Liquid sloshing in an upright circular tank under periodic and transient excitations

In this paper, the liquid sloshing problem in an upright circular tank undergoing an oscillation in a single degree of freedom is considered. A fully nonlinear time-domain harmonic polynomial cell (HPC) method based on overset mesh is developed to delve into the flow physics, and the comparison is made with the weakly nonlinear multimodal theory. Both time-harmonic (or periodic) and transient-type oscillations are considered. For the time-harmonic oscillation, planar and swirling waves (including time-harmonic and periodically modulated swirling waves) occur when the excitation frequency is close to the lowest natural frequency. The swirling direction is dependent on initial conditions. The periodically modulated swirling waves switch back and forth between swirling waves and planar waves. The occurrence of swirling waves results in lateral hydrodynamic force and roll moment on top of in-line force and pitch moment components acting on the tank. The sloshing response in terms of surface elevations contains higher harmonic components, while the hydrodynamic forces and moments on the tank are remarkably linear. A NewWave type of excitation is imposed which represents on average the most probable maximum excitation to be expected for a given sea state, and narrow-banded sloshing responses are observed. This allows application of a designer-wave type of excitation which would excite the most probable maximum response in sloshing. A focused-wave-type excitation is also considered, and an impulselike feature is observed in the in-line force and pitch moment components. The total in-line force and pitch moment are further decomposed into inertial component, which is induced by the acceleration of the external motion, and residual component, which is closely associated with the liquid sloshing. It is found that under a focused-wave-type excitation the inertial component contribution is more dominant, while under a designer-wave-type excitation the residual component dominates instead.

Figure 1

Definition of dimension and coordinate system of an upright circular tank.

Figure 2

A hexahedral cell with 27 nodes with the 27th node defined as the stencil center.

Figure 3

Interpolation of the node on the fringe of a circular grid system.

Figure 4

The overset mesh in the upright circular tank.

Figure 5

Schematic of ghost nodes outside of the computational domain.

Figure 6

Time histories of the free-surface elevation ${\zeta }_1$ (top), in-line force $f_x$ (middle), and pitch moment $m_y$ (bottom) for three different grid resolutions at $\omega ={\omega }_{1,1}$.

Figure 7

Time history of a periodic type of excitation (a) and the resulting responses in terms of surface elevation (b), normalized hydrodynamic force (c) and moment (d) on the tank determined by the fully nonlinear HPC method for $\omega =0.98{\omega }_{1,1}$.

Figure 8

The top and middle panels show the time histories of the envelopes of ${\zeta }_1$ and ${\zeta }_2$, respectively. The present fully nonlinear HPC results (red) are in comparison with the results obtained from the weakly nonlinear multimodal theory (black) and the linear modal theory (blue). The bottom panel displays the phase plot defined as the trajectory of ${\zeta }_1$ and ${\zeta }_2$ from $t=200$ s to $t=400$ s obtained from HPC (left) and the multimodal theory (right). All results are for $\omega =0.98{\omega }_{1,1}$.

Figure 9

Same as Fig. 8 but for $\omega =0.99{\omega }_{1,1}$.

Figure 10

Same as Fig. 8 but for $\omega ={\omega }_{1,1}$.

Figure 11

Same as Fig. 8 but for $\omega =1.01{\omega }_{1,1}$.

Figure 12

Same as Fig. 8 but for $\omega =1.02{\omega }_{1,1}$.

Figure 13

Time histories of ${\zeta }_1$ and ${\zeta }_2$ for $\omega ={\omega }_{1,1}$ in the steady state. The present fully nonlinear HPC results (red) are in comparison with the results determined by the weakly nonlinear multimodal theory (black).

Figure 14

Time history of the normalized hydrodynamic forces due to sloshing ($f_x=$ in-line force by gray line; $f_y=$ lateral force by red line. (a) $\omega =0.98{\omega }_{1,1}$, (b) $\omega =0.99{\omega }_{1,1}$, (c) $\omega =1.00{\omega }_{1,1}$, (d) $\omega =1.01{\omega }_{1,1}$, and (e) $\omega =1.02{\omega }_{1,1}$.

Figure 15

Time history of the normalized hydrodynamic moments due to sloshing ($m_x=$ roll moment by gray line; $m_y=$ pitch moment by red line). (a) $\omega =0.98{\omega }_{1,1}$, (b) $\omega =0.99{\omega }_{1,1}$, (c) $\omega =1.00{\omega }_{1,1}$, (d) $\omega =1.01{\omega }_{1,1}$, and (e) $\omega =1.02{\omega }_{1,1}$.

Figure 16

Comparison of ${\zeta }_1$ at $\omega =0.99{\omega }_{1,1}$ between HPC (top) and multimodal theory (bottom) in terms of the harmonic structure: total signal (gray), first harmonic (red), and second harmonic (difference and sum terms, black).

Figure 17

Comparison of ${\zeta }_2$ at $\omega =0.99{\omega }_{1,1}$ between HPC (top) and multimodal theory (bottom) in terms of the harmonic structure: total signal (gray), first harmonic (red), and second harmonic (difference and sum terms, black).

Figure 18

Individual phases of the first harmonic components of ${\zeta }_1$ and ${\zeta }_2$ (top) as well as the phase difference (bottom) as a function of time at $\omega =0.99{\omega }_{1,1}$. ${\zeta }_1^{\left(\omega \right)}=$ first harmonic component of ${\zeta }_1$; ${\zeta }_2^{\left(\omega \right)}=$ first harmonic component of ${\zeta }_2$; ${\phi }_1^{\left(\omega \right)}=$ phase angle of ${\zeta }_1^{\left(\omega \right)}$; ${\phi }_2^{\left(\omega \right)}=$ phase angle of ${\zeta }_2^{\left(\omega \right)}$.

Figure 19

Same as Fig. 18 but for the second harmonic components. ${\phi }_1^{\left(2\omega \right)}=$ phase angle of ${\zeta }_1^{\left(2\omega \right)}$; ${\phi }_2^{\left(2\omega \right)}=$ phase angle of ${\zeta }_2^{\left(2\omega \right)}$.

Figure 20

Same as the bottom subplot of Fig. 18 but with the initial condition (24), respectively.

Figure 21

Individual phases of the first harmonic components of ${\zeta }_1$ and ${\zeta }_2$ (top) as well as the phase difference (bottom) as a function of time at $\omega ={\omega }_{1,1}$. ${\zeta }_1^{\left(\omega \right)}=$ first harmonic component of ${\zeta }_1$; ${\zeta }_2^{\left(\omega \right)}=$ first harmonic component of ${\zeta }_2$; ${\phi }_1^{\left(\omega \right)}=$ phase angle of ${\zeta }_1^{\left(\omega \right)}$; ${\phi }_2^{\left(\omega \right)}=$ phase angle of ${\zeta }_2^{\left(\omega \right)}$.

Figure 22

Same as the bottom subplot of Fig. 21 but with the initial condition (24).

Figure 23

Same as the bottom subplot of Fig. 21 but for $\omega =1.01{\omega }_{1,1}$.

Figure 24

Time history of a NewWave type of excitation (a) and the resulting responses in terms of surface elevation (b), hydrodynamic forces (c) and moments (d) on the tank.

Figure 25

Comparison of RAO derived from the NewWave type of excitation (solid line) and the periodic type of excitation (squares).

Figure 26

Time history of a designer-wave-type excitation (a) and the resulting responses in terms of surface elevation (b), hydrodynamic force (c) and moment (d) on the tank.

Figure 27

Time history of a crest-focused type of excitation (a) and the resulting responses in terms of surface elevation (b), hydrodynamic force (c) and moment (d) on the tank.

Figure 28

Time history of excitation and sloshing response in ${\zeta }_1$ (top) and decomposed hydrodynamic force and moment into inertial and residual components (middle and bottom) for focused-wave-type excitation.

Figure 29

Same as Fig. 28 but for NewWave-type excitation.

Figure 30

Same as Fig. 28 but for designer-wave-type excitation.