Added resistance and vertical hydrodynamic coefficients of oscillating cylinders at speed

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801085

TECHNIScHE HOGESCHOOL DELFT

AFDELING DER SCÑEEPSBOUW- EN SCHEEPVAARTKUNDE LABORATORIUM VOOR SCHEERSHYDROMECHANICA

ADDED RESISTANCE AND VERTICAL HYDRO-DYNAMIC COEFFICIENTS OF OSCILLATING CYLINDERS AT SPEED.

W. Beukeiman

September 1980

Rapport no. 510

Deift University of Technology Ship Hydromechenics Laboratory

Mekelweg 2

2628 CD DELFT

The Netherlands Phone 015 -786882

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Contents: Slunimary.. Nomenclature. Introduction. Experiments. 3.. Calculations.

Discussion of the iesults. 4.1. Added resistance.

4. 2.. Hydrodynamic coefficients.

Conclusions and recommendations.

6.. Acknowledgements.

References.

Appendix.

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ADDED RESISTANCE: AND VERTICAL HYDRODYNAM1,C COEFFICLENT,S OF OSCILLATING CYLINDERS AT SPEED

by

W. Beukeiman

Surnmry:

With a rectangular and triaigular cyiïnder forced oscillation

tests in the vertical mode have been carried out for two

speeds of advance.. The add'ed resistance due to oscillation has been measured while at the same time the hydrodynamic coeffi-cients have been. derived from the measured vertical forces. The results have been compared. with computations for both resistance increase and the hydrodynamic coefficients.

It appeared, that the resistance increaseriat

comparable with the predicted values The difference. between

experiment and calculation for the hydrodynamic coefficients has shown to be very srnaiJ for thé triangular cylinder, but very larçe for thé damping of the rectangular cylinder.

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Nomen'ciattire.. A,B,C,b,E,G a,,b ,c ,d,e , g B e: CB Fn g H k yy L' L m T

Te

V V

_a

Xbl 'b , Z

We p o Subscripts Superscripts }

hydrodynamic coefficients of the equations of pitch and heave respectively.

beam

i.ec.g&tiO"

boòk coeffjc:ient Froude number

acceleration due togravity depth

vertica]. loÌgitudïnal radius of inertia of model

effective length of model length of model

added mass

draught of model

period of oscillation forward velocity

vertical relative velocity with respect to the water

right hand coordinate system fixed to the model with tie origin situated in. the

water-line of the model and the port side positive

i.alf width of waterline (z=O)

heave displacement phase angle

volume of displacement of model circular f requency of oscillation density of water

ptch angle

amplitude of denoted parameter

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-1-1. Introduction.

In the past extended osdiliation tests have been carried out by

Ueno et al Li,2,3,4,5j , to determine the added resistance for

different ship models and for several modes of motion. These authors refer to the calculation method of Maruo'[ 6] for the detérmination of the part of the' added resistance in waves due

to the ship's oscillations only. Also Goeman t7] performed a

ver-tical oscillation test with a ship model to measure the ad'ded

re-sistance.

'-Many experiments have 'been carried out with respect tothe

resistance increa'e of a ship model in waves. Measurements nd

a simple calculation procedure for the determination of the added resistance in waves have been presented by Gerrtsma and'

B'eukel-man

in8).

With respect to the determination of the hyd'rodynamic coe:ffiients many oscillation tests have been carried out from ihich the,

great majority was related to model's of ships '[9,10,1]].

'Only a few number of oscillation tests have been perf'drmed for

cylinders, s'o foI' the two-dimensional

were mostly restricted to zero speed of advance. For this c'ase it is important to mention the work of Vugts [12) 'whose tests are related to cylinders with different beam/draught ratio's for circular, triangular and Lewis-form cylinders.

For rectangular sections thé experiments of Keuning and Beukelman

[i3] may be used. TheIr oscillation tests with 'a brge at zero

speed delivered hydrodynam-ic coefficients for both deép and shallow water.

For two-dimensional shipl±ke bodies' oscillation tests at shallow

water have also been carried out by''Taaki, [14]

The methods for the 'calculation of vertical hydrodynamic coeffi-cients for a ship at speed are weÏl-knówn [10,15,16] and mainly based on the work of Urseli [17] for oscillating cylinders in a free surface.

The main purpose of this work is to measure 'the 'added resistance

due to vertical oscillation for a reótangular and triangular

cylinder a't speéd, for which difféient types' of dping'y"be

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from a caidulation procedure. on the method presented.

in t15].

Furthermore the vertical hkdrodynamic coêffióients have been der ived from the hdrodynamic forces. Cömparison. with the

calculated results shows the. diffèrence in damping, the influence of speed, ampiit'of oscillation and fórm of the cylinder

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2. Experiments.

For the experiments two cylinders consisting of polyester reinforced plate had been manufactured..

One of these cylinders had a rectangular transverse section and the other a triangular one, An equal section form was maintained over 2 m, while fore and aft the ends h4d a length of O . 25 m. At these ends the breadth was linearly reduced to zero. For the dimensions of these cylinders see table 1 and

figure 1.

-Table 1.

The length L' may be consïderd as the effective length for determination of the Froude number and the dimensionless hydrodynamic coefficients.

The models have been forced oscillated by the PMM. (Planar Motion Mechanism.) for vertical motions as described in £ 9] The models were connected to two rods of the oscillator by means of two sträin gauge dynamometers for each rod., One of these dynamometers was sensitive in only the. vertical direction while the other was sensitive in the horizontal direction for the determination .of the resistance.

Two speeds of advance for both cylinders were taken into

con-sideration viz. y = .0.743 and 1.238. rn/s. For the rectangular

cylinder these speeds ag.reed with n = 0.16 and O.26 and fòr

the txiangular cylinder with Fn = 0.16 and 0.27. .

Rectangular i cylinder Triangular cylinder 2.50

m

. 2.50 m B 0.25 m . :O m

o.25

p' H

p T 0.15

m

0.15 m H 0.25 m .0.25 m V _{0.. 8438 m3} _{0.2419 m3} k / yy'L .0.25 0.272 L' 2.333 rn 2.167 m

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Tbethiioi'xg circular frequencies were adjusted fcr the heave and pitch oscillation

3,5,7,8,10 and 12

The amplitude of oscillation was for heave

r = 0.01, and 0.03 m

and

&rpitch

r- = .-01 and 0.02 ni

At first the stili water resistance 'R had been ,measuredr for

both cylinders as shown in fig. 2 and 3.

Afterwards the models were Orced t carry out a heaving and

pitching oscillation fOr the above- mentioned frequencies and amplitudes. Fcr these situations the total resistance RT had, been measured and after reduction -of the still water resistance: the added ±esistance RA could be established'.

The me-aured values for the added resistance are for both, cylinders.

shöwn in fig. 4- - ii..

'The measured vertical förcês- were reduced, into an in--phase

component and a :9Ø degrees out-of-pháse cOmponent with thç adjusted motion by means of an analoque Fourier-transformer, mechanically

(iii e- th the oc1]!ator

--_---

-'From these foice-components -and the known particulars of the, model the hydrodynarnic coefficients could be defied as denoted.

i-n appendix 1.

-- They are shown in a dimensionless way-- for both cylinders in fig.

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3. Calculations.

According to 8 it should, be possible to determine the added

resit-ance- for a ship oscillating in sti:1l water. This methOd is based on the principie that the energy radiated by the damping waves is equal to the energy performed by the. added resistance tO maintain the adjusted speed of advance.

The following general equation should therefore be used:

RAVT

=

/fTe

_b,:V2dtd

in which:

the relative -vertical speed of a cross section Xb is given by:

V

= Vza còs-(-w t

+ C) = -Z- -

XbO + VO (2)

and:

RA = added resistance

Te = period' of oscillation

b-' N' -- V =- the -sectional damping,

'b

coefficient a-t speed

N' = the sectional damping coefficient for zero s-peed

z z- cosw t = t'he heave motion

-a e

-O = -O cosw t = the pitch motion

a -e-

-From (1) follows the added resistance for the- geñeral cas.e o.f

oscillation: -

-= W

f

b'

y

2 -dx

(3)-L

- AccordIng t-o appendix i it may be evaluated- from -(3) that for

a púre heáve oscillation the added resistance will. be:

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-iñwhi-ch, - b =- the- dam-pi-ng coefficient for heave -of the total

model at speed.

in the same -way the added-resistance for thépure pJtching

oscilla-- tion may- -be dç-r.ived- from- (:3) a-s shown i-n- appe-n'dix i and i.s Written

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:-a

- (i)

2V e 2 B + 2VD ) + V2b } (.5) In which:

B the damping coefficIent for pitch

D = the mas coupling coefficient for p:itcb

The calculatiOns of the added resistance have been carried for the pure heave- and pitch oscillations in agreement with (4) and

(5). These calculations were related to both the calculated and

measured, hydrodynamic coeffic lents..

The results' for both cylinders are shown in fig. 4 - Il.

Wit respect to the calculation of the hydrodyn'amic cóeffïcients

reference should be made. to i5 where a method is presented

to determine these coefficients for two versions according to

the. strip theory.

The expressions for the differênt hydrodynamic coeffidients

accodiñg to

i5J are preented in appendix 2.

The calculated hydrodynamic coefficients are for both cylInders

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a. dimensionless as RA

.rpg

/L-frequency parameter;

7

4'. Discussion o.f the rèsuits.

4.1. Added -resistance.

The calculated and measured added resistance have been plotted in two ways:

on basis of the dimensionless

vg

See fig.- 4 - 7.

b. as- a percentage of the measured. still: water resistance, so

R

x 100 also on. basis of the frequency parameter

ù.«--g! FUr these results see fig. 8 - ii.

The experimental results for added resistance show remarkably low values with one exception for the case of pitching at the lowest speed.

Table 2 gives an impression of the áverage measured resistance increase as a percentage .of the still water resistance for both cylinders and speeds.

-i.t is quite cl-ear that the highest values may be expected a-t the lowest speed, because of the relatively low still water resistance.

Comparison between measured and calculated -values shows

unreasona-bly high differences. Evidently one shóu-ld establish that the

f low -of energy is directly from thé Hoscillator into t-hé radiated

damping wave-s so that no resistance Increase will be caused.

-The small values which h-ave been observed Îna-y be: due toncn

-linearity because-of viscous influence.

Table 2 R -A x 100%. ¡Rs -Heave -- :- -- HPitch 0 16 27 Fn=O 16 Fn=° 26/o 27 Rectangular cylinder Triangular cylinder -2.1 0.9 --:0.4 0.1 I 7.4 - 8.6 H 2.4 -1.3

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The same tendency has been observed by Goeman in 6]whe.re no,

sometimes even small negative added resistance was found with a

ship model forced pcil1ated in still water This model was

equal to the one described in with high added. resistance in

waves.

This ail means that it is impossible to determine the oscillatory part of the added resistance bymeans of forced oscillation of ship models.. Therefore it is worthwhile to consider the measured results of teno L1,2,3,4,53 in this respec.t.

The deviation between the values of added resistance determined with calculated - and experimental hydrodynamic coefficients is very significant for the rectangular cylinder and is caused by the high viscous damping for this cylinder as shown in fig. 18..

4.2. Hydrodynamic coefficients.

For added mass and the cross coupling coefficïents for added mass it appears from fig. 12 - 17 that in generai there is a good

agreement for both cylinders between expetiment and calculation. The strongest deviation but stIl rather small could be. established for the rectangular cylinder with respect to the added mass moment of inertia.

From the figures also follows no :referénce for one of the two versions as result of comparison with the experiments.

Furthermore it appears from the figures that there is a good

proportionality for added mass with respect to amplitude of oscillation.

The speed influence has shown tò be negligible.

Considering the damping of the rectangular cylinder it is clear from fig. 18 and 20 that there is no question of any agreement between experiment and calculation as already said before. Also a rather strong proortionii Influence of Ehe

oscillation may be observed from fig. 24-25at lower speeds. Damping increases with the amplitude at increase of f requency of oscillation. The viscous damping because of the sharp edges of the rectangular cylinder appeared to be. most impottant.

The speed influence is also rather significant for the rectangu-lar cylinder and may be characterized as an increáse with speed.

In general it may be observed from fig.,2that or the

rectangu-lar cylinder the experïmentaI data tend to the calculated urve,

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speed decreasetO zero.

This phenomenon is in agreement with the calculated and., measured

values presented in [12j and 13]thr a pontoon at zero speed and

deep water.

in a rversed way one may put it so that .f órthe rectangular

cylinder the viscous damping increases with amplitude of. oscillation

and forward velocityjquared (fig,. 24. - 25 26)

The agreement between experiment and calculation for the damping cross coupling coefficient is somewhát better. than. for the damping coefficient in the case .of t'he rectangular cylinder, see fig.

22-23. The.. proportionality with amplitude and freqency of

oscillation is also smaller .thár for h& normai, dámping coefficient

as shown in f ig. 24-25.

With respect to::the triangular cylinder a nice agreement between experiment and calculation could be observed fr the damping. at

heave from fig1 19.. It is obvious that for this case there is only

potential damping and almost no vïscous influence.

The (,jproportionaiity of damping with amplitude is very small for

the wholé frequency ange.

For pitch the agreemert between measurements and caiculations is worsè just as for the damping cross coupling coefficients as shown

in fig. 21-23..

It is remarkable that the damping cross coupling coefficients show a better agreémerit between exper±ment:ànd calcuiation for the.

rectangular cylinder than for thé triangular one.

Concerning the speed influence it is clear from fig. 18-19 that for the triangular c.yiinder there is a small decreáse of damping, with speed just reversed in tendency as for the rectangular cylinder.

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lo

-5. Conclusions and recommendations.

From the preceeding experiments and calculations the following conclusions and recommendations may be derived with respect to the added resistance due to forced vertical oscillation and with respect to the hydrodynamic coefficients:

Almost no or only a small added resistance may be measured for a model forced oscillated at still water, which values are not comparable with those of the calculations and those measured in waves.

The flow of energy appears to be directly from the oscillator into the damping waves so that no resistance increase arises. The measured and calculated values of added mass and the

mass cross coupling coefficients are in good agreement.

The results of the experiments and the calculations agree very well for the damping coefficient of the triangular cylinder,. which means that for this case viscous influence is almost

negligible.

The measured damping coefficients of the rectangular cylinder are very high in comparison with the calculated values with a

strong proportionality related to the amplitude at the

higher frequencies and lower speeds.

The large differences between measured and calculated damping coefficients for the rectangular cylinder should be due to viscous influence on account of the sharp edges.

The speed influence is mainly measured for the damping coeffi-cients with a strong increase with speed for the rectangular cylinder and a moderate decrease with speed for the triangular cylinder.

The viscous damping for a rectangular cylinder increases with the amplitude of oscillation and with forward velocity squared. The damping cross coupling coefficients for the rectangular cylinder show a better agreement between experiment and cal-culation than those of the triangular cylinder.

lO.It should be valuable to gain a better insight in the flow

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6. Acknowledgements.

The author owes a great debt to the variousmembeii of the staff of the Ship Hydromechanics Labóra:tory of the Deift University of Technology for their assistance in running and conducting the described'.experimentSpecial thanks are due to mr. A. y. Strien who carried out the experiments.

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-7. References.

E']

Ueno, K. et aI,

Some experiments of heaving effect on ahead resistance of ships,

Journal of the Soc±ety 'of. Nava]. Architects. of West Japan, No., 37, Fèbruary 11969 and 12th. ITTC,, p. 112, 1969.,

E2J Ueno, K.et ai,

Some. experiments of pitching 'effect on ahead resistance 'of ships,

Journal of the .Society of 'Naval Architect's of West Japan,

'no. 37, 'February 1969 and 12th ITTC, _. 114, 1969.

[3

_J

Ueno, K. .ò.t al,

Sorne experiments of yawing, effect on ahead re'sistaice of ships,

Journal of the Society cf Naval Architects of West Japaln',,

'No. 23, Maròh i962..

Ueno, K. et al,

Further experiments of yawing. effect on ahead resistance of ships,

Journal of the Society of Naval Architect's .of West Japan,,, No.. 27, March. 1.964.

Uëno, K. et al,

Some experiments of roiling effect cn ahead resistance 'öf

ships,

Journal of t'he Society of 'Naval Architects of West Jlapan,

'no. 31,. March 1.961,.

Maruo, H...,

On th.e increase f the 'resistance 'of a ship in rough seas,

Journal of the Society, of Naval Architects of Japan,, no. 101,,

19517.

[73

Goem'an, A.,

Weerstan'ds- ei vóortst'uwingsproeven met eén model van de

S.A. vä'n der 'S.tel bscilieend in vlakwater'.'(in Dutch),:

h'ip Hydromechanics Laboratory of the Deiff University Of

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13

-Gerritsina, J. and W. Beukelman,

Analysis of the resistance increase in waves of a fat cargo ship,

International Shipbuilding Progress, voi. 19, no. 217.,

1972 and 13th iTTC, vol.. 2,, 1972.

9] Gerritsma, J. and W. Beukelman,

The distribution of the hydrodynamic forces on a heaving.

and pitching shipmd1 in still water,

Fif:th Symposium Naval Hydrodynamics, 1964.

t'° Gerritsma, J. and W. Beukelman,

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

International Shipbuilding Progress, vol. 14, noo 156, 1967.

Beukelman, W.,,

Pitch.- and' heave öharacteristics of a destroyer, International Shipbúild'ing Progress, no. 192, 1970.

Vugts, J..H.,

The hydrodynamic coefficiènts for swaying,heaving and

rolling y1inder.s in a free surface,

Netherlands Ship. Research Centre TNO, Report no. i12S',

May 1968.

E13] Keuning, J.A. and W. Beukeiman,

fldodnarn±, coefficients of rectangular bar ges' in

shallow water,

Second International Conference 'on Behaviour of Off-Shore Structures (BOSS'7'9),. August 1979, vol. 2, paper 55.

t14] Takaki., M.,

Wave induced forces and momerts as acting on the two dirnen-si.onal bodies oscillating in shàlÏow water,

Res. Inst. of Appi. Mech:., Kyushù University.,Japan, vol. 25, No. 78, 1977.

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-

14

-ti:.

_:1 Gerritsma, J., W'. Beukelman and C.C.. Giansdorp,

The effec.t of beam on the hydrodynamic characteristics of

shiphulis,

Tenth Symposium Naval Hythodynarnics, 1974.

t

] Saivesen, N.., .E.O. Tuck and: O. Faitinsen,

Ship motions and sea loads,

Transactions of the SNA.ME, 1970,.

t17 Ursëi.l, F:..,

On the heaving motion of a 'circü:lar cylinder on the surface

of afluid,

Quarterly Journal of Mech. and AppI. Math.,

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Appendix 1.

As generai equation for, the added resistance

due

to oscillation

is 'given: 1 2V o and with: =

Vcos(wt

+ =

-

ve

(2;) follows:

RVT

₌

.J:f'b'

Vcos2(t ±

E)d(Wt

+

e)

= .,

Ib!

v2 dxb

Further evaluation yields the general expression for the added

resistance due :to osciliatÎon:

Ib'

V (3)

For pure heave oscillation i.t holds that in

(2:)

e = :0 so that

V ₌

a

ea

Obviously the added rèsistance due to pure heave osc:illation may be written as: RA ₌ _2V L W.Za2

i

b'dx or: _:27 ₂ e :a RA ₌ _2v 'b ' (4')

FOr pure' pitch oscillation it holds that' in (2)

z

0 SO that

V

V S]fl6 . = X. (i) O..

za.

y

bea

15.

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from which follows:

and

V2

_za

16

-= (weXbOa) 2 +

Substitition:'o'.(6): into (3;) delivers for the added resistance

due to pitch oscillation:

RA = (u

f

_b'xb.dxb y2

fb'd

) (7)

with:

b' N' - V see: 3) and thé hidrodynamïc coefficients.

B

_f,Xb2d:b

- 2V

_fm'xbdxb

=

vf

_xb2dxb

Xb dxb

(VO)2. (6)

(see appendix 2)

it is possible to reduce (7) to:

RA

02

+ 2VD}

+V2b

(5)

This expression, is valid for both verioi-is as presented in

appen-dix 2.

I

D = J m'

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17 -Appendix 2.

For convenience the hydrodynamic coefficients as they are derived

in appendix i of

[15]

for 'both mentioned versions are presented

here':

Heave

a

₌

_Jm'd.xb

J

_{r dxbl}

f,

din' b. =

i

(N -V---)dx

L

dxb

b C '

2pg

j

dx

d' =

fm'

_xbdxb

ÍN'dxb

_-

f

dxb+

r

fdN.'

,+

(Le

I ;--

_

xbdxb

L

'b1

f

è

=

J

N'xdx

_b

- 2V

.j

In'dxb - V

J

_{-- xbdxb}

+ L L Xb g =

2pg

_yw

xdx

Pitch

A =

fmt:2dx+[:]\T

2

.xbdxb

=

Ç

Xbd:Xb + + _Xb .

dxb i

B =

f

:

f.m'xbdxb

V

_b2d

_±

rV2

f

-

L b

.dxb]

C

2pg

xbdxb

D

fIn'xbdxb

1 Xb

dxbj

E.

f

N" x

dx

- V

f

--

x

dx

b

'b.

_d.xb b b

G= 2pg,

y. X

dx.

L

w.b

b.

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18

-When the terms between the brackets are left out one finds the hydrodynarnic coefficients according to version 1; if not these coefficients according to version 2 are' presented.

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