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Effects of excitation angle and coupled heave–surge–sway motion on fluid sloshing in a three-dimensional tank

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Abstract

Sloshing waves in moving tanks is an important engineering problem, and most studies of this phenomenon have focused on tanks that are excited by forcing motion in a limited number of directions and with fixed excitation frequencies throughout the forcing. In practice, the excitation comprises multiple degree of freedom motion that potentially couples surge, sway, heave, pitch, roll, and yaw motions. In the present study, a time-independent finite difference method is used to simulate fluid sloshing in three-dimensional tanks filled to an arbitrary depth for various excitation frequencies and multiple degree of freedom motion. The numerical scheme developed here was verified by rigorous benchmark tests. The coupled motions of surge and sway are simulated for various excitation angles, frequencies and water depths. Five kinds of sloshing waves found under coupled surge–sway motions: diagonal, single-directional, square-like, swirling, and irregular waves. The effect of excitation angle on the frequency responses of different sloshing waves is analyzed and discussed in the present study. Further, the components of horizontal force of various sloshing waves are also presented. The coupled effect of surge, sway and heave motions is also discussed, and the results show that unstable sloshing occurs when the excitation frequency of the heave motion is twice the fundamental natural frequency. Moreover, the effects of heave motion on the different types of sloshing waves are explored. It is found that heave motion causes all of the sloshing waves to change type.

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Acknowledgments

This study is partially supported by the National Science Council under the grants NSC 96-2221-E-110-107-MY3 and NSC 98-3114-P-110-001, and is also funded by the Ministry of Education under the grant Aim for the Top University Plan. The second author also would like to thank Prof. Faltinsen for providing valued supervision during his visit to the Norwegian University of Science and Technology, Norway.

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Authors and Affiliations

  1. Department of Marine Environment and Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan

    Bang-Fuh Chen & Chih-Hua Wu

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  1. Bang-Fuh Chen
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  2. Chih-Hua Wu
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Correspondence to Bang-Fuh Chen.

Appendices

Appendix 1

The dimensionless momentum equations can be written as

$$ {\frac{\partial V}{\partial T}} + C_{11} C_{14} {\frac{\partial V}{\partial Y}} + C_{1} C_{13} U{\frac{\partial V}{\partial X}} + C_{2} C_{14} U{\frac{\partial V}{\partial Y}} + C_{5} C_{14} V{\frac{\partial V}{\partial Y}} + C_{8} C_{14} W{\frac{\partial V}{\partial Y}} + C_{9} C_{15} W{\frac{\partial V}{\partial Z}} = - g/g - \left( {C_{5} C_{14} {\frac{\partial P}{\partial Y}}} \right) - \ddot{Y}_{CTT} $$
(25)
$$ \begin{aligned} & {\frac{\partial W}{\partial T}} + \left( {C_{11} C_{14} {\frac{\partial W}{\partial Y}}} \right) + C_{1} C_{13} U{\frac{\partial W}{\partial X}} + C_{2} C_{14} U{\frac{\partial W}{\partial Y}} + C_{5} C_{14} V{\frac{\partial W}{\partial Y}} + C_{8} C_{14} W{\frac{\partial W}{\partial Y}} + C_{9} C_{15} W{\frac{\partial W}{\partial Z}} \\ & \quad = - \left( {C_{8} C_{14} {\frac{\partial P}{\partial Y}} + C_{9} C_{15} {\frac{\partial P}{\partial Z}}} \right) - \ddot{Z}_{CTT} . \\ \end{aligned} $$
(26)

Continuity equation:

$$ C_{1} C_{13} {\frac{\partial U}{\partial X}} + C_{2} C_{14} {\frac{\partial U}{\partial Y}} + C_{5} C_{14} {\frac{\partial V}{\partial Y}} + C_{8} C_{14} {\frac{\partial W}{\partial Y}} + C_{9} C_{15} {\frac{\partial W}{\partial Z}} = 0. $$
(27)

Kinematic free surface condition:

$$ {\frac{\partial H}{\partial T}} + UC_{1} C_{13} {\frac{\partial H}{\partial X}} + WC_{9} C_{15} {\frac{\partial H}{\partial Z}} = V. $$
(28)

Poisson equation:

$$ \begin{aligned} & {(C_{1}^{2} C_{13}^{2} ){\frac{{\partial^{2} P}}{{\partial X^{2} }}} + \left( {C_{2}^{2} C_{13}^{2} + C_{5}^{2} C_{14}^{2} + C_{8}^{2} C_{15}^{2} } \right){\frac{{\partial^{2} P}}{{\partial Y^{2} }}}} \\ & \quad + (C_{9}^{2} C_{15}^{2} ){\frac{{\partial^{2} P}}{{\partial Z^{2} }}} + 2(C_{1} C_{2} C_{13} C_{14} ){\frac{{\partial^{2} P}}{\partial X\partial Y}} + 2(C_{8} C_{9} C_{14} C_{15} ){\frac{{\partial^{2} P}}{\partial Y\partial Z}} = - \Upomega \\ \end{aligned} $$
(29)

where

$$ \begin{aligned} \Upomega & = \left [ U(C_{1}^{2} C_{13}^{2} U_{XX} + C_{2}^{2} C_{14}^{2} U_{YY} + 2C_{1} C_{2} C_{13} C_{14} U_{XY} \right. \\ & \quad + C_{2} C_{5} C_{14}^{2} V_{YY} + C_{1} C_{5} C_{13} C_{14} V_{XY} + C_{2} C_{8} C_{14}^{2} W_{YY} + C_{1} C_{8} C_{13} C_{14} W_{XY} \\ & \quad + C_{1} C_{9} C_{13} C_{15} W_{XZ} + C_{2} C_{9} C_{14} C_{15} W_{YZ} ) + V(C_{2} C_{5} C_{14}^{2} U_{YY} U_{ZZ} + C_{1} C_{5} C_{13} C_{14} U_{XY} \\ & \quad + C_{5}^{2} C_{14}^{2} V_{YY} + C_{5} C_{8} C_{14}^{2} W_{YY} + C_{5} C_{9} C_{14} C_{15} W_{YZ} ) + W(C_{2} C_{8} C_{14}^{2} U_{YY} + C_{1} C_{8} C_{13} C_{14} U_{XY} + C_{1} C_{9} \\ & \quad C_{13} C_{15} U_{XZ} + C_{2} C_{9} C_{14} C_{15} U_{YZ} + C_{5} C_{8} C_{14}^{2} V_{YY} + C_{5} C_{9} C_{14} C_{15} V_{YZ} + C_{8}^{2} C_{14}^{2} W_{YY} + C_{9}^{2} C_{15}^{2} W_{ZZ} \\ & \quad + 2C_{8} C_{9} C_{14} C_{15} W_{YZ} )] + [ (C_{1} C_{13} U_{X} + C_{2} C_{14} U_{Y} )^{2} + (C_{5} C_{14} V_{Y} )^{2} \\ & \quad + (C_{8} C_{14} W_{Y} + C_{9} C_{15} W_{Z} )^{2} + 2(C_{1} C_{13} V_{X} + C_{2} C_{14} V_{Y} ) \times (C_{5} C_{14} U_{Y} ) + 2(C_{1} C_{13} W_{X} + C_{2} C_{14} W_{Y} ) \times \\ & \quad \left. (C_{8} C_{14} U_{Y} + C_{9} C_{15} U_{Z} ) + 2(C_{5} C_{14} W_{Y} ) \times (C_{8} C_{14} V_{Y} + C_{9} C_{15} V_{Z} ) \right ] . \\ \end{aligned} $$
(30)

The finite-difference expressions of \( \wp_{i,j,k} ,\Re_{i,j,k} ,\;\aleph_{i,j,k} \) and \( \Im_{i,j,k} \) in Eqs. 17–20 are listed as follows:

$$ \begin{aligned} \wp_{i,j,k} & = - (C_{11} C_{14} U_{Y} + C_{1} C_{13} UU_{X} + C_{2} C_{14} UU_{Y} + C_{5} C_{14} VU_{Y} + C_{8} C_{14} WU_{Y} + C_{9} C_{15} WU_{z} ) - \ddot{X}_{CTT} \\ & = - \left( {C_{11} C_{14} {\frac{{U_{i,j + 1,k} - U_{i,j - 1,k} }}{2\Updelta Y}} + C_{1} C_{13} U{\frac{{U_{i + 1,j,k} - U_{i - 1,j,k} }}{2\Updelta X}} + C_{2} C_{14} U{\frac{{U_{i,j + 1,k} - U_{i,j - 1,k} }}{2\Updelta Y}}} \right. \\ & \quad \left. { + C_{5} C_{14} V{\frac{{U_{i,j + 1,k} - U_{i,j - 1,k} }}{2\Updelta Y}} + C_{8} C_{14} W{\frac{{U_{i,j + 1,k} - U_{i,j - 1,k} }}{2\Updelta Y}} + C_{9} C_{15} W{\frac{{U_{i,j,k + 1} - U_{i,j,k - 1} }}{2\Updelta Z}}} \right) - \ddot{X}_{CTT} \\ \end{aligned} $$
(31)
$$ \begin{aligned} \Re_{i,j,k} & = - \left( {C_{11} C_{14} {\frac{\partial V}{\partial Y}} + C_{1} C_{13} U{\frac{\partial V}{\partial X}} + C_{2} C_{14} U{\frac{\partial V}{\partial Y}} + C_{5} C_{14} V{\frac{\partial V}{\partial Y}} + C_{8} C_{14} W{\frac{\partial V}{\partial Y}} + C_{9} C_{15} W{\frac{\partial V}{\partial Z}}} \right) - \ddot{Y}_{CTT} \\ & = - \left( {C_{11} C_{14} {\frac{{V_{i,j + 1,k} - V_{i,j - 1,k} }}{2\Updelta Y}} + C_{1} C_{13} U{\frac{{V_{i + 1,j,k} - V_{i - 1,j,k} }}{2\Updelta X}} + C_{2} C_{14} U{\frac{{V_{i,j + 1,k} - V_{i,j - 1,k} }}{2\Updelta Y}}} \right. \\ & \quad \left. { + C_{5} C_{14} V{\frac{{V_{i,j + 1,k} - V_{i,j - 1,k} }}{2\Updelta Y}} + C_{8} C_{14} W{\frac{{V_{i,j + 1,k} - V_{i,j - 1,k} }}{2\Updelta Y}} + C_{9} C_{15} W{\frac{{V_{i,j,k + 1} - V_{i,j,k - 1} }}{2\Updelta Z}}} \right) - \ddot{Y}_{CTT} \\ \end{aligned} $$
(32)
$$ \begin{aligned} \aleph_{i,j,k} & = - \left( {C_{11} C_{14} {\frac{\partial W}{\partial Y}} + C_{1} C_{13} U{\frac{\partial W}{\partial X}} + C_{2} C_{14} U{\frac{\partial W}{\partial Y}} + C_{5} C_{14} V{\frac{\partial W}{\partial Y}} + C_{8} C_{14} W{\frac{\partial W}{\partial Y}} + C_{9} C_{15} W{\frac{\partial W}{\partial Z}}} \right) - \ddot{Z}_{CTT} \\ & = - \left( {C_{11} C_{14} {\frac{{W_{i,j + 1,k} - W_{i,j - 1,k} }}{2\Updelta Y}} + C_{1} C_{13} U{\frac{{W_{i + 1,j,k} - W_{i - 1,j,k} }}{2\Updelta X}} + C_{2} C_{14} U{\frac{{W_{i,j + 1,k} - W_{i,j - 1,k} }}{2\Updelta Y}}} \right. \\ & \quad \left. { + C_{5} C_{14} V{\frac{{W_{i,j + 1,k} - W_{i,j - 1,k} }}{2\Updelta Y}} + C_{8} C_{14} W{\frac{{W_{i,j + 1,k} - W_{i,j - 1,k} }}{2\Updelta Y}} + C_{9} C_{15} W{\frac{{W_{i,j,k + 1} - W_{i,j,k - 1} }}{2\Updelta Z}}} \right) - \ddot{Z}_{CTT} \\ \end{aligned} $$
(33)
$$ \Im_{i,j,k} = UC_{1} C_{13} {\frac{\partial H}{\partial X}} + WC_{9} C_{15} {\frac{\partial H}{\partial Z}} = UC_{1} C_{13} {\frac{{H_{i + 1,j,k} - H_{i - 1,j,k} }}{2\Updelta X}} + WC_{9} C_{15} {\frac{{H_{i,j,k + 1} - H_{i,j,k - 1} }}{2\Updelta Z}}. $$
(34)

Note that if the finite-difference expression is applied near the boundaries, the forward or backward finite-difference expression is implemented in Eqs. 31–34.

Appendix 2: The coefficients due to coordinate transformation

$$ \begin{aligned} C_{1} & = d_{0} {\frac{{\partial x^{*} }}{\partial x}} = {\frac{{d_{0} }}{{b_{2} - b_{1} }}} \\ C_{2} & = d_{0} {\frac{{\partial y^{*} }}{\partial x}} = {\frac{1}{{h^{2} }}}(y + d_{0} ){\frac{\partial h}{\partial x}} = {\frac{1}{{H^{2} }}}(y + d_{0} ){\frac{\partial H}{\partial x}} \\ C_{3} & = d_{0} {\frac{{\partial z^{*} }}{\partial x}} = 0 \\ C_{4} & = - d_{0} {\frac{{\partial x^{*} }}{\partial y}} = 0 \\ C_{5} & = - d_{0} {\frac{{\partial y^{*} }}{\partial y}} = {\frac{{d_{0} }}{h}} = \frac{1}{H} \\ C_{6} & = - d_{0} {\frac{{\partial z^{*} }}{\partial y}} = 0 \\ C_{7} & = d_{0} {\frac{{\partial x^{*} }}{\partial z}} = 0 \\ C_{8} & = d_{0} {\frac{{\partial y^{*} }}{\partial z}} = {\frac{1}{{h^{2} }}}(y + d_{0} ){\frac{\partial h}{\partial z}} = {\frac{1}{{H^{2} }}}(y + d_{0} ){\frac{\partial H}{\partial z}} \\ C_{9} & = d_{0} {\frac{{\partial z^{*} }}{\partial z}} = {\frac{{d_{0} }}{{b_{4} - b_{3} }}} \\ C_{10} & = \sqrt \frac{{d_{0} }}{g} \frac{{\partial x^{*} }}{\partial t} = 0 \\ C_{11} & = \sqrt \frac{{d_{0} }}{g} \frac{{\partial y^{*} }}{\partial t} = \sqrt \frac{{d_{0} }}{g} \frac{1}{{h^{2} }}(y + d_{0} )\frac{\partial h}{\partial t} = \sqrt \frac{{d_{0} }}{g} \frac{1}{{H^{2} }}(y + d_{0} )\frac{\partial H}{\partial t} \\ C_{12} & = \sqrt \frac{{d_{0} }}{g} \frac{{\partial z^{*} }}{\partial t} = 0 \\ C_{13} & = {\frac{\partial X}{{\partial x^{*} }}} = [1 + k_{1} (x^{*} - \lambda_{1} )(2x^{*} - 1)]{\text{e}}^{{k_{1} x^{*} (x^{*} - 1)}} \\ C_{14} & = {\frac{\partial Y}{{\partial y^{*} }}} = [1 + k_{2} (y^{*} - \lambda_{2} )(2y^{*} - 1)]{\text{e}}^{{k_{2} y^{*} (y^{*} - 1)}} \\ C_{15} & = {\frac{\partial Z}{{\partial z^{*} }}} = [1 + k_{3} (z^{*} - \lambda_{3} )(2z^{*} - 1)]{\text{e}}^{{k_{3} z^{*} (z^{*} - 1)}} \\ \end{aligned} $$

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Chen, BF., Wu, CH. Effects of excitation angle and coupled heave–surge–sway motion on fluid sloshing in a three-dimensional tank. J Mar Sci Technol 16, 22–50 (2011). https://doi.org/10.1007/s00773-010-0111-0

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