 0.6439

C.SO7

17.8380 0.9725 0.SnCO 0.5465 O.467c 20.3809 O.9s36 0,4000 0.3901 0.316 23.5564 0.8674 0.3100 0.1353 C'.1164 71,79!37 0.3073

r)ATA TI f)Y5EEREiGNTNG PRflPELLPOS I Sjor:

n 500

OYSFLENG)E : (),15c3 SPAITAVSTAND : 0.0010 X D / L 0.00 0.17 0.33 0. ü C. 67

0.3

1 .00 X0J1 C.flc 0.00 0.01 0.02 0.03 0.05 0.06 0.08 0.10 0.12 0.15 0.17 0.20 0.23 0.27 0.30 44 0.38 0.'42 0.47 0.;i 0. '6 0.61 0.67 0.72 0.78 0.84 0.90 0.97 OR 0.00 -rl.74430 0.00 -0.303 0.00 fl.1100 0.00 -0,07440 0.30 -0.0630 C'DO -C.04'40 0.00 -'.0230 TY DTY 0.3005 6.6250 0.0016 2.3550 0.0026 1.2100 0.0036 0.8500 0.00441 0.6520 0.0056 0.S'400 0.0065 C.14450 0.0û75 0.3700 0.0086 0.3030 0.0095 0.2470

00103

0.1970 0.0109 0,1530 0.0115 0.1180 0.0120 0.3880 0.0123 C.0520 0.0124 0.0200 0.0123 -n.Oolo 0.0121 -C.02S0 0.0118 -0.O4sa 0.01144 -0.0650 0.0108 -0.0730 0.0103 -0.0800 0.0097 -0.0860 0.0092 -0.0920 0.00844 -0.0980 0.0076 -0.1020 0.0067 -0.1060 0.0058 -0.1100 O0047 -0.1180 0.0037 -0.1180 ROY 0.1810 0,1682 3.1635 0.1618 0, 1598 0. 1585 0.1576

ST= 1 .1367.+0s ANTATT TURUIENT STRØMNING PA ÇUGESTOF

RESUI TATF XD/L FRA flYS1fERE(IN1N(, VXDp VkFp VXDGI

W

miti G1)Y5 TYKX TYKR (PS 0,0rD2F 0.2659 -0.4fl93 j.h14'4 1.9752 32.9733 29,6534 26,2079 0.3025 -0.1222 -1,1291 O.0025n 0.2e,76 -0.4110 1.R14'4 1.9162 33.4876 10.3381 8.2725

01147

0.0071 -2,6209 0.O06?t4 0.2708 _0,14145 1.0144 2.O'92 35.3913 6.7239 5.3448

0079

0.0354 -3.3693 0.01361 0.2756 -0.4199 1.8114% 2.0459 37.9230 5.3578 3.7533 0.0647 0.0431 -3.8794 0.022Sj 0.2817 -0.427'4 1.8j94 2.098'3 40.6752 4.7346 3.1219 0.0750 0,0375 -4,2750 0'033t1 0.2891 -0.4372 1.8j44 2.1683 43.10S 4.4429 2.8272 0.1389 0.0000 -4.6890 0.04694 0'297 fl.9496 1.814% 7,7563 44.SR34 4.3242 2.5904 0.2122 -0.0397 -5.3121 0.06250 0.306/.

0.4S1

1.8144 2.3606 '4'4.5129 4.2967 2.4163 03109 -0,0888 -6.0515 0.08028 0.3167 rJ.4140 1.8.14 2.4769 42.4151 4.3127 2.2280 o'211 -0.1379 -6.9080 0.10028 0.3260 -0.5070 1.8144 2.5981 38.1143 4.3384 2.0099 0.5426 -0.1838 -7.7997 0.12250 3,3355 -fl,534% 1.8144 2,7150 3j.8368

43476

1'729

0.6524 -0,2160 -8.6321 0.14699 3.3943 -0.5670 1.8144 2.8175 24.0725 4.3250 1.3879 0.7251 -0.2408 -9.2460 0.I'34j 3.3s84 -0.6058 1.8144 2.8970 15.7678

.0ç2

1.0'460 0.8136 -0.1950 -9.2599 0.20250 3'3850 n.6520 1.8144 2.9476 7.8124 4'0452 0.7061 v.8559 -0.1741 -9.5845 -0.0660 0.23361

0o31

751

1.8149 2.9673 0.9222 4.0117 0.3313 0.8358 -0.1527 -9.5566 -0.0844 0.26694 0.4014 fl.7659 1.8144 2.9ç73 -4.6208 3.9112 0.0039 07735 -0,1294 -9.1070 -0.0985 0.3025d

0.3851 -.R52

1.8149 2.9734 -90549 3.7507 -0.2282 0.7185 -0.1090 -8.4511 -0.0148 0.39028 0.3411 -3.9308 1.81414 2.8577 -12.998 3.5512 -3.4577 0.6399 -0.0957 -7.6386 3.38028 0.2526

l'0ì3

1.8144 2.7615 17.082 3.2537 -0.6536 0.5462 -0.0871 -6.6016 0.4225e 0.1557 -1.2082 1.8144 2.6454 -21,463 2.8419 -0.8333 09i50 -0.0769 -5.2859 0.%669'4 0.0481 1.2829 1.81%'4 2,qnB9 -25.940 2.3469 -0.9j%R 0.2856 -0,0668 -3.8841 0.51361 -0.0864 -1.?'48 1.H!44 2.3,19 -27.498 1.7527 -0.9330 0.1550 -0.0601 -2,4963 0.56253 -0.2519 -1.1l% 1.8144 2,I29 -25,976 1.3214 -0.8682 O'0428 -0.0526 -1.4712 0.61361 -0.3868 -1.0689 1.8144 1.9232 -20.290 1.0017 -0.7256 -0.026? -0.0468 -0.7793 3.66694 3i4%51j 0.96t59 j.R!'44 1.7932 -12.125 0.7437 -0.5559 -0.0929 -0.0430 -o.3655 0.72250 -3.5n62 -0.8637 1.81144 1.7730 -5.3805 0.5428 -0.4332 0'0285 -0,0416 -0.1591 0.78328 -0.5186 ..Q.7655 1,8144 1.6P2 -3,5852 0.3881 -0.9061 -0.0329 3,3397 -0.0379 0.84028 -3.5215 -0.6733 1.8144 1.6'493 -4.9859 0.2666 -0'4204 -0.0745 -0.0359 0.1021

0.9025e -0'13

-0.5832 1.81144 1.6141 -0.9834 0.1706 -fl.3773 -0.1085 -0,0320 0.2196 0.96694 -0.4945 1j.5l13 1.8144 1.6416 3.2045 0.0911 -0.3638 -0.2212 -0.0298 0.2958

IDIEL IDFEL rVSETRUST p: T[j1= 2B.? DVSETRUSTKOFF,: KnII= 0,075v,

T12

KD12 27.4 0.0735 DYSENS DRAGKOFF, CP

O.n16

REELL r)YSETRUST T KP: Toi = 27.7 102 = 26.9 REFLL DVSETRUSTKOUE.

izri

0.074' vn2 = 0.3722 O 1.8637.04 1.0i44.O0 5.10F3.-01 1.6509,+o0 i 1,673p,f04 1.8I'4,DJ 5.7OO,-01 l.5667,00 2 i.322),04 7.26R6,-01 1.4159,+00 3 i.0152.0

1.fl1,00

9.5542,-01 1.2943,00 4 1.4984,C3 i.814'i,C 1.33S4,00 1.2000,+00 R/PP yAKS 0.95 1.6509 0.90 1.5667

C.0

1.4159 0.70 1,2943 0.60 1.2000 0.50 1.1284

0.40

.0750

0.30 1.0366 0.?E1 ,030S

REL-CAy. SIGM,8 LIFT RADIUS SAFETY 0.28' 87.147 '4.d22 0.30fl 85.1 4.287 9.40n 78.16 2.S3S 0.S0 74.81 1.7114 0.300 O.60n 72.78 1.212 O.2'1 O.70n 71.3' 0.898 0.226 3.80 7l.l 3.689 0.192 o,900 72.51 0.514M 0.158

o.5»

73.31

0.90

0'l' RELATIV RADIUS

3

0.60

STRESSES DUE TO THRHST AND TORQUE FORCES

136.1

71.0

STRESSES DUE TO CENTRIFUGAL FORCES

23.2 10.3 TOTAL STRESSES 159,3 81,14 CALCULATED THICKNESS 0.00149 0.0022 THICKNESS REQUIRED Y VERITAS 0.0086 0.0040

RFSULT ING THRUST POWER CONVERTED L F F I C t E r'1 C Y F4LADE AREA RLADE AREA RATIO (AV .NIINIRER MEAN EVr.PITc4 MEAN NOPi.PIT(H

.*RFSUI TS.e.

LENGTH

THICK-I/L CAlIBER PITCH LENGTH LENGTH NESS FORE 3.033

0.052 0.0121 0.232O.0a00

0.177 0.026 0.056 0.0118 0.2124 0.0006 0.187 0.028 '.3! 0.071 0.0102 0.1432 0.0016 0.228 0.036 71 I 5.9 0.289 0.04 0.524 i .025 3 28 3.264 0.086 0.0086 0.1001 0.0019 0.098 0.0070 0.0714 0.0020 0.104 0.0059 0.0520 0.0020 0.134 0.0039 0.0371 0,0020 0.091 0.0023 0.0255 0.0018 0.075 0.0016 0.0208 0.0015 WEIGHT BLADES

GDZ BLADES IN AIR GD2 TOTAL IN AIR GD2 TOTAL IN WATER

BPNUMBEP

DELT A-NUMBER ACT. BURRIL

0.2S 0.093 0.264 0,049 0.273 0.052 0.278 0.052 0.276 0.0146 0.266 0.038 1.3 0.0 0.0 0.1 129.50 351.3 0.222

R/R BE A L r A T ALFAO STI(,VN STIGVEF O

18.1289

32,8085

0.9682

0'000

33,7767

33.7767

O

j0

I 7.0007

32.379

1.1156

1.0057

33,4946

34.5003

o '409 12'91 13

30.1394

0.900

2.4401

31.1294

33.5695

Û sOo

10.3903

27.171g

0.7337

2.3116

27.9056

30.2172

O 600 8 '6873

74.4399

a.5590

2.0151

24.9989

27.0140

0.709

7 ''4615

22.0273

0.4497

1.7421

22,4770

24.2191

6 5 37 '4

19,8483

0.3735

1.4812

2Q,2218

21.7029

o .909

S'Bl 62

17.6834

0.3575

1.2176

18.0409

19.2584

O 95

5.5120

16.156g

9.3714

1.1219

16.5274

17,6492

KT

0.190'4i

K Q

c.03o1

J

9.2889

A i .OnDo A N ALFA! i 1. MALFA

1.000ri

M2 1.C5O

*1* H

R/R Y D R O

D VN A

Nl T cP (K C D T CALF,A

A '

Cr, CFS CFT

O 280

r. 01483

0.02031

O. 0C0O O.0351'4

0.00584

0.00320

0.300

0,01 346

0.91727

0 '00000

0.03072

0.00569

0.00300

O. 400

0.00977

O 0089 9

0.00000

0.01876

0.00513

0.00232

o

0.00863

r) .O04o

0 '00000

0.01353

O,O475

0.00241

0. oo

O 09829

0.00265

0. 00000

0.01094

0.00448

0.00276

0.700

0.00798

0.00211

0.00000

0.01008

0,00430

0,00292

o 809

o. 0tj772 O .00O6

0.00000

0.00868

0.00420

0.00299

O '900

0,00756

0.00109

O .00000

0.00865

0.00421

0.00298

0.950

o Oj753

o 00131

O .00000

0.00886

0.00433

0,00290

R/R CL! CL EPS KKT KALFA KC

0.280

0.00000

0.00033

o .00oQO

0.41772

1.67646

1.34504

0.309

1, 13030

9'13030

.23576

0.38708

1.60474

1.27619

0. '400

9.316)5

9.31615

0.05935

0.26129

1.36321

1.07630

O.SCû

o 295

O 29951 0' 04517

0.18113

1.31687

1.11706

0.600

0.26110

r.261 10

0.04190

0.12472

1.34874

1,19264

O 700

0.22572

0.22572

o 04468

0.08383

1,'42367

1.30422

o 809

0.19191

0.19191

0.94523

0.05651

i.siO3i

1.47398

(J 900

0.15776

3. 15776 o .05486

0.03894

1.81598

1.85744

0.959

, 14536 O a 1 't 536

0.06094

0.03380

2.05995

2.12931

R/R ..e p R O F NACA 16 100.03 90.00 T L T

A).8

80.00 A B E L 7i.00

.*

60.00 50.00 'io'OO o.ûo 20.00

1.00

5.oO 2.50 0.00 11'95 UX -0.8 -0.5 -0.2 --.1 -0.0 0.3 -o.o -0.1 -0.3 -0.5 -0.7 -0.8 -0.8 IX 0.7 1.1 1.4 1.5 1.6 1.5 1.4 1.2 0.9 0.7 0.5 LENGTH 75.2 67.7 60.2 57.6 45.1 37.6 30.1 22.6 15.0 7.5 3.8 t. 0.0 3.90 ux -0.6 -0.9 -0.2 -2,1 -0.0 0.0 -0.0 -0.1 -0.3 -0.5 -0.6 -o.7 -0.6 TX 1.0 1.6 7,!J 2.3 2.3 2.3 2.1 1.8 1.3 1.0 LENGTH 91.4 82.3 73.1 64.0 54.8 '457 36.6 27.4 18.3 9.1 4.6 2.3 0.0 3.83 uX -0.0 -0.1 -0.0 .0 0.0 0.0 -0.0 -0.1 -0.2 -3.3 -0.3 -0.3 -0.0 TX 1.6 2.7 1.4 3.8 3.9 3.8

35

3.0 2.2 1.6 1,2 LENCITH 104.4 94.0 83.5 73'l 62.6 52,2 41.8 3i,i 20.9 10.4 5.2 2.6 0.0 3.73 uX 0.7 0.3 0.2 .1 0.0 0.0 -0.0 0.0 -0.0 0.0 0.1 0.2 0.7 TX 2.3 3.8 4.8 5.3 5,4 5.3 4.9 4.2 3.1 2.3 1.6 LENGIH 1o4'4 94.0 p3,5 71.1 62.6 52.2 41.8 31.3 20.9 10.4 5.2 2.6 0.0 fl.63 UX 1.5 0.8 0.4 8.2 0.1 0.0 0.0 0.1 0.2 0.4 0.6 ü.7 1.5 TX 2.9 '4.9 A.1 6.8 7.0 6.8 6.3 5.4 4.0 2.9 2.1 LENGTH

98.0

88.2

78.4 68.6 58.8 '49.3 39.2 29.4 19.6 9.8 4.9 2.4 0.0 n.5 ux 2.4 1.3 0.7 -p.3 0.1 0.0 0.0 0.2 0.4 0.8 1.1 1.4 2.4 TX 3.6 6.0 7.5 8.3 8.6 8.4 7.7 6.7 4.9 3.6 2.6 LENGTH 85.8 77.2 bB.6 62.1 51.5 '42.9

343

25,7 17.2 8.6 '4.3 2,1 0.0 0.40 (jX 3.5 1.9 1.1 .4 0.1 0,9 0.1 0.3 .7 1.3 1.8 2.2 3.5 TX 4.3 7,1 R.9 9.9 10.2 9.9 9.2 7.9 5.9 4.3 3.1 LENGTH 71.2 64.1 57.0 '42.7 35,6 28.5 21.9 14.2 7.1 3.6 1.8 0.0 0.30 UX 5.3 3.0 1.6 '.1 0.2 3.0 0.1 o.5 1.1 2.2 3.0 3.6 5.3 TX 4.9 8.3 12.4 11.5 1 .8 11.5 10.7 9.2 6.8 '4.9 3.6 LENGTH 55.6 50.0

'4q5

3.9

33.9 27.8 22.2 16.7 11.1 5.6 2.8 1.4 0.0 n.28 ux 3.5 1. .7 0.2 0.0 0.1 fl,6 1.4 2.6 3.5 4.2 6.j TX 5.1 8.5 1".6 11,8 12.1 11.8 10.9 9.4 7.0 5.1 3.7 LENGTH 52.2 4/.0 '41.9 36.5 31.3 26.1 20.9 15.7 10.4 5.2 2.6 1.3 0.0

*.(AV.SAFCIY

PRESURE SIDE

DISTRIBUTI0J...

RADIUS

0.950

fl.9çj

0.8n0

0.700

0.60cj

0.500

0.430

0,300

0.280

2.5

110.14

109.16 107.58

105.55

103.47

101.36

98.92

94.92

92.70

t09'5

108.q 106.70 109.61 102.51

100.39

97.98

93.50

91.76

10.0

108.62

107.'18

10./S

l03e7

IQj.3'4

99,22

96,83

92.37

90.61

20.0

108.06

10e,î7

104.99 102.71

100.56

98.'43

96.06

91,60

89,83

30.0

107.71 106.'47 109.47

102.19

100.02

97,88

95.50

91.03

89.29

40.0

107.15

105.u6

103.76

101.92

99.23

97.08

94.72

90.2'4

88,99

50.0

106.3

105'i1

103.'e6

101.09

98,88

96,72

94,5

89.86

88.04

60.0

106.71

105.36

10i.i6

100.75

98.53

96.35

93,98

89.47

87.63

70.0

107,33

106.03

103.93

101.57

9936

97.19

94.80

90.27

88.94

80.0

109.18

108.07 106.77

104,10

101.95

99,79

97.34

92,78

90,99

90.0

10.27

105.82

105.10

109.07

102,97 101.81

100.96

98.05

97.18

95.

108.06

138.20

IOR.41

108.19

107.52

106.71

105.59

103,55

102,99 SUCTION STnE

2.5

77.27

76.95

76.37

77.10

78.78

8,94

84.22

90.61

92.70

5.0

7,52

76.12

75,443

76,39

77.67

79.82

83,12

89.63

91,76

10.0

75.63

75.12

74.74

7'4,76

76.39

78,46

81.79

88.42

90.61

20.0

75.03

74.'6

73,90

75,45

77,54

80.89

87.61

89.83

30.0

74.52

73'90

12.1

73.20

74,72

76.81

80,18

84,99

89,24

40.3

73,92

73.23

72.3

72.34

73.82

75.d9

79.27

86,16

88.94

53.0

73.56

72.83

7i.8

71.85

73.32

75.38

78.78

85.73

88.04

60.0

73.19

72.94

1'12

71.36

72.82

79,88

78.29

85.33

87.63

73.0

73.73

73.0'4 71.c45

72.17

73.68

75,76

79,18

86,14

88.94

80.0

7c.61

75.15

74.3

7'4.91

76.54

78,71

82.09

88,79

90.99

90,0

8,73

F8.66

H.ç5

89,07

9o.O

91.25

92.97

96,15

97.18

95.

97.22

97.62

98.78

99,04

99.77

100.8

101,25

102.51 102.94

.*.PRESURE PRESUPE STDE

r)ISTRT UTj0N.,.

RADIUS

0.95c

2.5

Q,95

0.95

10.0 20,0

0.96

30.

0.96

0.

0.95 0.95 0.96

0.6

0.96

O 80

o .95

0.95 0.96

3,97

3.97

0. 7o0

0.95

0.96 0,97

0.98 0.98

0.600

0.96

0.97

0.98

o.99

1.00

0.500

0.98

0.99

i .01

1.03

1,04

0,400

i .03

i .0 1.08

1.10

1.12

0.300

i.2'4

1.28

1.33

1,36 1.38

50.

0.97

6Q.

.97

70.a

0.9ò

80,0

90.0

0.97

95.

0.96

3.97

0.97

0.97

0.7

0.96

0'97 0.96

0.97

o

0.98

3.97

0.96

0.6

0.9'4

0.99

0.99 0.99 0.99 0,96

0.96

0.93

1 .01 1 .01 1 .02 1.01

_0.98

.96

0.91

1 .05

1.06

1.06

i ris

1.00

0.97

0.89

1,14

1.16

1.16

1.13

1.07

_{0.99 0.86}

1.42

1.'43

1.45

1.'42 1.31

1.08

0.85

SUCTION STDE

2.5

1.11 _5, O

1.12

13.0

1.12

23.

1.12

30 O 1 12 4O .0

1.13

SO

1.13

60.0

1.13

7010

1.13

1.12

9010

1.06

1.13

1.13

I 14 1' 1'4

i '14

1'lS

1.15

i

is

i 15 1'4 i

.16

1,16

1.17

1. l3

1.18

1.19

i. i9

1.20

1 .20 I 19

1.18

i .08

1.21 i 22

1.23

1.23

1.24

1.25

1.25

1.26

1.25

1.23

1q10 I .26 1.27

_1.29

1.30

1.31 1.32 1 132

1.33

1.32 1.28

_1.12

1.33

1 35 1.37 i 38

i .40

i .41

1 '42

1.43 1,42

1.37 1.15

1.41 1 .'4'4

1.47

1 '9

1.51

i.5'4

1.55 1.56

1.54

1.'46

1.18

i '40

.'4'4 1 .50 1 .53 1 .56 1 .59 1,61

1.63

1 .59 1 '48

1.16

1.31 1.31 1 1.01 1 .00

0.99

0.97

0.89