Vertical motions and added resistance of a rectangular and triangular cylinder in waves

824825

TECHNISCHE HOGESCHOOL DELFT

AFDELING DER MARITIEME TECHNIEK

LABORATORIUM VOOR SCHEEPSHYDROMECHANICA

VERTICAL MOTIONS AND ADDED RESISTANCE OF A RECTANGULAR AND TRIANGULAR

CYLINDER IN WAVES

W. Beu'kelman

Reportno.: 594

July 1983

Delft University ofTechnology Ship Hydromechanics Laboratory

Mekelweg2 2628 CD DELFT

The Netherlands Phone 015-786882

Summary.

The present work may be considerèd tò be a continuation of an earlier investigation related to forced osciil.atiön tests

with a rectangular and triangular cylinder to measure the

hydrodynamic coefficients- and added resistance for two speed of advance. For the same speeds and also for zero speed

vertical motion's and added resistance have been measured in regular head waves, with respect tq the mentioned cylinders. The experimental results are compared with computed ones based 'on strip theory. Also frequency response calculations. with experimental hydrodynamic coefficients have been car-ried out.

It appeared that the agreement is very satisfactory for the triangular cylinder but insufficient for the rectangular cylinder in particular for the highest speed.

It may be concluded th'at this is due to viscous dampïng which occurs for the' rectangular cylinder only'.

-i-Nomenclature.

A, B, C, D, E, G hydrodynamic coefficients of the

a, b, c, d, e, g equations of pitch and heave respectively

B _{beam of the cylinder}

B' _{breadth of the cylinder at the waterline}

CB b]ockcoeffïcjent

V

Fn _{Froudenuniber (Fn =}

_{___}

gL'

g _{acceleration due to ravity}

H _{depth of model} k _{wave number} ( 2:'ir/X) k yy L L Rs RAW T V V

_z.

Xb,

_b'

Zb z

-2-longitudinal radius of mass-inertia

of the model

length of model

effective length of model to determine

Froude nthther

s:tjll water resistance added resistance in waves

amplitude of oscillation

draught of model

perIod of oscillation or wave,

period of encounter

forward veioctiy

vertical relative water velocity

right hand coordinate system fixed

to the model with the origin situatéd

in the waterline of the model and the.

port side pos'itive heave displacement

C phase angle between motion and wave

V volume of displacement of model

e pitch angle

wave length

w wave frequency or frequency of oscillation, circular wave frequency of ençounter

p density of water

displacement of water surface

Subscipts

a amplitude of parameter considered

Superscripts

1. Introduction.

Recently forced oscillation experiments have been performed to measure the.added resIstance and the hydrodynamic coef

fi-cients ofa rectangular and triangular cylinder at two for-ward speeds

Both modelsare shown in figure 1. ,

The results of the fOrced oscillation experiments' together

with calculated values have been presented in port 1 .

One of the main objectives of this study was to show that the

added resistance experiencedby a forced oscillating model

is small or almost zero. An important conclusion of this investigation was that the added resistance in waves cannot be dètermined from forced oscillation experiments.

Moreover it appeare.dfrom comparison between measured and

calculated hydrodynamic coefficients that satisfactory agree-ment could be established for the triangular cylinder.

See for the damping coeffibient ligure 2 (i.gure 19 from ti]. ).. For the rectangular cylinder, however, it was obvious that especially with respect to the damping coefficients

considerable differences exist. See for this case the damping coefficient in figure 3. (figure 18 frorn1 ). The large damping in this case is caused. by viscous effects because of the sharp edges of this cylinder under water. it was possible to distinguish by analysis of the experimental results both the vertical soeed influence dependent on the oscillation amplitude and frequency and the forward speed influence on this. viscous.pàrt of the damping.

A plot of the total damping. coefficient for cöns tant speed. and frequency on a base of oscillation ,ampJitude shôws tha:t the damping coefficieñt is a linear function: of the amp.li tude of oscillation which increases with speed reduction. See figure 4 and 5 (figure 24 and 2.5 from tu )..

Extrapolation to zero amplitude of oscillation delivered: such values of the damping coefficient thát linearity with the forward speed squared could be shbwn, while these values for zero speed equalled thé calculated values f:or pbtentiai damping. See figure 6. (figure 26 from 1] )..

significantly influence the vertical motions as well as the added resistance in waves it was decided to carry out expe-riments in waves to measure these parameters for both cylin-ders.

The calculations of the vertical motions heave and pitch are based on the well-known strip theory as presented a.o. in

E 2, 3 and 4)

For the calculation of thè added resistance in waves use has been made of the method presented by Gerritsma and Beukelman in 5]

Comparison of experimental and calculated vertical motions and added resistance in waves showed a good agreement for the triangular cylinder, but a strong deviation could established for the rectangular cylinder especially in case of the highest speed considered.

Also the vertical motions and the added resistance in waves have been calculated by using measured hydrodynainic coeffi-cients.

A better agreement between experiment and calculation has been achieved for both cylinders if for the calculation the measured hydrodynamic coefficients are taken into account.

2. Experiments.

For the experiments in waves the same cylinders have been used as for the oscillation tests which are reported in

E i)

For the sake of completeness a short description of the test models follows hereafter.

Both models are manufactured of reinforced polyester plate.

One of these cylinders had a rectangular transverse section

and the other a triangular one.

An equal section form was maintained over 2m, while fore and

aft end had a length of 0.25 m. At these ends the breadth was linearly reduced to zero together with the depth for the

triangular cylinder.1 For the dimensions of these cylinders see table i and figure 1.

Table I

The length L' may be considerd to be. the effective length for determination of the Froude number and the dimensionless hydrodynarniC coef fi cien ts.

The breadth B' is the width on the waterline and is used for the dimensionless added resistance coefficient..

To determine the frequency response functions and the added resistance in head waves two speeds of advance were taken into consideration viz.: V = 0.743 and 1.238 rn/s.

For the rectangular cylinder these speeds agree with Fn = 0. 16 and 0.26; for the triangular cylinder with Fn = 0.16 and 0.27. As .àn addition for both cylinders also zero speed has been

considered.

The following range of wavelength-modelle.ngth has been ad-jus ted:

for-3 wave amplitudes viz.:

O.O2, 0.03 and 0.04. in

a

At firs.t the stili water resstance R had been measured for

s

b.th cylinders as shown in figure 7.

Ln the mentioned waves thé heave- and pitch amplitudes to-gether with the. accessory phâse- angles h-ave beér de-terfnined

Rectangular cylinder . Triangular cylinde.r L 2.5-0 rn 2.50 m B 0.25 m 0.25 m B'

_0.25m

0.15 in T .

0.15m

0.15m

H

0.25m

0.2-5m V .. 0.0-84.38 m3 . . 0.0-2419m3 kyy/L 0.25 0.25 L.' . 2.-333 m 2.167 rn KG 0.12 m . 0.12 m X/L =0.50, 1f30, 3.40, 0.60, 1.40, 3.70,. 0.70, 1.60,. 4.00 0.80, 2.00., 0.9Ó, 2.30, 1.00,. 2.70,. 1.10, 3.00,.

as weil as the total resistance RT,.

After reduction of this total resistance RT wïth the still water resistance R5 the added resista.nce RAW in waves could be established.

Surging motion was possible. but has nòt been measured. The measured values for heave and' pitch are presented on basis of the dimensionless wave frequency of encounter

lLt

WeV

¡g and' are shown respectively as and'

in the figures 8, 9 and 10 for the three speeds:

con-side:red,.

The measured phase angles fo.r.:heave and pitch are plotted on basis of the same dimensionless wave frequency of encounter respectively äs and in the figures 11, 12 and 13.

The added 'resistance in waves determined from the measurements has been presented as. dimensionless coefficients

3...iculations.

For, both cylinders the vertical response functions have been calculated according. to the modified strip theory as

'presented in [2] and

[4)

, while the added resistance in waves has been determined wit.:h the method presented in [5], The amplitude and phase angles of the heave and .pitch motion 'have been computed according. to version 1 of

[1]

. For this

version the forward speed influence is ta'ken into account in such 'a way that although the'syrnmetry relations in the dam-ping cross coupling, coefficient are 'lacking, thismethôd generally showed better agreement wIth the' meàsurements especially for thé higher speéds [4 ].

In these calculations no viscous influence has been taken. into account. The calculated values of thé héave 'and pitch amplitude. have ben plotted in a dimensionless' way as

respectively Za/Ç and'

0/k

as function of thé

'dimension-a a Ça

,

.less wave: frequency of encounter L /'g for the three

speeds in 'the figures' 8, 9 and' 1û

'The 'calculated pháse 'angles of, heave and pitch respectively

e _and aré presented in the' 'same way in.the 'f igures

;,p g on basis of the frequency coefficient L,,/

11, 12 and 13.

The computed values of the added resistance in waves are

2

shown as dimensionless coefficients _RAWli

/PgB

in the figures 14, 15 and 16.

As an addition the frequency response functions of heave and pitch have been computed with the experimentally

deter-mined hydrodynamic coefficients as presented in [i]

This means that for this case the important viscous influence has been taken into consideration.

Because the distribution of the hydrodynamic coefficients over the modellength has not been measured it was assumed that this distribution is similar to the calculated distri-bution on the understanding that the calculated sectional value should be multiplied by the ratio of the measured and calculated total value of the related coefficient.

The calculations with the experimental coefficients have been carried out for both cylinders, for two forward speeds viz. Fn = 0.15 and Fn = 0.26/0.27 and for three amplitudes of oscillation viz.: r = 0.01, 0.02 and 0.03 m.

The differences in response function because of these three amplitudes of oscillation appeared also to be small and

negligible. For this reason only the results of r = 0.02 m

amplitude of oscillation are shown in the figures 9, 10, 12,

13, 15 and 16. In these figures the amp1itudehase angle

and added resistance are presented in the same way as des-cribed for the pure calculated values.

4 . Discussion of the results.

Comparison of the measured and pure calculated results in the figures 8 - 16 shows satisfactory agreement for the

triangular cylinder with an exception for the added resistance in waves in the case of zero speed. See figure 14. In general

the calculated values are higher than the measured ones for this case. This might be due to the small values measured for the triangular cylinder at zero speed. For the rectangu-lar cylinder it is obvious that there is good agreement be-tween calculation and experiment for the case of zero speed as well as for fhephase angl-s-.---Howevcr,

diffcreeesin-creasing with forward speed may be observed for both the,

heave, and pitch amplitude responsé fun'tions and for thern

added' resistance in waves

The figures 9, 10, 15 and 1,6 show what was expected viz.

a significant influence of the viscous damping on the motions

aid added resistance especially for the highest forward

speed. 'For Fn =0.15 this differences are still rather small

for t'he motions although they are more importart for the

added resistance in waves.

it is worthwìle to remark that fôr t'he increased resistance

in waves two opposite a'cting factors play an important role:

the increased damping

oefficient causesan increase

of the added resistance (at equal amplitude of motion,

,)

the. increased damping.ca'uses lower, motions and the

squared values of these motions with rès.pect to the

watersurface result in a strong decrease of the added

resistance.

Above mentioned effects föllows from the' expres:s'ion for the

calculation of the add'ed resistance in waves as presented

;

f b', Va

'b

o

in which.:

b' = the sectional damping coefficient at speed

wave nuiriber

'

circular wave frequency-of encounter

.4

x O ± VO - Ç = vertical re:latïve 'water

'b

_velocity

From flgure '10" it is clear that especially the part of the

viscous d'antping caused. by the. forward speed influence results

into a strong reduction of the measured motions compared to

the calculated values. From these strong motion reductions

a decrease 'of the add:ed resistance follows too ('see figure

I 6i in spite' 'of thò .ef-fect of' t'he-i-ne.rea.ed ddmp±ny euef fi-'

cients.

k

= W

e

=' V z =

io

-Comparison of figure 15 and 16 shows thatfor the low speed

the increased damping coefficient' dominate the added resis-tance resulting in higher measured values than calculated ones. This part of the viscous damping is mainly dependent on

the vertical speed due to the amplitude and frequency of os-cillation.

For the highest Froude number the situation is just reversed: the calculated values for the added resistance are higher th'an the measured ones.

For this case the motion reduction resulting from the forward

speed influence on the vis'cous damping dominate t'he added' resistance in waves.

It is obvious form the preceding results that for some.

cases where viscous damping play ari important role this part of the damping' should not be neglected for the calculation :Qf the

vertical 'motionandthe added resistance in waves especially

for h'igher Fro.ud'e numbers. Such a case may beexcept sharp rectangular ship-or barge sections e.g. bilge keels which' for high forward speeds may show a signific'ant influence ,on the vertical motions and the added resistance.

in future it should also be worthwïle to. investigate the influence of the radius of the bilge on the 'viscous damping. This may also be of interest for. the 'rolling motions.

For.practical use it is also important to investigate the scale influence on viscous dampiñg.

Better agreement between calculated and measured values hàs been achIeved by introducing thé experimental

hyd.ro.-dynamic coefficients obtained by thè forced oscillation

tests-as described in

[i

j.

From theufIgi.res '9, 10,. 12, 13, 15 and 16 it is cleár that t..he' improvement n calculation éspecially hbl,d:s for. the rectangular cylinder a,t the highést forward speed,. This meàns that it is ss'ential to take''into account thè. viscous

dàmping for th'i's speed.

5. Conclusions arid recommendations.

From. t'he presented invesi- i gM- i öns re'latd to' thé e-as-u-remcnt

and calculation f the vertical motion and added res'istance in waves for t'he cylInders consIdered the'' following

conclu-sions and' recommendatiöns may be derived:

For the triangular cylinder the vertical mot: ions and added resistance in waves are to determïne very well with the potential theory only, but for the rectangular

cylinder viscous damping should be added 'at least for forward speeds higher than Fn = 0.15.

This viscous damping appeared to be linear proportio-nal to the squared value of the forward speed'.

For the rectangular cyiinder the influene of the ver

tical speed dependent on thé oscillation amplitude and frequency incrases with a rediction of the forward' speed and is also maximum for zero. speed. Even f or, zero speed this influence on the vertical motions and added resistance appeared to be rather small. This viscous part of the

damping showed to be linéar proportional to the amplitude of oscillation.

Determination of viscous damping either by calculation

or by systematic experiments is recommended These svste-mât'ic experiments should be related to investigations

with respect to the influence of bilge keels and bilge 'radii: include the influence of forward speed,, vertical

6. References.

[i] Beukelman,

W.,

Added resistance and vertical hydrodynamic coefficients of oscillating cylinders at speed, report nr. 510,

September 1980,.

Ship Hydromechanics Laboratory of the Deift University of Technology, The Netherlands.

12

-Salvesen, N., E.O. Tuck and O. Faltinsen, Ship motións and sea loads,

Transactions of the SNA, 1970.

Gerritsma, J., W. Beukelman and C.C. Giansdorp,

The effect of beam on the hydrodynamic characteristics of ship hulls,

10th Office of Naval Research Symposium, 1974, Boston, U.S.A.

Gerritsina, J. and W.. Beukelman,

Analysis of the resistance increase in waves of a

fast cargo ship,

irternational Shipbuilding Progress, vol. 1:9,

no. 217, 1972 and 13th ITTC vol. 2, 1972.

[2]

Gerritsma, J. and W. Beukeiman,

Analysis of the modified strip theory for the calculation of ship motions and wave bending moments,

International Shipbuilding Progress, vol. 14, no. 156,

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