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18 H.AKYILDIZ

thin enough. The height of the baffle (hB) is established by the ratio to liquid depth which varies
from 0.2 to 1.0. The pressure transducers are installed on the left side in the center plane of the
beam and one location on the top wall.

Present numerical code is set up to handle a simple harmonic forcing function. Thereafter, it
advances the velocities in time explicitly using the two momentum equations. First, the angular
displacement and its derivatives are calculated. The apparent acceleration terms are then
calculated and finally the advective, diffusion and pressure gradients terms are calculated
yielding an estimate of the velocity at the new time level. The tank motion is the pitch
oscillations about y-axis only which follows the sinusoidal function given as where θ0 and ω are
the rolling amplitude and the frequency, respectively. The rolling amplitude is chosen as 40 and
80 in this study.

In order to testify and verify the discretization of the numerical model, three different grid
systems are chosen for various fill depth (16×31, 23×31 and 46×31) for the liquid fills 50% and
75% of the tank height. Generally, the choice of ‘Δt’ is of extreme importance. In explicit
schemes, ‘Δt’ will govern the stability and also the accuracy, while in implicit schemes it will
affect the accuracy. In this study, the value of ‘Δt’ can be automatically calculated by the
program and dynamically modified to insure stability and also optimize the pressure solution.
When the excitation is harmonic rolling, it has been found that the normalized time step is
somewhat independent of the forcing period. Therefore, about 200 time steps are required per
forcing period in this program. This seems to hold regardless of discretization. A relatively fine
discretization (46×31) with a finer enforcement of the velocity divergence requires three
seconds of computer time per time step comparing a coarse discretization (16×31). Therefore,
(46×31) grid system is chosen to obtain results in a reasonable time considering stability and
accuracy. Furthermore, as the liquid responds violently increasing the period and the amplitude
of the excitation, the numerical solution becomes unstable. So, for the sake of avoidance of the
instability, the rolling amplitude is chosen as 40 to testify the grid dependence.

Fig. 2 and Fig. 3 show the reasonable agreement of the time variations of the pressure for
different grid systems at T1 (50% liquid fill) and T3 (75% liquid fill). To estimate the limited
impact pressure on the tank top and to demonstrate the capability of the numerical code in
computing impact-type loads, the liquid sloshing at 75% fill depth with the rolling amplitude 80
are chosen for all cases.

GiDB|DERGi Sayı 1, 2014