An investigation into the difference between nominal and effective wakes for two twin-screw ships

TECHNISCHE UNIVERSITEIT Laboratorium 'mor Scheepshydromechanlca frrchief Mekelweg 2, 2628 CD De!ft TeL: 015 786873 - Fax: 015-' 781838

MONOGRAPH PUBLISHED BY THE NETHERLANDS MARITIME INSTITUTE

i,

q

An investigation into the

_difference

between nominal and

_effective

wakes for two twin-screw ships

M. Hoekstra

M22

An investigation into the

_difference

between nominal and

_effective

wakes for two twin-screw ships

CONTENTS

page

Summary 3

i Introduction 3

2 Application of the diffuser test to twin-screw ships 4

3 Experiments 4

4 Results and discussion 6

5 Conclusions 11

References 11

Appendix 12

AN INVESTIGATION INTO THE DIFFERENCE

BETWEEN NOMINAL AND EFFECTIVE WAKES FOR TWO TWIN-SCREW SHIPS

by

M. HOEKSTRA

Netherlands Ship Model Basin, Wageningen

Swnmary

With the aid of the diffuser test method, the effects of propeller-hull interaction ori the wake field were studied for two twin-screw ship models with different afterbody shape. lt appears that interaction effects tend to reduce the non-uniformity of thewake field.

Calculations of the unsteady shaft force and moment components show a better correlation with experimental data when the effective, instead of the nominal, wake field is used as input.

i

Introduction

When dealing with wake fields in the propeller plane

of ships, two kinds must be distinguished: the nominal

wake field and the effective one. The nominal wake

field is the velocity field behind the ship without pro-peller. The effective wake field is the flow field actually

experienced by the propeller; it is the flow field

obtained by subtracting the propeller-induced velocities

from the flow through the propeller plane.

The differences between the nominal and the effective

wake distribution are due to propeller-hull interac-tion. Propeller theories assume the flow in which

the propeller operates to be a potential flow and

the presence of any body in its environment is

ne-glected. This means that the inflow to the propeller is found by simply superposing the original potential wake and the propeller induction. In a viscous flow with vorticity, produced by a body, the propeller will interact with the vorticity and with the solid

bound-aries of the body to maintain the no-slip condition.

All these interaction effects have to be incorporated in the wake, resulting in what one calls the effective wake, in order to calculate the propeller performance

correctly with existing theories.

Recently, the effect of propeller-hull interaction on

the flow field in the propeller plane of a ship has received increased attention. This is not surprising

when it is realized that for a big tanker with a heavily loaded propeller, the difference between the nominal and effective wake fraction can amount to 2O0/ of the

nominal wake fraction, the effective wake fraction

being the smaller one.

Evidently, the interaction plays a less important role for twin-screw ships. The clearances between hull and

propeller are relatively large for these ships, and so,

the difference between nominal and effective wake will be less significant. However, hardly anything is known

about the nature of this difference and whether,

although small, it is still important for the propeller design. Therefore, an investigation of this difference

was initiated within the scope of an extensive research

project on two twin-screw dredgers, one with a con-ventional and the other with a pram-type afterbody

shape {3].

The problem in such an investigation is the deter-mination of the characteristics of the effective wake field. Since the propeller-induced velocities are not

contained in the effective wake field, the latter is

essentially unmeasurable. Hence, in order to arrive at the effective wake distribution, we must follow a procedure, such as, for example:

measuring the velocity field closely ahead of the

propeller.

calculating the propeller-induced velocities and

subtract them from the flow field measured. Unfortunately, when a Pitot tube is used as a device

to measure the velocity, the desired measurements

cannot be made sufficiently close to the propeller. In order to overcome this difficulty, it was decided to

replace the propeller by a diffuser with circular cross-section, mounted co-axially with the shaft axis to the

ship model with the orifice at the propeller plane.

Such an axisymmetrical diffuser may be conceived as an annular hydrofoil-at-incidence. When it is fitted behind the towed ship model, a flow circulation will be established which causes the velocity in the orifice

to become greater than the advance speed of the

diffuser. Thus, ahead of the orifice the sink-like action

of the propeller is simulated. The diffuser-induced

velocities, which have to be subtracted from the flow field measured in the orifice of the diffuser, must be

calculated.

The obvious advantage in using a diffuser instead of the propeller is that the velocity measurements can be made in the propeller plane with the standard Pitot

apparatus. On the other hand, it should be realized

completely correct. In fact, every ship-propeller com-bination has a unique effective wake field. With the application of the diffuser, it is assumed that only the

mean loading of the propeller to be simulated

is

relevant for the interaction. Interaction effects asso-ciated with the details of the propeller geometry and load distribution are lost in this approach.

In this paper, the application of the diffuser test to twin-screw ships will be discussed. Results will be shown for the two subject vessels and a comparison

will be made between nominal and effective wake

distributions.

2 Application of the diffuser test to

twin-screw ships

A description of the diffuser test method for

single-screw ships has been given in Ref. [I]. The application of the method to twin-screw ships calls for some addi-tional remarks.

The diffuser-induced velocities can be computed when the diffuser is represented by an annular sheet vortex. Assuming uniform inflow to the diffuser, this

vortex model has a relatively simple form, since

varia-tions of vorticity only occur in longitudinal direction and not in circumferential direction. This approach is justified as long as the non-uniformities in the inflow and the deviations of the flow direction from the axial direction are small. This model was actually used for

diffusers behind single screw ships. Unfortunately, the

condition of small inclination of the flow to

the diffuser axis is not satisfied for most twin-screw ships.

As a result, a circumferential variation of the vorticity

distribution of the diffuser surface occurs. Hence, such

a circumferential variation of the vorticity must be

accounted for in the calculation

of the

diffuser-induced velocities. En order to avoid the cumbersome evaluation of a surface integral, inherent to this

prob-lem, we shall apply the following analysis.

It will be assumed that the inflow to the diffuser is

uni-form but inclined to the diffuser axis. The vorticity

strength distribution on the surface will be re-presented by t, which we define as

r = hm

(2.1)

ö5 being the vorticity vector and ör the infinitesimal

thickness of the vortex sheet. Introducing a cylindrical

co-ordinate system x-r-O with the x-axis aligned with the diffuser axis, the circumferential component of T will be assumed to be of the form

ro = A(x)+B(x) cosO (2.2)

A is the contribution to the vorticity strength

asso-ciated with a uniform inflow in axial direction, which

4

can be calculated by means of the procedure given in

Ref. [I . BcosO is the perturbation of the vorticity

dis-tribution caused by the flow inclination. The angular

coordinate O is zero at the location of maximum angle

of incidence. The distribution of the induced axial velocity in the orifice resulting from (2.1) is (see

Appendix):

VA1fld(r, O) = A 1(r) + B(r) cosO (2.3)

The contribution A is related to the circumferentially

uniform part of the loading and B1

is due to the

variable part.

Any vorticity field has to satisfy the condition

V5 = 0

For the present sheet vortex this implies 8w 13w0

+-

=0

8x

r80

or, equivalently, ¿3T

+=0

i 8f0 8x

r80

Equation (2.6) shows that the circumferentially varying vorticity implies the presence of a longitudinal

component of t. More specifically, recalling (2.2), f'

will be of the form

I = a(x)sinO (2.7)

This longitudinal component of T gives rise to an ap-proximately uniform "downwash" in the diffuser

ori-fice as shown in the Appendix. Thus, the

tangen-tially and radially induced velocities of the diffuser can be approximated by

VT1d = - B, sin O (2.8)

VR1fld = A2(r) + B2 cosO (2.9) where the contributions B2sinO and B2cosO result

from f and A2 is related to the uniform part of the

loading.

The effective wake field can be determined from the

flow measurements in the orifice of the diffuser, once the coefficients A1, A,, B1 and B2 specifying the

dif-fuser-induced flow field, are known. A1 and A, can be

calculated according to Ref. [I]. B1 and B2 will be

estimated on the basis of the test results. As is shown in section 4 below, this is fairly easy.

3 Experiments

The ship models used for the present investigations were NSMB ship model No. 4984 with a more or less conventional stern and No. 4984A with a pram-type

(2.4)

(2.5)

stern. Both models represent a twin-screw dredger. Their body plans are presented in Fig. 1. The

shaft-strut configurations and the shape of the afterbody

frames are shown in Fig. 2. Both models have a length

of 12 m.

The test programme consisted of nominal wake field measurements and velocity measurements in the orifice of a diffuser, carried out for both models. The nominal

wake measurements were made in the propeller plane

on port side. The three velocity components were

measured in 144 locations specified by r/R = 0.40,

0.48, 0.64, 0.80, 0.88, 1.04 and çb = 0,20, 40, 60, 80, 100,

20

MODEL NO, 4984

MODEL NO. 49A Fig. I. Body plans of the subject models.

850

measurements in the orifice of the diffuser were made

at the same stations, except for those on the outer

radius, being located outside the diffuser orifice. The

dimensions of the diffuser used behind both models are given in Fig. 3. The diffuser replaced the port propeller;

the starboard propeller was normally driven during

the diffuser tests.

All measurements were made at a model speed of 2.19 m/s, the measuring device being a 5-holes Pitot

tube.

BASE LINE

MODEL NO 4984A

Fig. 2. Shaft-strut configurations.

O A.PP

Fig. 3. Geometry of diffuser.

MODEL NO. 4984

DETAIL ' LEADING EDGE DIFFUSER

NSMB NOZZLE iBA PROFILE

120, 140, 160, 170, 180, 190, 200, 210, 220, 230, 240,

250, 260, 270, 280, 300, 320 and 340 degrees. The

o

- va-/v Va-/v V0, /V 1.00 0.50 o 1.75 1.50 1 00 0 50 1.50 1.00 0.50

EFFECTIVE FLOW FIELD

Fig. 4. Circumferential distributions of axial velocity compo-nent (model No. 4984).

4 Results and discussion

The circumferential distributions of the three velocity

components VA, VT and VR as measured in the nominal

wake fields are plotted at the top of Figs. 4-6 (model 4984) and Figs. 7-9 (model 4984A). Corresponding results of the measurements in the diffuser orifice, as well as of the derived effective wake fields, are added for a direct comparison. As usual, the velocity

com-ponents are given as a fraction of the model speed. In addition, plots of the lines of equal axial velocity

are presented in Fig. 10 and vector diagrams of the transverse velocity components in Fig. Il.

Let us first consider the axial velocity component.

In the nominal wake field, this axial component is

almost uniform over a large part of the propeller disk.

Distortions of the flow are found primarily around

the ç = 2000 position. They are caused mainly by the

presence of the propeller shaft. The position of the

shaft wake peak is slightly different for the two models

6 Vt/v Vt/v 0.5 o -0,5 0.5 -05

DIFFUSER FLOW FIELD

EFFECTIVE FLOW FIELD

Fig. 5. Circumferential distributions of tangential velocity com-ponent (model No. 4984).

owing to differences in the hull shape. lt is striking

that the non-uniformities in the wake distribution are much less for the pram-type stern (4984A). This must be attributed partly to the smaller hull inclination and partly to the favourable strut location. Behind model

4984 the strut wakes interfere with the shaft wake, while

for model 4984A the strut wakes develop outside the shaft wake, at least at the outer radii. For both models

the variations of the velocity distribution with the radius are small, as appears from Figs. 4 and 7. In

the flow field measured in the diffuser orifice, this is evidently not the case due to the non-uniform diffuser induction. The axial velocity distribution in the

effec-tive wake fields was obtained by subtracting VA1 as

given by (2.3), after assessment of the contribution

B1cosO. This has been accomplished as follows. From

the direction of the transverse component, it can be

concluded that the heaviest loading of the diffuser occurs around 'p = 20°, hence O = 20°. The value

-I

DIFFUSER FLOW FIELD

lì--lì

NOMINAL WAKE FIELD NOMINAL WAKE FIELD

go,

Vr! ¡V -0.5 o -0.5 0.5 o -0.5 0°

DIFFUSER FLOW FIELD

90 180° 270° 360°

of B1 can then be derived when it is assumed that

interaction effects are negligibly small at ( = 20°

(O = 0°), which is reasonable in view of the distance to

the hull being relatively large. This means that after

subtraction of (2.3) the nominal velocity distribution should be recovered at p = 20°. Ef the axial velocity

component at q = 20° in the nominal wake field is denoted by VN(r) and the same component in the

diffuser flow field by V0(r), we can plot

B1 = VDVNAI

as a function of the radius. The actual value of B1(r) is

determined from a faired curve through the plotted

points.

Following this procedure, the distribution of the

effective velocity field can be constructed leading to the results shown in Figs. 4, 7 and 10. These results

indicate that the circumferential variations of the

Vol /V o 10 0.5 o

DIFFUSER FLOW FIELD

180° 270° 360° Fig. 6. Circumferential distributions of radial velocity compo- Fig. 7. Circumferential distributions of axial velocity

compo-nent (model No. 4984). nent (model No. 4984A).

axial velocity are smaller in the effective than in the nominal wake. This is also clearly illustrated by the harmonic analysis of these circumferential axial velo-city distributions, given in Tables I and 2.

Turning now to the transverse velocity components,

it is shown at the top of Figs. 5, 6, 8 and 9 that, in the nominal wake, the familiar, almost sinusoidal, distri-butions occur. In the diffuser orifice, the magnitude of

the tangential velocity components is found to be decreased and the radial components vary with the

radius, as anticipated in section 2, eq. 2.8 and 2.9. For

the derivation of the effective transverse flow fields, the

coefficient B2 has to be determined. For this purpose

a procedure analogous to that used for B1 can be applied, B7 being independent of the radius. The

effective transverse flow fields are compared with the corresponding nominal fields in Fig. 11. It appears

that the transverse velocity components are hardly

affected by the interaction.

_u

____-0.88

EFFECTIVE FLOW FIELD EFFECTIVE FLOW FIELD

NOMINAL WAKE FIELD NOMINAL WAKE FIELD

90° cf 0.5 Vn 'V 1.0 Va/v 05 0.5 Vr/ 20 Va-/V 1.5 o

Vt/ 'V Vt/ ¡V Vt' 1v 8 0.5 -0.5 0.5 0 -05 05 o -0.5 00 900

effective wake field

DIFFUSER FLOW FIELD

EFFECTIVE FLOW FIELD

1800 2700 _{3 6Ö0} Vr! ¡V Iv 05 -0.5 05 o -0.5

DIFFUSER FLOW FIELD

Fig. 8. Circumferential distributions of tangential velocity corn- Fig. 9. Circumferential distributions of radial velocity

cornpo-ponent (model No. 4984A). nent (model No. 4984A).

Table 1. Harmonic analysis of axial velocity component in nominal and effective wake field ol model No. 4984

VA/V= A0

+Acos(i+)

EFFECTIVE FLOW FIELD

r/R0 40

7.-nominal wake field

nR A0 A1 a1 A2 a, A3 a3 A4 a4 0.40 0.744 0.143 15.2 0.083 225.3 0.067 69.0 0.028 267.9 0.019 108.6 0.48 0.756 0.138 14.4 0.075 224.3 0.060 70.5 0.025 270.9 0.015 119.1 0.64 0.772 0.129 18.0 0.061 229.4 0.047 75.2 0.018 269.8 0.015 141.2 0.80 0.774 0.135 20.5 0.063 235.5 0.047 89.9 0.021 326.5 0.018 171.4 0.88 0.770 0.139 19.5 0.064 236.3 0.045 88.2 0.020 317.2 0.023 167.8 1.04 0.760 0.157 19.9 0.076 244.7 0.054 98.3 0.031 319.4 0.031 168.6 nR A0 A1 a5 A2 a3 A3 a3 A4 a4 A5 a5 0.40 0.831 0.042 21.0 0.035 221.0 0.038 67.8 0.014 251.6 0.001 139.2 0.48 0.845 0.028 33.1 0.025 235.5 0.034 71.9 0.0 18 269.3 0.009 153.6 0.64 0.8 57 0.033 30.3 0.019 260.3 0.030 103.5 0.021 297.3 0.0 17 147.0 0.80 0.842 0.070 28.4 0.026 231.3 0.022 106.1 0.0 17 322.3 0.017 144.7 0.88 0.800 0.127 32.0 0.052 252.1 0.023 122.7 0.012 8.5 0.010 161.3

NOMINAL WAKE FIELD NOMINAL WAKE FIELD

0 go. 1800 _{2 70 36d0}

05

Vr,!

o

.85 75

MODEL NO. 4984

NOMINAL WAKE FIELD

85

DIFFUSER FLOW FIELD

EFFECTIVE WAKE FIELD

.80

MODEL NO.

4984A

NOMINAL WAKE FIELD

DIFFUSER FLOW FIELD

EFFECTIVE WAKE FIELD

1.10 1.15 j.20 1.25 t30 35 1.40 45 1.50

MODEL NO. 4984

NOMINAL WAKE FIELD

As an illustration of the effect of the differences be-tween nominal and effective wakes, as derived for the subject vessels by the diffuser test, on propeller per-formance, calculations were made of the fluctuations of shaft force and moment in both the nominal and effective wakes. The employed calculation procedure is based on lifting surface theory [2]. The results of

these calculations are compared with measurements in

Fig. 12. The correlation with the measurements is, in general, slightly better for the effective than for the

nominal, wake field. The correlation is improved, especially in the case of model 4984. Apparently, in

lo

EFFECTIVE WAKE FIELD

Fig. 11. Transverse velocity components.

MODEL NO. 4984A

NOMINAL WAKE FIELD

EFFECTIVE WAKE FIELD

spite of the assumptions made in the analysis of the diffuser test results, the derived effective wake fields are a better representation of the flow field actually experienced by the propeller than the nominal wake

fields. Nevertheless, the interaction effects are so small for twin-screw ships that the restrictions on measuring

accuracy and the approximations involved in the diffuser test may lead to errors of the same order of magnitude as these effects. The diffuser test

ap-proach is therefore considered more useful when the

interaction effects to be determined are more

Table 2. Harmonic analysis of axial velocity component in nominal and effective wake field of model No. 4984A

VA!V= A0 +A1cos(ic+1)

5 Conclusions

The effective wake distributions of two twin-screw

dredgers were derived from the results of diffuser tests

and a direct comparison was made with the results of nominal wake measurements. Moreover, the nominal

and effective wakes were indirectly compared with the

aid of shaft force and moments calculations.

Although the diffuser test method is insufficiently

sensitive to find all interaction effects in detail, we can

conclude with confidence from the present work that the propeller-hull interaction effects on the wake dis-tribution of twin-screws ships is such as to reduce the

circumtèrential velocity variations. The transverse flow

field of the present ships is hardly different in the

nominal and effective wakes. The calculations of shaft l'orce and moments fluctuations, based on the effective

wake field, correlated better with experimental data

effective wake field

20-FORCE (kgl io3) ¡ io 05 O E 20 MOMENT 15 N lID M E34 N EM05 T FH FV N N E M M O NIH MV

FIRST HARMONIC AMPLITUDES OF THE THREE FORCE AND MOMENT

N

NE

M M

FM MH FH MM

STATIC TRANSVERSE SHAFT FORCE AND MOMENT

COMPONENTS OF THE FLUCTUATING SHAFT

Fig. 12. Comparison of calculated and measured shaft force and moment.

than those based on the nominal wake. Nevertheless, one should bear in mind that the diffuser test is

pri-niarily useful when interaction effects on the wake are more pronounced, as for most single-screw ships.

References

I. HOEKSTRA, M.: An investigation into the effect of

propeller-hull interaction on the structure of the wake field, paper

presented at the Symposium on Hydrodynamics of Ship and

Off-shore Propulsion Systems, Det norske Ventas, Oslo,

March 1977.

GENT, W. v: Unsteady lifting-surface theory for ship

screws: derivation and numerical treatmentof integral

equa-tion, Journal of Ship Research, Vol. 19, No. 4, December

1975.

BERG, W. VAN DEN and Kooy, J. VAN DER: The effect of a

pram-type aftbody shape on performance, cavitation and

vibration characteristics of twin-screw dredgers. Nether-Lands Maritime Institute, Monograph, M 18, June 1977. nominal wake field

nR A0 A1 A: (i, A3 (03 Ag (04 A5 0.40 0.788 0.099 10.0 0.063 213.0 0.046 47.7 0.014 232.3 0.002 106.0 0.48 0.803 0.086 5.3 0.051 205.5 0.044 38.0 0.016 227.7 0.008 52.9 0.64 0.812 0.085 359.2 0.040 199.5 0.030 35.0 0.009 178.1 0.006 3.9 0.80 0.818 0.077 357.9 0.026 205.1 0.017 49.1 0.005 105.9 0.008 239.1 0.88 0.815 0.084 358.2 0.028 194.5 0.012 48.8 0.006 67.2 0.008 223.3 1.04 0.810 0.095 352.6 0.028 187.0 0.009 43.6 0.008 64.1 0.010 219.8 nR A0 A1 a A2 a: A3 a3 A4 (04 A5 a5 0.40 0.854 0.014 307.6 0.024 195.6 0.025 51.8 0.011 242.9 0.002 10.7 0.48 0.864 0.020 299.3 0.017 151.5 0.015 39.6 0.008 187.8 0.002 216.4 0.64 0.881 0.017 266.1 0.013 114.6 0.004 151.4 0.003 89.2 0.007 208.1 0.80 0.874 0.013 342.6 0.013 151.0 0.002 165.4 0.008 30.5 0.006 176.8 0.88 0.846 0.052 2.2 0.020 156.4 0.001 230.1 0.003 59.1 0.005 227.2 15 to-D5 D N 05 N E1 IM '!.t' T FM FV 0 MM MV MODEL NO 4984 MODEL NO 4934k N: RESULTS OF CALCULATIONS BASED ON NOMINAL WAKE

E: RESULTS OF CALCULATIONS BASED ON EFFECTIVE WAKE

M: RESULTS OF MEASUREMENTS

FV MV PV MV

20- 20

FORCE MOMENT

APPENDIX

The induced velocity field of the diffuser is determined

from a vortex sheet representation of the diffuser. The velocity induced by a vortex sheet is given in vector

form by

--

i

xt(')

-

J dA(x) (A.l)

4rr S

where is the position vector of the point where V is calculated, ' is the position vector of the point where

the integrand is evaluated,

s=

and dA is

an element of the surface A of the vortex sheet (Fig. A-1). The vector t has no component perpendicular to the sheet. So, if the vortex sheet is axisymmetrical,

t is in the Cartesian co-ordinate system x-y-z given by

r = jF+j(T0sjn 0')+ k(T0cos 0')

(A.2)

where F and r0 are the components of T in axial and

circumferential direction respectively. The vector can be written as

= (xx')+J(rcos0r'cos0')+

+ k(rsin 0r' sinO')

(A.3)

Hence the axial, radial and tangential components of

xF are

( x

=F0[(r cosO r' cos 0') cosO' +

+ (r sinO - r' sin 0') sin 0'] (A.4a)

( x

= T[(r sinO - r' sin 0')cosO +

(r cosO - r' cos 0') sin 0] +

- T0(x - x') (cos Ocos O' +sinO sin 0') (A.4b)

(ixfl =

T[(rsin0r'sinO')sin0+

+ (r cos 0 r' cos 0') cos O] +

+ T,(x x') (sin O cosO' - cosO sin O') (A.4c)

Expressing speeds, f' and F0 as a fraction of some

representative velocity V0 and lengths as a fraction of a representative radius R, the non-dimensional axial,

radial and tangential components of the induced

velocity V are given by

2ir F0[rcos(0O')r']

VA=

_--JJ

r dx dO (A.5a)

47ra0 R

VR =

b 2,t_{FXr' sin (0 0') F0(x - x') cos (OO')}

r' dx' do'

-

R (A .5 b) VT

-

SS

TX[r'cos(0O')r]

4mao R3

F0(xx')sin(O 0)

+ R3

-} r' dx' dO' (A.5c) 12 in which

R = [(x_x')2+r2+r'22rr'cos(0_0')]t

(A.6)

We now assume that

T0 = A(x')+B(x')cosO' (A.7)

Then from the conservation of vorticity, expressed by

äT

i aï0

(A.8) x r' ao'

we deduce

= a(x')sinû' (A 9)

Notice that an integration constant does not appear, since at the leading edge of the sheet (x' = a) both 1.

and a(x') are zero. Besides we observe that, whereas U9

is rstricted to the actual diffuser surface, TX extends

to infinity.

Inserting the above expressions for F and T in

equa-tions (A.5) and replacing 0-0' by4i,we arrive at

b 2,t [A + B cos (4' 0)] [r cos '1' r']

VA---5J

r'dx'di,li 4ltaO R3 (A.lOa) i

_{27rarsin('4/_0)sin'4J,d,d'4,+}

R3 4m o

b 2,, [A+Bcos('4íO)](xx')cos'4'

+SS

_R3 (A.lOb)

asin('4iO)[r'cos'4ír]

r' dx' d'4'+ VT= 5$ 4m R3 b 2,, 14 -4- R rnc(i/, - 1)V1 ( - y", in $ J L - \T -. r' dx' d 4mao R (A.lOc)

The function hR3 is symmetrical on the interval

O-2m with respect to 4i= m. When it is multiplied by

an antisymmetrical function such as, for instance, sin4', the integral of the product over 4i vanishes.

With this in mind, equations (A.lO) can be simplified to

i b2lr(A+BcosOcos'4J)(rcos'4,...r')

VA - - -

r'dx'dçlí 4m R3 (A.l la)

1 2,, ar'cosOsin2IJ,,,4,

VR=--$J

_R3 4m a o

i b27r(A+BcosOcos)(x_x)cos'4,

r' dx d'4i

+sS

_{4m o R3} (A.11b)

asinOcos'4i(r'cosJir)

V=

₅₅

r'dx'd,fi+

4m0o R3 b 2,t Bsin0sin2'4í(xx') r' dx' di4i R3 (A.iic)

R ' _{B(cos2Ocos2/j + sin2Osin2i)(xx')}

'dx' di

+-J$

r

47ta0

R3 (A.13a)

2ir

(V)x=a = - _s

ar sinOcosO(sin2icos2i)

ol

R3

or sinO cosO cos

R3

r'dx'd+

+

b2,r

B sin O cos O(sin2i cos2Jí) (x -x')

r' dx' di

R

(A.13b)

Fig. A-1. Definition of co-ordinate systems.

Recalling eq. A.6 with OO' = çli, it is readily shown

that C(r) tends to become independent of r for r'»r.

Under the same condition, the integral in (A.15a) and

(A.15b) tends to zero. In our case r'>r, so that

('í)

is approximately constant and (V2) is almost

zero.

lt follows that the axial component of the induced

velocity in the plane x = a becomes

Since

2n sin2i 2,r cos2i

(VA)x=a =f1(r)+f2(r) cosO (A.12)

The velocity components at x = a in y and z direction caused by r and the O'-dependent part of 10, are

di

di

2,r _{arcos i/i} (A.14) R3 we find sin2O I 2ir ar'(cos2Osin2(/+sin2Ocos29'í) + (A.15a)

(V)==C(r)+

4m R3 r'dx'd,1í

()x=aJJ1

R3 arsin2Ocos!1J1

_{r'dx'dí +}

sinOcosO 2,r

a?cosl,d,d,

R3 (A.15b)

()x=a =

_4m X b X X a

PUBLICATIONS OF THE NETHERLANDS MARITIME INSTITUTE

Monographs

M 1 Fleetsimulation with conventional ships and seagoing tug/

barge combinations, Robert W. Bos, 1976.

M 2 Ship vibration analysis by finite element technique. Part

III: Damping in ship hull vibrations, S. Hylarides, 1976.

M 3 The impact of Comecon maritime policy on western

shipping, Jac. de Jong, 1976.

M 4 Influence of hull inclination and hull-duct clearance on performance, cavitation and hull excitation of a ducted

propeller, Part 1, W. van Gent and J. van der Kooij, 1976. M 5 Damped hull vibrations of a cargo vessel, calculations and

measurements, S. Hylarides, 1976.

M 6 VLCC deckhouse vibration, Calculations compared with measurements, S. Hylarides and R. van de Graaf, 1976. M 7 Finite element ship hull vibration analysis compared

with full scale measurements, T. H. Oei, 1976.

M 8 Investigations about noise abatement measures in way of ship's accommodation by meansoftwo laboratory facili-ties, J. Buiten and H. Aartsen, 1976.

M 9 The Rhine-Main-Danube connection and its economical

implications for Europe, Jac. de Jong, 1976.

M 10 The optimum routeingofpipes in a ship's engine room, C. van der Tak and J. J. G. Koopmans, 1977.

M 11 Full-scale hull pressure measurements on the afterbody

of the third-generation containership s.s. "Nediloyd Delft", R. A. P. J. Schulze, 1977.

M 12 Cavitation phenomena and propeller-induced hull pressure fluctuations of a third-generation containership, A. Jonk and J. van der Kooij, 1977.

M 13 Hull vibration measurements carried out on board the

third-generation containership s.s. "Nediloyd Delft",

R. A. P. J. Schulze, 1977.

M 14 Hull vibrations third-generation containership, S.

Hylari-des, 1977.

M 15 Influence ofhull inclination and hull-duct clearance on performance, cavitation and hull excitation of a ducted propeller. Part II, J. van der Kooij and W. van den Berg,

1977.

M 16 The determination of the acoustical source strength of

propellers of two merchant vessels. A. de Bruijn, 1977. M 17 Experiments on acoustic modelling of machinery

excita-tion, J. W. Verheij, 1977.

M 18 The effect of a pram-type aftbody shape on performance,

cavitation and vibration characteristics of twin-screw dredgers. W. van den Berg and J. van der Kooij, 1977.

M 19 investigations into the effect of model scale on the perfor-mance of two geosim ship models, Part 1: Flow behaviour

and performance in calm water, A. Jonk and J. van de

Beek, 1977.

M 20 Investigations into the effect of model scale on the

perfor-mance of two geosim ship models, Part II: Behaviour and performance in waves, M. F. van Sluijs and R. J.

Dommershuijzen, 1977.