The heave damping coefficients of bulbous cylinders, partially immersed in deep water

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DEPARTMENT OF THE NAVY

DAVID TAYLOR MODEL BASIN

WASHINGTON, D.C. - OD7

THE HEAVE DAMPING COEFFICIENTS OF

BULBOuS CYLINDERS,. PARTIALLY

IMMERSED IN DEEP WATER

by

W. Frank

TechnicalNotes are working documents subject to revision or expansion and are not to be referenced

in formal publications. They are intended for

recording current pertinent technical inforniation.

External distribution is limited and controlled

by Head, Hydromechanics Laboratory.

May 1966 . Hydromechanics Laboratory

.. Technical Note 47

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The heave damping coefficients of several bulbous cylinders

are computed, exhibiting zero damping at: some frequencies.

It has been shown by Ursell [1], Porter [2] and others that

semi-submerged circular, elliptic and Lewis-form cylinders give rise to non-zero

damping forces for heaving oscillation at finite frequencies in deep water.

Motora and Koyaina [3] have measured the heave exciting forces on circular and elliptic cylinders with vertical struts in regu),ar waves. Their results

indicate the existence of almost vanishing minimum forces for some of their

test models. Newman [4] has shown that for finite wave'numbers the heave

dmping coefficient is proportional to the-square of the heave -exciting

force Motora and Koyama, therefore, conjectured that corresponding to

these minimum exciting forces on the bulbous forms tested, the damping

coefficients for the respective wave numbers must be practically zero. it

is the purpose of thIs paper to show by direct computation that the damping

coefficients do in fact go to zero for those bulbous shapes at these wave

numbers.

Two dimensional waterwave problems for cylinders of arbitrary shape,

may be so].vd in the following manner. The cross section of the cylinder is

defined bya finite number of offset points. The velocity potential is

represented by a distribution of wave sources over the boundary of the sub

merged part of the crosssection, the varying strergth of which is an

unknown function to be determined from an integral equation found by applying

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equation it is assumed that the source strength _{is constant over discrete}

small elements of the boundary curve, and the resulting matrix of influice

coefficients Is inverted. This method is applicab.è to a large variety of

shapes - even those with severe corners, bulbs apd bilge keels -. and becomes

more accurate as the number of offset points is increased. The method. will

be more fully described In a future publication.

The IBM 7090 computer program developed for this problem is limited to

46 offset points and 50 wave numbers. The output includes the added mass

and damping coefficients and the pressures in phase with the acceleration

and the velocity. The computer time for an Input of 21 offset points is

app]dxImately 20 seconds per frequency.

FIgures 1 through 4 exhibit the added mass and damping coefficiints -together with the geometry of the cross-seCtins _{- of twelve circular}

cylinders with vertical struts. _{The indicated dimensions of the water lire}

beams and diameters of the cross-sections are relative to a draft of one.

Figures 1, 2, 3 and 4 represent shapes with beam to diameter ratios of 1,

3/4, 1/2 and 1/4, respectively; hi1e parts. A, B and C of these figures

depict diameter to draft ratios of 1, 4/5 and 2/3, respectively.

The added mass coefficients - represented by the solid curves - are

non-dimensionalized by , while the damping coefficients - the dashed

cues - are divided by rp1w , where T is

the

draft, p is the density of water and w the circular frequency of oscillation. The abscissae are

wT/g

, g being the acceleration of gravity; i.e., the wave numbers are

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non-ditnensionalized with respt to the draft.

Except for the non-bMlbous cylinders of Figure 1, all the damping

curves vanish for some wave number. _{In Figure 2 the zeros of the damping}

coefficients are located between the _{wave numbers l.-0 and 2.0, in Figure 3}

the zeros are in the range of wave numbers between 0.5 and 1.0, and _Figure 4 shows the zeros to be between _{0.2 and 0.4.}

The author thanks Drs. T. F. _{Ogilvie, 3. N. Newman and B. 0.}

Tuck for valuable suggestIons.

BIBLIOGRAPHY

Cl] F... tlrsell, ttOn the Heaving Notion _{ofa Circular Cylinder}

on the Surface of a Fluid," Quart. 3, Mech. Appi, _{Math,, 2, 1949.}

W. R. Porter, "Pressure Distributions,

Added-Mass, and Damping

Coeffici-ents for Cylinders Oscillating in a Free Surface," University of

California, Institute of Engineering _{Research, Berkeley; Calif.,} _Series

82, Issue 16, July 1960.

S. Motora and T. Koyama, "On Wave-Excitatjon Free Ship Forms," Journal

of.Zosen Kiokai, (The Society of Naval Architects of Jpan) Vol, 117,

June 1965.

J. N. Newman, "The Exciting Forces _{on Fixed Bodies in Waves," Journal}

of Ship Research, Vol. 6, 1962.

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0.50 0.25 0.50 0.25 0.50 0.25 0 WAVE NUMBER I '4 .4 .4 .4 .4 0.50 100 1.50 FIGURE 1A 0 0.50 .4 .4 "4% DAMPING 1.00 FIGURE 18

4*-ADDED MASS -

_I_

-2.0 I

P..

-0.50 0.25 0.50 0.25 I

i--0.50 1.00

10200 0

FIGURE 1C .4 .4 .4 .4 0.50 1.00 FIGURE 2A

I-.

--.

---..---0.50 1.00 FIGURE 2B 1.00 FIGURE 2C 1.50

j---1.50 2.00 2.00

Figure 2 - Beam/Diameter

- 3/4

Figure 1

_{- Beam/Diameter}

₁

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0.50 0.25 -o:so 0.25 0

--I

0.50

___1--1.o0 -FIGURE 3B F-2/54 1.50

Figure 3 - Beam/Diameter

- 1/2

5 0.50 1.00 FIGURE 4A 1.00 FIGURE 4B 1.50 1.50 2.00 0.50 1.00 FIGURE 3C 1.50 1.00 FIGURE 4C

Figure 4

- Beam/Dianieer

1/4

0.50 i.00 FIGURE 3A