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16 H.AKYILDIZ

F  g  d V  2Ω  V  d Ω  r  Ω  Ω  r (3)

dt dt

where g, V and Ω are the gravitational vector, the translational velocity and the rotational
velocity vector. In addition, r is the position vector of the considered point relative to O. On
the free surface, both the kinematics and dynamic conditions should be satisfied:

  U  (  z)  0 (4)
t

P = Patm (5)

where  represents the free surface profile and Patm is the air pressure or ullage pressure inside
the tank. The surface tension is ignored in this study. Therefore, a no-shear is needed on the
free surface. But, proper wall conditions are necessary on the tank walls and the internal
members.

2.1 Numerical Computation

For the analysis of the sloshing flow inside a partial filled tank, a finite difference method is
applied to the governing equations. A FDM (finite difference method) is useful when there are
internal structures inside the tank or the fluid contacts the tank ceiling frequently. As the
internal structures exist, the viscous effects may be dominant. In this study, the method
concentrates on the global fluid motion, so some local effects, such as turbulence and wave
breaking have been ignored. In some cases, these local effects are important, but the simulation
of global flow plays a more critical role in many sloshing problems, due to the slosh-induced
moment in the ship cargo (Kim, 2001).

The scheme adopted in this study is the SOLA method (Hirt and Nichols, 1981). Tank volume
is discredited into Cartesian staggered grid cells. A single layer of fictitious cells (or boundary
cells) surrounds the fluid region. The fictitious cells are used to set the boundary conditions so
that the same difference equation can be used in the interior of the mesh (Lee et al., 2007; Liu
and Lin, 2009; Eswaran et al., 2009).

Fluid velocities are located at the centers of the cell boundaries and pressure (P) and the volume

of fluid function (F) are computed at the center of the cell. The solution algorithm works as a
time cycle or ‘movie frame’. The results of the time cycle act as initial conditions for the next

one. At each step, suitable boundary conditions must be imposed at all boundaries.

There are two alternatives for the wall conditions; when the viscosity effect on the tank
boundary is significant, the no-slip condition should be imposed. However, in most sloshing
problems, the viscous effect is not significant and the boundary layer thickness is much less than
the cell size. Therefore, the free slip condition is applied in the present study. For example, if
the left boundary of the computing mesh is to be a rigid free slip wall, the normal velocity will
be zero and the tangential velocity should have no normal gradients, i.e.

GiDB|DERGi Sayı 1, 2014