Nv ⎛ ⎞ 

The roll damping coefficient of conventional three- dimensional hulls, with the assumption of the strip the- ory (Newman 1977), can be obtained by integrationof

W (S )    iv ln⎜ S  Sv ⎟   m() lnS  d

the sectional damping coefficients along the ship length。

The first term on the r。h。s。 of Eq。 7 refers to an array of

_

Nv, and their images located at Sv and Sv, respectively。 The second term refers to the source distribution of strength m()。 The infinite integral associated with the source distribution occurs because the transformed computational domain is periodic。

The solution proceeds by an apparent time solution。 At each time step the vortices are convected away by the cross flow computed from the complex velocity

dW (s) v iw ds

3。

Preliminary Calculations of Barge Sections

In the early phases of its development, the model ap- plied to a barge section with beam to draught ratio of

2。5。 Two-dimensional barge section of the experimental case (Ikeda et al 1977) was computed with 41 nodes girthwise and 25 nodes in radial direction for the roll center CR = 0。0。 Two vortices were introduced at each time step to satisfy the Kutta condition at shedding edges。 The Kutta condition ensures smooth separation at a shedding edge, and is enforced by inducing zero ve-

locity in the working plane at the points corresponding to the shedding edges。

The results are given after theend

Park (2001) describes the method based on the Cloud- in-cell technique in detail。

2。3 Calculation of Forces

Under the assumption that the responses are linear and harmonic, the equation of roll motion of a floating body can be expressed as

of one roll cycle at T/Tp = 1。0 for an amplitude of 0。175 radians and frequency of 6。283 rad/sec。 The figure shows that at least two sets of vortices separate from the sharp bilge corners after completion of one roll cycle。 The flow visualization experiment by Cozen (1987) also showed the existence of these vortices。

Fig。 5 shows the flow visualization of a forced roll ex- periment carried out on a model barge, which is taken

 

(M 44 A44 )  B44   C44  F4

B44 BWBF BV

fromDownieetal(1988)。Theexperimental figure

clearly showed the one pare of shedding vortices under thebargesection。Thesimilarflowpatternsare also

The force coefficients of mass M44, added mass A44, wave damping BW, restoring C44 and the exitingforces F4 can be calculated using a potential based panel method。 is the amplitude of response in roll motion。 The frictional damping BF can be estimated by conven- tional ideas of boundary layer theory (Himeno, 1981)。

The vortex induced viscous roll damping coefficient Bv, which is difficult to estimate by conventional potential methods, is calculated by the developed discrete vortex method。 The vortex-induced moment on the rolling hull is calculated from integration of the pressure P over the hull section H usingshown in the present study。

n is the unit normal into the hull section and Rc is the length between roll center and hull surface。 The pres- sure distribution on the hull surface is found from un- steady Bernoulli’s equation as

q is the hull surface velocity and is the disturbance velocity potential, which is the real part of the complex potential。 The two-dimensional roll damping coefficient

Fig。 4: vortex evolution of the barge section

(Frequency = 6。238rad/sec, Amplitude =0。175)

In the present calculation, there are no clear vortex interactions between two edges but with decreasing of the beam to draught ratio, the separated vortices from each edge will affect each other。 If the ratio is very small, the flow pattern will be more like the oscillating plate with one edge separation