Each hull section and associated mesh in the Z-plane (Z

= y + iz) is mapped into the computational plane, the S- plane (S = +i), by a two-stage conformal transforma- tion as seen in Fig。1。 First, the transformation maps the ship transverse section in the complex Z-plane to the

the complexity of its shape。 The first three terms only, for example, lead to the Lewis forms。 For the realhull as the Series 60 hull, eleven terms (N = 10) have been used in the series approximating the real geometry well, Fig。2。

2。2Numerical Model for Roll Motion

At each time, the roll motion of the ship section can be represented by a distribution of sources as seen in Fig。3。 The rate of sectional hull surface change in roll motion can be described as follows,

exterior of a circle in the S1-plane by means ofa Laurent series。 The circle in the S1-plane is then trans-

where CR is roll center, is frequency and Ris the roll

amplitude。 The normal velocity on the body in Z-plane is due to the rate of the change of body section, the roll motion as time calculation progresses。 The source strength at the point = 0 in the transformed S-plane can be specified by satisfying the impermeablebound-

r is theradius ofacylinder, =/(nk –1)and=

2/R in the S-plane, in which the computational mesh is rectangular and periodicand of length R。

ary condition on the body surface。 The relationship between the normal velocity Vn on the hullsurfacein the Z-plane and the source strength m() in the S-plane is

∂Z/∂S is the derivative of transformation and overbar indicate a complex conjugate。

Fig。 1:Two Stage Conformal Mapping

Fig。 2: “ Series 60 hull”: Original and conformal mapping

The working plane has the advantage of being both rectangular and invariant for all of the hull sections。

The coefficients an in Eq。3 are determined by the hull geometry, the number of terms required dependingon

-0。05 0 0。05

Y

Fig。 3:Representation of roll motion for barge section

As the time calculation progresses, vortices are intro- duced around sectional hull surface。 If the hull section has well defined shedding edges, as for barge sections, the single vortices are introduced at the points in the working plane corresponding to the shedding edges in the Z-plane so as to satisfy the Kutta condition at each time step。 In the case of the sections with continuous surface, with no well-defined separation points, as for the Series 60, a continuous layer of discrete vortices are distributed around section so as to satisfy a pointwise no-slip condition on the hull surface, and no separation point is specified。 These approach is not intended tobe a detailed model of the boundary separation, of the boundary layer itself, but has been found to reproduce the gross features of the flow reasonably well in two- dimensional bluff body flows (Downie et al, 1989)。

The impermeable condition is satisfied by the simulta- neous introduction of image vortices, which are located in the interior of the hull section, for every vortex intro-

duced。 In the transformed S-plane, the image vortices are introduced at the conjugate points of the realvortex

Bv induced by vortex shedding is related to the vortex moment by

position。 The hull section is modeled as a doublebody 2 T

and each vortex has a reflected and a periodic image。 The sources are lie on the real axis of the computational

BVT Mv (t) sin(t)dt

plane and are also periodic。 The complex potential in the working plane is given by: