The effect of a nozzle on steering characteristics

r2

I tIEl

ARCHIEF

Lab,

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Scheepsbouwkund

Technische Hogeschaol

Deift

DRUNEN - HOLLAND

oth.ek van-

ciElY

Onderafdelu e.psbouwkunde

Jeehnische Hogescho.

,

'eift

DOCUMENTATIE

_E

_f.

DATUM:

THE EFFECT OF A NOZZLE ON STEERING CHARACTERISTICS

jointly by

L. A. van Gunsteren Lips N.V. Propeller Works

Drunen - HoI land

and

F. F. van Gunsteren

Sea Transport Engineering N.V. Amsterdam - Holland

Paper prepared for the

Second International Tug Conference, London, October 25 - 28, 1971.

Table of contents

PaQe

Abstract 5

Introduction

Mathematical model of turning 6

Turning capacity 8

Response time 9

Rudder-nozzle force coefficients 10

Prediction of lift forces on rudder and nozzle 11

Numerical results 18

Ful 1-scale tests 19

Turning circle tests 19

Discussion 20 Conclusions 20 Acknowledgement 21 Nomenclature 21 Symbols 21 Subscripts 25 References 25

List of tables

Table 1 - Down-wash coefficients and b1

Table 2 - Comparison of calculations for isolated nozzles and rudders with known results

Table 3 - Particulars of the tugs

Table L1 - Experimental results

4

List of figures

Figure 1 - Definition of symbols

Figure 2 - Turning rate as a function of time

Figure 3 - Factors for calculation of axial velocity U

Figure 14

-

_{System of coordinates}

Figure 5 - Influence of nozzle on rudder lift characteristics

Figure 6 - Influence of rudder on nozzle lift characteristics

Figure 7 - Lift forces on rudder and nozzle for = 00

Figure 8 - Lift forces on rudder and nozzle for f 50

Figure 9 - Lift forces on rudder and nozzle for = 100

Figure 10 - Stern arrangement of Tug i (picture)

Figure ii - Stern arrangement of Tug 2 (picture)

THE EFFECT OF A NOZZLE ON STEERING CHARACTERISTICS

Abs t rac t

A method is presented for predicting how the steering characteristics of a ship

are affected by fitting a fixed nozzle. The presence of a nozzle upstream appears to have a significant effect on the rudder forces. Full-scale manoeuvring trials carried out with two twin-screw tugs, one with open propellers and the other equipped with nozzles, confirm the predicted trends. It is concluded that

propeller, nozzle and rudder should be designed in an integrated way to ensure

that an optimum solution is obtained with regard to both propulsive and steering qualities.

Introduction

A fixed nozzle not only affects the propulsive characteristics of _{a ship but also}

its steering properties. Accordingly, when a nozzle is considered for its well-known propulsive features - for instance, an increase in thrust at low speeds

-it is imperative to pay careful attention to the effect this will have on the rnanoeuvring characteristics of the ship. The purpose of this paper is to provide a means of predicting how characteristics will be affected by fitting a nozzle,

assuming the behaviour without a nozzle to be already known. Propeller, nozzle and rudder can then be designed in an integrated manner, taking account of both

propulsive and manoeuvring requirements.

Although the theory can also be used to predict transverse forces on steering nozzles with or without stabilizers, we shall confine ourselves here to fixed

nozzles. From a manoeuvring point of view, the fitting of a nozzle is comparable

to increasing the lateral area of skegs and the like, this having an adverse effect

on the turning diameter and a favourable influence on response time and stability

6

This can be - and generally should be - compensated for by decreasing the latera] area of the afterbody, particularly directly in front of the nozzle. A second effect of the nozzle ¡s that it influences the flow at nearby lifting surfaces, especially the flow and consequently the forces on the rudder downstream.

Since ¡t ¡s not our intention to predict the actual manoeuvring characteri;tics,

but only to determine the differences in steering characteristics due to the

presence of a nozzle, a relatively simple mathematical model can be used, ef. [i]. The motion in the horizontal plane is described by the 1 ¡nearized equation of

motion regarding moments with respect to the centre of gravity (Nomoto's equation). The moments are split up into:

Moments due to transverse forces on the propeller-nozzle-rudder

con f ¡ gura t ion.

All other moments.

Available theory for the prediction of lift forces on nozzles and rudders is

limited to an isolated nozzle (without propeller, ref. [21 [3], or with propeller, ref. [4]) or rudder. Since it may be expected that coupling effects between the nozzle and the rudder are significant, a linearized theory has been developed

which allows for these effects. The theory is applied in a computer program for

predicting lift forces on a nozzle-rudder configuration at arbitrary angles of attack. The results of calculations in which the coupling terms are disregarded, are in agreement with known results of isolated nozzles, ref. [3], and rudders, ref. [5]. Full-scale manoeuvring trials with two twin-screw tugs, one with open propellers and the other equipped with nozzles, illustrate that the procedures

given are indeed adequate for analysing how steering characteristics are afected

by fitting a nozzle.

Mathematical model of turning

In order to predict how turning qualities are affected by fitting a nozzle, we may describe the manoeuvre by the linearized equation of moments about the centre of gravity in the horizontal plane, ref. [i]:

whe re:

= mass moment of inertia, including hydrodynamic effects

r = rate of turning

N6 . 6 = moment, due to the lateral forces on rudders and nozzles Nr . r = all other moments in a steady turn

As explained in the next section the rudder-nozzle moment can be written:

N6 . 6 = l

R - C2 6

where:

= angle of attack of the rudder

6 = angle of attack of the nozzle 6 = rudder angle

According to figure 1, we have:

6

= + 5

The angle of attack at the rudder or rudder-nozzle configuration is proportional to the rate of turn:

6

= C3r

The coefficient C3 depends on the distance, X, from the rudder-nozzle configuration

to the pivotal point at which the centrel me of the ship is perpendicular to the radius of turn, and on the axial velocity U at the rudder-nozzle configuration,

including the effect of the propeller: X

C3 =

57.3

u

For the present purpose, X can be approximated by:

X =

0.75

L ₍₆₎

where:

8

When the rudder is moved from zero to a constant angle the solution of

equation (8) gives the turning response as a function of time (see figure 2):

r = K (i -

et/T)

The turning motion is determined by two quantities:

K, representing the turning capacity

i.e. the rate of steady turn per degree of rudder angle,

which is reciproca] to the turning diameter.

T, representing the response time

i.e. the time taken to reach 63°/e of the steady rate to turn.

A method will now be given for calculating these two turning qualities, ''turning

capacity'' and ''response time'', for any rudder-nozzle configuration considered, assuming that K or T is known for an existing rudder-nozzle or non-shrouded

configuration ona similar hull.

Turning capacity

If the turning capacity of a certain rudder-nozzle configuration, K, is known,

the turning capacity of a new rudder-nozzle configuration, , can be calculated

by applying equation (io) to both configurations.

Substitution of equations (3), () and (2) into equation

Cl (1) gives: r ₍₇₎ i + (C + C ) C r 1 2 3 N + r (C 1 + C ) C 2 ₃

This is equivalent toNomoto's equation:

T + r = K ₍₈₎ where: T - ₍₉₎ Nr + (C1 + C2) C3 Cl K (io) N _{+ (C + C2) C3} r

For the new configuration we have:

Cl

K=

Nr + (C1 + C2) 03

The total less-rudder-nozzle-moment coefficient, N, and 03 may be assumed to be

¡ndependent of the rudder-nozzle configurations, at least for a fishing vessel hull design, ref. [5]:

=N

r r

C3 = C3

If necessary, corrections can be introduced for differences ¡n hull design. The new turning capacity is obtained by substitution of equations (io) and (12)

into equation (13):

Cl

K=K[

]

C1 + K {C3 (C1 + C2) _{- 03 (Ci + C2)}}

The coefficients C1 and C2 can be calculated for both configurations with the

method given in the next section.

Response time

The response time can be calculated for a new rudder-nozzle configuration from one of the two known turning qualities, K or T, of a certain configuration by

substitutionof equation (flor (8)

intoequation (13),whichgives:

If T is known: =

T[

I I + T (C3 (C1 + 2) - 03 (C1 + 02)) If K is known: T 1K C1 + K (03 (cl + 02) - C3 (Cl + C2) .(12) (17) (13)

(1)

For the quai itative prediction of the response time, the mass moment of inertia is assumed to be independent of the rudder-nozzle configuration.

Rudder-nozzle force coefficients (C1, C2) The rudder-nozzle moment can be written as:

N 6 = CL U2 (D 1 + b c) . Xf .(18) in which: CLTQT = LR = LD = = D = b = c = Xf = U = p =

The formula for CLTOT can be written in the form:

CL = a0 + a1f) - b0 (S - blaR) (20) w h ere: d CLR bc a0 d bc + 1D + L0 .(19) U2 (1 D + b c)

lift force on the rudder (positive in turning circle) lift force on the nozzle (negative in turning circle) length of nozzle

diameter of nozzle height of rudder

chord length of rudder

distance from the rudder-nozzle force to the centre

of gravity of the ship

velocity at rudder-nozzle configuration including effect of prope]ler (see next section)

In the theoretical prediction of the transverse forces on propeller, nozzle and rudder, the fol lowing assumptions are made (see also ref. [7]

- The fluid is incompressible and non-viscid

- Body forces are neglected

- The flow is steady

- Thickness effects may be neglected

For our purpose we may neglect the propeller-nozzle interaction and use the superposition model. Then the lift of the duct is over-estimated, but this is compensated for by neglecting the transverse force on the propeller, thus yielding an approximately correct total lift (see reference [Li] ).

b0 d CLD (22)

d3

bc+1D

dC

dC

Here d LR (b/c) and d LD

(l/D) are the lift gradients for the isolated rudder and nozzle respectively, and can be obtained from references

_[51

and [3].

The coefficients a1 and b1 represent down-wash terms due to the interaction between nozzle and rudder. The influence of the downstream lifting surface on the upstream lifting surface is small, but the reverse is not the case.

Consequently, b1 may be neglected in approximate calculations. The coefficient a1

depends on the aspect ratio of the duct and the distance to the rudder, but the

influence of both these parameters is small (see table i). According to equations

(2), (18) and (20) the coefficients C1 and C2 are:

C1 = Xf U2 (ID + bc) (a0 + b0b1) . NR (23) C2 = Xf . p U2 (1D + bc) (b0 - a3a1) . ND (2h) wflere: NR = number of rudders N3 = number of nozzles

12

This reduces the problem to the prediction of the lift forces on a nozzle without

propeller at an angle of attack

(= -)

due to the yaw and drift of the ship, and a rudder at an angle of attack which represents the rudder angle corrected

for the effect of yaw and drift. The main stream velocity U is taken to be:

U V (1 - w) . (i + F (CT) s T T speed of ship wake fraction

correction on the main stream al lowing for the axial velocity due to the propeller

(25)

The correction on the main streani extends the validity of the method to ''moderate' propeller loadings. In the case of a nozzle, the velocity far downstream is taken to be representative and according to the axial momentum theory described in ref. [8J this yields:

C F (CT) T - 1 . ..(26) w h e r e: CT = thrust coefficient; T1 U1 - D

= total thrust of propeller and nozzle

= ratio of propeller thrust to total thrust

As far as the rudder is concerned this is a good approximation, but Jit1 regard to the nozzle approximately half this value would be correct.

An estimate of the errors introduced by al lowing for propel 1er loading ¡n this

rather crude way can therefore be made by recalculating with F half as large as in equation (26). lt is only if little difference is found that the results are trustworthy. In the case of an open propeller we have (see ref.

[91

)

CT where: V = w = F (CT) =

where:

x = downstream distance to propeller plane

F = factor according to figure 3

The lift forces on nozzle and rudder are calculated with the quarter point method

in which the lifting surface ¡s replaced by a single, concentrated bound vortex at the chord point, and the linearized boundary condition ¡s satisfied at the

chord point. _{In order to estimate the accuracy of the method, we assume in our} computer program that the lifting surface representing the rudder is split up into

an arbitrary number of vertical strips with a concentrated bound vortex at the chord point, and pivotal points in which the boundary condition is satisfied at

the chord point of each strip. For the sake of simplicity the derivation is

given for the case of only one concentrated bound vortex on the rudder. From the

linearized theory of ducts ¡n oblique flow, ref. [31, is known that the lift coefficient per degree of angle of attack on the duct depends only on the chord

diameter ratio. The camber and diffuser angle have no effect on the net lift

force. The problem can now be formulated as fol lows. G i ven:

D = diameter of cylinder representing the duct;

D=2R

= chord length of the duct

b = span of the rudder

c = chord of the rudder

a _{distance from trailing edge of duct to leading edge of rudder}

= angle of attack of duct

= angle of attack of rudder

Required:

CLD lift coefficient of duct;

lift on duct CLD -U2 I R = density of fluid x/R F (CT) (-i + _{+ CT)} (l + F (x/R) . _{. .(27)}

+ (x/RY

CLR = lift coefficient of rudder; ¡ ft on rudder CLR

-u2 c

b

In order to find these, the strength of the bound vortex distributions must be

calculated. The required lift coefficients then follow from the Kutta-Joukowsky

law. The rectangular coordinates x, y, z, and the cylindrical coordinates x, r, e, are introduced as indicated in figure . The bound circulation of the duct

is expressed in a Fourier series. The series contains only cosine terms having angles 0, 3 0, 5 e . . . , as can be derived from symmetry considerations:

FD (e) = 2 1 u Am cos [ (2m - i) e _]

m=l

Similarly, the circulation of the rudder can be written as:

co FR (i4i) = 2 c U B sin L (2m - i) ] m m=l whe re: = Glauert coordinate;

y=

(30)

Free rectilinear vortices are shed downstream from the duct having a strength per

unit arc length according to the law of continuity of vortex strength:

i d F0 (e)

1D (e) = - R d e

which gives with equation (28):

1D (e) = - 2 1 U 1R (y) = d FR

dy

(2m - i) A sin [ (2m - i) e ] m m=1 - . (31) (32)

Similarly, the free rectilinear vortices shed downstream from the rudder have a strength per unit of span of:

Vr (1/2, R, e) 1 (_....L\2 F1) = 2R1 u 'R = Rudder self-induced: y (d c/2, y, Q) z F u o., c2b R

8ir

o, '71

-

c U (2m - 1) B cos [(2m - 1)

I

b sint m=1

To apply the boundary condition at the duct, we have to consider the radial velocity

V (1/2, R, e) at the chord point of the duct induced by the entire vortex system.

With regard to the rudder the upwash y (d + c/2, y, O) must be considered. Because

of symmetry, only pivotal points in the first quadrant of the nozzle and in the

upper half of the rudder need be taken into account. After some manipulation,

application of the Biot and Savart law gives the fo] lowing resul ts. Where necessary,

coordinates referring to vortex elements are distinguished from pivotal point

coordinates by the subscript y. The symbol c indicates that the integral has to be taken in the sense of the Cauchy principal value.

Duct self-induced: 2R - 2 cos (o - e ) y singularity at: e ₌ 2n cos (e - e ) L _{A cos [(2m - i)} e I d e V m=1 m V V + 2 - 2 cos (e - e L 2R y Vr (1/2, R, o) sin (O - e ) (2m - i) A sin [(2m - i) e ]

= -

y m=l m y U [i - cos (e - e ) oJy

de

y sin B sin [(2m - i) ] m=1 m [(c/2)2

_{+ y2 +}

₍

cos)2

₊ y b cosi»]3/2

(3h)

(36)

dp

(37)

16 y (d + c/2, y, o) z C 1R y_z (d + c/2, y, o) F u u u D

nduced by rudder on duct:

[(c/2)2

+ y2 + (-

cosrp)2 + y b cosrjj

singularity at: y = - cost

Induced by duct on rudder:

iR (d + c/2) vr (1/2, R, e) _{b (d - 1/2) c} 1 u b

m1

(2m - i) B cos [(2m - 1)ì1

(y + --

cosr.)

[2

y _{+ (-- cos)2 + y b cosrp]}b c/2

m1

Am cas [(2m - i) O

I

cose [(d + c/2)2

+ y2 +

R0 - 2yR

_sinS]32

-TT/2 11/2

ml

(2m - i) Amsin [(2m -

i)eJ

(y -

R sine)

(d + c/2)

1+

[(d + c/2)2 + y + R2 - 2 _{y R sine]} o -Tr lT r r (1/2, R, e) c b (2m - i)

B cas [(2m - i)] cos cose

m=l m

=

[r2

+ ( cos)2 + R b cosy

smC]

(d - 1/2) [(d - 1/2)2 ₊

(

cos)2 + R

b cos

sin]

m1

Bmsin [(2m -

i)e]

sin cosE

[(d - 1/2)2+ R2+ (

cos)2+

R b cossin&]312 de

(Lo)

(2)

(yo +

R2 - 2 y R

sine)

ofTt y (d + c/2, y, O) u lT -dq

(38)

dO (3 d

(1)

We truncate the series at N terms:

m =

N pivotal points have to be selected in the first quadrant of the nozzle and

N pivotal points on the upper half of the rudder, for instance at:

Equations (1+5) and (1+6) form a set of 2 N 1 mear equations from which the 2 N unknown coefficients Am and B (m = i, 2 . . . N) can be obtained. Because of the singularity at s

= in equation (36), a small interval around the singularity

should be excluded from the integration: e-AO rO+A0 2n

dO

+

dO

+

dO

V y V

o e-AO e+Ae

lt can be shown that the middle term does not contribute to the induced velocity, provided AO is taken small enough (for instance Ae = 0.05 radians). Equation (38) should be evaluated in a simi lar way. Once the coefficients Am and Bm have been found, the lift on nozzle and rudder fol low from the Kutta-Joukowsky law:

'2Tr

lift on duct: _LD = 2 p 1 U2R

m=i

Am cos [(2m - i)o] . cosO dO (47)

1,

i

2 .... N for the duct . . . (L3)

= _{m =}

m

-1

=

m = i, 2 .... N for the rudder . . . (11+)

-y_m = ₂ cos _m

The boundary conditions give:

y (1/2, R, 0m) r duct: on the 0) FD, 'D' FR, _{'R =} 5'D'

FR, 1R

D cosüm =

-(1+5) (1+6) u y (d c/2, z on the rudder: 'm' u

18

i ft on rudder: _{LR = p} c U2 b B sin [(2m - i)p] . sin d

m

m= i

Only the first term of the Fourier series contributes to the lift, so that the

lift coefficients become:

LD CLD = = Lt Tf A1 u2 i R L c R = B LR

-u;

bc

Numerical results

(8)

A computer program based on the theory of the preceeding section has been developed

to predict the lift forces on nozzle and rudder. The input consists of the

para-meters uR, D, l/R, c/R, b/R, a/R, the number of Fourier terms and the number of concentrated bound vortices on the rudder. The output consists of the Fourier

coefficients Am and Bm and the lift coefficients CLR and Results of

calculations for isolated nozzles are compared with known results, ref. [3], [5], in table 2 which also shows the effect of the number of lifting lines on the rudder. lt can be concluded that the agreement is good and that the number of li ftig i ines

on the rudder has little influence. Two lifting lines have therefore been taken

throughout all the calculations.

Results of calcuations pertaining to the configurations of the full-scale tests described in the next section are given in figures 5 and 6. The interference effects appear to be significant. The response time is characterized by the lift

gradients at zero yaw given in figure 7. lt can be seen that the presence of the nozzle reduces the lift force of the rudder to some extent, but that the total

lateral force ¡s increased, thus yielding an improved response time. An impression

of the lateral forces on nozzle and rudder during turning can be obtained from

figures 8 and _9, in which the lift gradient of the isolated rudder has also been

Ful I-scale tests

In order to verify our method for the prediction of turning qualities for various rudder-nozzle configurations, full-scale manoeuvring experiments were conducted

with two twin-screw tugs, one with open propellers and the other equipped with

nozzles (see figures 10 and Ji).

The particulars of the tugs are given in table 3.

Turning circle tests

After putting the rudder in a certain posi tion, the fol lowing quantities were measured on board during steady turning at given time intervals:

- course angle i .e. direction of the ship with respect to an axis of reference.

- distance from the ship to a buoy laid before the experiment; this distance was measured with a coincidental distance meter and by radar.

- heading relative to the buoy.

By plotting the ship's position as a function of time on polar paper, the rate of turn per degree of rudder angle, i.e. the parameter K, and the drift angle-turning rate ratio, C3, can be obtained. The response time T was graphically determined by plotting the course angle as a function of time.

Integration of equation (ii) gives the yaw angle :

= K (t - T) + K T e

After some time, say:

t >

3T

we have: -t/T K _{(t - T)} >> K 60 T e and: K _{(t - T)} t /T (52)

20

The graphical determination of T is indicated in figure 12.

lt can be concluded that K T and - provided the measurements are sufficiently accurate - even C3 can be obtained by simple turning circle tests.

The results of the turning experiments are given in table

Discussion

The turning qual i ties of tug 2, equipped with nozzles, calculated according to the present procedure from the test results for tug i with op'en propellers, are

compared with the full-scale test results for tug 2.

The influence of the ship's size is allowed for by using the non-dimensional coefficients: K' = K ( C 2 Ci C'

-LV

s C2

LV

s L V s Vs C3 = C3 (

-r-The results are given in table 5.

Although the effect of a nozzle on the turning quality indices K and T is

over-estimated by the calculations, ¡t can be concluded that the full-scale manoeuvring trials confirm the predicted trends i.e. a considerable decrease in turnincj

capacity K and response time T.

Apparently, the nozzles have the same effect on turning capacity as propellers and

skegs: ''They tend to resist any turning of the stern, having a decided òbjection to being moved sideways'', ref. [io].

Conci us ions

T' =T

A method is given for predicting how the steering characteristics of a

ship are affected by fitting a fixed nozzle.

3. Propeller, nozzle and rudder should be designed ¡n an integrated

way to ensure that an optimum solution is obtained regarding both

propulsive and steering qualities.

Acknowledgemen t

The authors wish to express their appreciation to the owner of the two tugs: Terminales C.A., Maracaibo, Venezuela, to the shipbuilders: D.W. Kremer Soim, Elmshorn, Germany, and especially to: Jonker & Stans, Hendrik Ido Ambacht, Holland for thei r cooperation in the manoeuvring trials.

Nomenc I a tu re

Symbols

A _{Fourier coefficients for the nozzle circulation}

a _{distance from trail ing edge of nozzle to leading edge of rudder;}

a

= d-l-c

a0 lift gradient for the isolated rudder;

LR a0 =

u2 (b c + I u)

a1 _{correct ion term due to down-wash of nozzle}

B _{Fourier coefficients for the rudder circulation}

b span of rudder

b0 I ift gradient for the isolated nozzle;

LD b3

22 CLD -CLR -CT LD u2 1 R LR p u2 b c thrus t coefficient Thrust i 2 -,2 Ç) L) propeller diameter nozzle diameter

d distance between quarter points of nozzle and rudder

F _(CT) factor for mean induced velocity

F1 (x/R) factor relating mean induced velocity in propeller si ipstream to velocity induced at the centreline by actuator disc

horizontal mass moment of inertia of the ship, including hydrodynamic effects

K turning capacity;

K' - K

(L.)

vs

b correction term due to down-wash of rudder

C chord length of rudder

C1, C2, C3 coefficients of turning

CLD lift coefficient of nozzle;

CLR lift coefficient of rudder;

CT

L length of ship

length of nozzle

LD lift force on nozzle

LR 1 i ft force on rudder

m Fourier term index

N number of Fourier terms

ND number of nozzles

NR number of rudders

Nr total less-rudder-nozzle-moment coefficient

N rudder coefficient R propeller radius nozzle radius r turning rate radial coordinate T response time; TI = vs thrus t t i me

axial velocity at the rudder-nozzle configuration

24

speed of ship

wake fraction

x downstream distance to propeller plane

Xf distance from the rudder-nozzle force to the centre of

gravity of the ship

X distance between pivotal point and rudder-nozzle configuration

x, y, z Cartesian coordinates

x, r, O cylindrical coordinates

aD angle of attack of the nozzle

angle of attack of the rudder

- D ( is positive in steady turning)

ci rcu la t ion

'Y vortex strength per unit length

rudder angle

e angu lar coord i na te

p dens i ty

-t ratio of propeller thrust to total thrust

G lauert coord mate;

b y 2 cos. yaw angle V s w

Subscripts

D nozzle

R rudder

y vortex element

due to bound vortices of duct

due to bound vortices of rudder

1D due to free vortices of duct

due to free vortices of rudder

Barred symbols indicate new configuration.

Symbols with apostrophe indicate non-dimensional coefficients. A dot on a symbol indicates time derivative.

R e f eren ces

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Technological University, Delft, 1967.

[21

Weissinger, J.: ''Zur Aerodynamik des Ringflügels, I. Die Druckverteilung

dünner, fast drehsymmetrischer Flügel in Unterschallstr6mung'',

Deutsche Versuchsanstalt für Luftfahrt, E.V. Bericht Nr. 2, Mülheim, September 1955.

[31

Morgan, \'.B. and Caster, E.B.: 'Prediction of the Aerodynamic

Characteristics of Annular Airfoils'', David Taylor Model Basin, Report No. 1830, January 1965.

Greenberg, M.D., 0rday, D.E. and Lo, C.F.: ''A Three-Dimensional Theory

for the Ducted Propel 1er at Angle of Attack'', THERM Inc. TAR-TR 6509, December 1965.

26

[51

Wicker, L.F. and Fehlner, L.F.: ''Free Stream Characteristics of a Family of Low Aspect Ratio Control Surfaces'', David Taylor Model Basin, Report

933,

May 1958.

[61

Bardarson, R.R., Wagner Smitt, L. and Chislett, M.S.: ''The Effect of Rudder Configuration on Turning Ability of Trawler Forms. Model and Full-Scale Tests with Special Reference to a

Conversion to Purse-Seiners", Transactions of the Royal Institution of Naval Architects, Vol. III, 1969, _p.

_283-310.

[71

Gordon, S.J. and Tarpgaard, P.T. : ''Utilization of Propeller Shrouds as Steering Devices'', Marine Technology, vol. 5, no. _3,

July 1968.

[8]

van Manen, J.D. and Oosterveld, M.W.C.: "Analysis of Ducted Propeller

Design'', Transactions of the Society of Naval Architects and Marine Engineers, vol. 71., _1966.

[91

van Gunsteren, L.A.: ''Eine Analyse des Einflusses der DickenverteiÌung von Fl]gelschnitten auf Kavitationseigenschaften'',

Schiffs-technik, Band 18, Heft

90, 1971.

[io]

Watts, P.: ''The Steering Qual ities of the Yashima'', Transactions of the Institution of Naval Architects, vol. f40, _1898.

TABLE i Down-wash coefficients a1 and b1

TABLE 2 Comparison of calculations for isolated nozzles and rudders with known results

Rudder aspect ratio

b/c = 1.52

b/D = 1.00

Nozzle aspect ratio

l/D a1 b1 .14Q _.73 .0945 2d/D = _{.1183 .50 .80 .0935} .60 .86 .0905 2d/D 1.00 _{.50 .85 .020} l/D = 0.8 reference [31 present method (3 Fourier terms) 0.1215 0.1216

b/c = 1.

_CLR/R°

_{number of lifting lines}

reference [5] _0.025

-present method 0.0251k i

(3 Fourier terms) _0.02551 2

28

TABLE 3 Particulars of the tugs

TUG 1 TUG 2 Length o.a. [ml

37.70

33.65

Length b.p. [ml

35.00

30.25

Breadth mid. _[m]

_9.15

_8.90

Oraught, mean [ml

L00

3.39

Power [HP]

2 x 1600

2 x 11400 Propeller diameter [ml

2.600

2.135

Number of blades '4 Pitch ratio .82

.975

Blade area ratio _.55

Propeller rotational speed [r.p.m.]

_237.5

260

during tests _{150 170}

Rudder area (i rudder) [m2] 14.32

2.98

Rudder aspect ratio 565

1.52

Number of rudders 2 2

Nozzle diameter [ml

2.28

TABLE 14 Experimental results

TABLE 5 Comparison of measured and calculated turning qualities

Tug Run Date Rudder

angle

Speed Rate of turn K

[degrees] [knots] [degrees/sec]

1 114/7/71 5 SB

8.60

0.78

0.155

2

-

lo

SB

8.35

1.55

2 ₃

2+/9/71

_{7 BB}

_8.8

_0.938

2 14 -

1.5

_SB

9.0

0.650

0.138

2 ₅

_-

_7.5

_SB

_8.7

_0.915

2 6

-

12.5

BB 8.14 1.81+0 TUG 1 TUG 2 K' T'

measured calculated measured

1.17

2.08

0.68

1.32

0.814

1.62

30

T

r = K0

Figure 1 Definition of symbols

- T)

T T

o

Figure 2 Turning rate as a function of time

TIME -..-i00% K

n

I.0

.8

Figure3

Factors for calculation of axial velocity U

Figure 14 _{System of coordinates}

a '/2 (1 F1(2/p) Vl+(x/R)2) x/p 2.0 1,5 .0 .5 0 - Io -1.5 -2.0 x/R

32

Figure 5 ftfìuence of nozzle on rudder lift characteristics

IO 15 R IO IS = <D degrees CLR -ce

-Figure 6 Influence of rudder on nozzle lift characteristics

Lp Y2p U2 bo Lg u2(bc ID) RUDDER - NOZZLE CONFIGURATION: b/c = .52 2d/D _.1183 RESULTS. DOWN-WASH TERM DUE TO NOZZLE o = .860 ISOLATED RUDDER LIFT GRADIENT 00=0180 CLD- LD '/2pu2 l/2D LD Y2eu2(bc LD) RUDDER - NOZZLE CONFIGURATION L/D = .6 b/c 1.52 2d/D = .1183 RESULTS. DOWN-WASH TERM DUE TO RUDDER r b .0905 ISOLATED NOZZLE LIFT GRADIENT : b0 .0385

5

Io

-.1

-.2

Figure 8 Lift forces on rudder and nozzle for 50

lo

Figure 7 Lift forces on rudder and nozzle for d = 00

NOZZLE 1 /3 00 CONFIGURATION ID .6 6/ _{= .52} 2d/D .1183 IS g dogro 20 O IS o /3=50 u _.4 CONFIGURATION: o I-o b/ = .6 1,52 -J Q- .L183 .3 .2

34

o

0 5 Io

O(p-drs

N0Z2L

-.4

Figure 9

Lift forces on rudder and nozzle for

3 =

loo

o 4

/3io

Ui Q .3 CONFIGURATION o U- .6 n b/c L52 Q- 2d/0 .1193 .2

Figure 10 Stern arrangement of tug 1 (picture)

36