﻿ 基于完全非线性边界元方法的楔形液舱入水问题研究
 舰船科学技术  2001, Vol. 44 Issue (6): 29-33    DOI: 10.3404/j.issn.1672-7649.2022.06.006 PDF

Water entry of a wedge tank into water based on fully nonlinear boundary element method
WANG Ning, LU Wei-chuan, SUN Shi-yan
School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China
Abstract: The hydrodynamic problem of a two dimensional wedge tank filled with liquid entering a calm water surface is analysed based on the incompressible velocity potential theory. The motion effect of inner liquid on the entry process is investigated through comparison with the result containing equivalent solid mass or the liquid being frozen. The problem is solved through the boundary element method in the time domain. Two separated computational regions are constructed. One is the inner domain for the internal liquid, and the other is the outer open domain for the open water. The former is solved in the physical coordinate system, and the latter is solved in a stretched coordinate system. The solutions of two separated domains are connected through the motion of the body with the help of the auxiliary function method, and in-depth discussions on the coupling effect between them are provided.
Key words: water entry     fully nonlinear boundary conditions     boundary element method     dual domains
0 引　言

1 数学模型和数值过程 1.1 物理参数的定义和笛卡尔坐标系

 图 1 液舱入水示意图和坐标系定义 Fig. 1 Schematic diagram of liquid tank entering water and definition of coordinate system
 ${\gamma _1} = \frac{\text{π} }{2} + \theta - {\gamma _{\text{0}}},{\gamma _2} = \frac{\text{π} }{2} - \theta - {\gamma _{\text{0}}}。$ (1)

 $\dot{s}(t)=W(t)\text{，}\dot{l}(t)=U(t)\text{，}\dot{\theta }(t)=\varOmega (t)。$ (2)
1.2 数学模型

 ${\nabla ^2}{\phi _i}{\text{ = 0}},\begin{array}{*{20}{c}} {} \end{array}(i = 1,2)，$ (3)

 $\frac{{{\rm{D}}{\phi _i}}}{{{\rm{D}}t}} = \frac{1}{2}{\left| {\nabla {\phi _i}} \right|^2} - g(y - s) \text{，}自由面，$ (4)
 $\frac{{{\rm{D}}x}}{{{\rm{D}}t}} = \frac{{{\partial} {\phi _i}}}{{\partial x}} - U,\begin{array}{*{20}{c}} \end{array}\frac{{{\rm{D}}y}}{{{\rm{D}}t}} = \frac{{\partial {\phi _i}}}{{\partial y}} + W \text{，}自由面，$ (5)
 $\frac{{{{\partial}} {\phi _i}}}{{{{\partial}} n}} = (U - \Omega y){n_x} + ( - W + \Omega x){n_y} \text{，}物面，$ (6)
 $\nabla {\phi _{\text{2}}}{\text{ = }}0 \text{，}外域远场。$ (7)
 ${y_i}(x,t = 0) = {H_i},\begin{array}{*{20}{c}} {} \end{array}{\phi _i}(x,{H_i},t = 0) = 0\text{，}初始时刻 。$ (8)

 ${\phi _2}(x,y,t) = s\varphi (\alpha ,\beta ,t),\alpha = x/s,\beta = y/s，$ (9)

 $\frac{{{{\partial}} \varphi }}{{{{\partial}} n}} = (U - s\Omega \beta ){n_\alpha } + ( - W + s\Omega \alpha ){n_\beta }\text{，}物面,$ (10)
 $\frac{{{\rm{D}}(s\varphi )}}{{{\rm{D}}t}} = \frac{1}{2}(\varphi _\alpha ^2 + \varphi _\beta ^2) - g(s\beta - s)\text{，}自由面,$ (11)
 $\frac{{\rm{D}}(s\alpha )}{{\rm{D}}t}={\phi }_{\alpha }-U\text{，}\begin{array}{c}\end{array}\frac{{\rm{D}}(s\beta )}{{\rm{D}}t}={\phi }_{\beta }+W \text{，}自由面。$ (12)

2个计算域的控制方程和边界条件确立以后，均可以采用边界积分公式进行求解，可参考文献[17]。

1.3 压力求解

 ${p_i} = - {\rho _i}\left[ {{\phi _{it}} + \frac{1}{2}{{\left| {\nabla {\phi _i}} \right|}^2} + g(y - s)} \right]，$ (13)

 $\frac{{{\partial} {\phi _{it}}}}{{{\partial} n}} = \left( {{{\dot {\boldsymbol{U}}}} + {{\dot {{\varOmega}} }} \times {\boldsymbol{{{x}}}}} \right) \cdot {\boldsymbol{{{n}}}} - {\boldsymbol{{{U}}}} \cdot \frac{{{{\partial}} \nabla {\phi _i}}}{{{{\partial}} n}} + {{{{\varOmega}} }} \cdot \frac{{{\partial}} }{{{{\partial}} n}}\left[ {{\boldsymbol{{x}}} \times \left( {{\boldsymbol{{U}}} - \nabla {\phi _i}} \right)} \right]。$ (14)

 ${\phi _{it}} = {\chi _{i0}} + \dot U{\chi _{i1}} + \dot W{\chi _{i2}} + \dot \Omega {\chi _{i3}} - {\boldsymbol{{U}}} \cdot \nabla {\phi _i} + {\varOmega} \cdot \left[ {{\boldsymbol{{x}}} \times \left( {{\boldsymbol{{U}}} - \nabla {\phi _i}} \right)} \right]。$ (15)

 $\begin{split}\frac{{{{\partial}} {\chi _{ij}}}}{{{{\partial}} n}} =& {n_j},{n_0} = 0,{n_1} = {n_x},{n_2} = {n_y},{n_3} = (x{n_y} - y{n_x}),\\ &(i = 1,2,j = 0, \cdots, 3)\text{，}物面，\end{split}$ (16)
 $\begin{split}{\chi _{i0}} = & - \left[ {\frac{1}{2}{{\left| {\nabla {\phi _i}} \right|}^2} + g(y - s)} \right] + U{\phi _{ix}} - W{\phi _{iy}} - \\& {\varOmega} \left[ {x( - W - {\phi _{iy}}) - y(U - {\phi _{ix}})} \right] \text{，}自由面，\end{split}$ (17)
 ${\chi _{ij}} = 0,(i = 1,2,j = 1,2,3) \text{，}自由面，$ (18)
 $\frac{{{{\partial}} {\chi _{20}}}}{{{{\partial}} n}} = \Omega ({n_x}W + {n_y}U) \text{，}远场，$ (19)
 $\frac{{{\partial} {\chi _{2i}}}}{{{\partial}} n} = 0,(i = 1,2,3) \text{，}远场。$ (20)

1.4 运动方程

 $\left[ {{{\boldsymbol{M}}_0}} \right]\left[ {\boldsymbol{A}} \right] = \left[ {{{\boldsymbol{F}}_1}} \right] + \left[ {{{\boldsymbol{F}}_2}} \right] + \left[ {{{\boldsymbol{F}}_{\text{G}}}} \right]。$ (21)

 $\left[ {{{\boldsymbol{M}}_0}} \right] = \left[ {\begin{array}{*{20}{c}} {{m_0}}&0&{ - {m_0}{l_{c0}}\cos \theta } \\ 0&{{m_0}}&{ - {m_0}{l_{c0}}\sin \theta } \\ { - {m_0}{l_{c0}}\cos \theta }&{ - {m_0}{l_{c0}}\sin \theta }&{{I_0}} \end{array}} \right]，$ (22)

 $\begin{split} {{\boldsymbol{F}}_i} = & - {\rho _i}\int_{{S_i}} \Biggr\{ {\chi _{i0}} + \dot U{\chi _{i1}} + \dot W{\chi _{i2}} + \dot \Omega {\chi _{i3}} - {\boldsymbol{U}} \cdot \nabla {\phi _i} + \\& {{\varOmega }} \cdot \left[ {{\boldsymbol{x}} \times \left( {{\boldsymbol{U}} - \nabla {\phi _i}} \right)} \right] + \frac{1}{2}{{\left| {\nabla {\phi _i}} \right|}^2} + g(y - s) \Biggr\}{\boldsymbol{n}}{\rm{d}}s \end{split}。$ (23)

 $(\left[ {{{\boldsymbol{M}}_0}} \right] + \left[ {{{\boldsymbol{C}}_1}} \right] + \left[ {{{\boldsymbol{C}}_2}} \right])\left[ {\boldsymbol{A}} \right] = \left[ {{{{\boldsymbol{F'}}}_1}} \right] + \left[ {{{{\boldsymbol{F'}}}_2}} \right] + \left[ {{{\boldsymbol{F}}_{\text{G}}}} \right]。$ (24)

 ${C_{ijk}} = {\rho _i}\int\limits_{{S_i}} {{\chi _{ij}}{n_k}{\rm{d}}s}，$

 $\begin{split} {{\boldsymbol{F'}}_i} = & - {\rho _i}\int_{{S_i}} \Biggr\{ {\chi _{i0}} - {\boldsymbol{U}} \cdot \nabla {\phi _i} + {{\varOmega }} \cdot \left[ {{\boldsymbol{x}} \times \left( {{\boldsymbol{U}} - \nabla {\phi _i}} \right)} \right] +\\ &\frac{1}{2}{{\left| {\nabla {\phi _i}} \right|}^2} + g(y - s) \Biggr\}{\mathbf{n}}{\rm{d}}s。\end{split}$

2 数值结果与讨论

2.1 内半角 ${\gamma _0}$ 为45°的楔形液舱入水

 图 2 水舱和冰舱入水过程中速度时历对比（ ${\gamma _0}$ =45°） Fig. 2 Comparison of velocity time histories between water tank and ice tank ( ${\gamma _0}$ =45°)

2.2 内半角 ${\gamma _0}$ 为60°的楔形液舱入水

 图 3 水舱和冰舱入水过程中速度时历对比（ ${\gamma _0}$ =60°） Fig. 3 Comparison of velocity time histories between water tank and ice tank ( ${\gamma _0}$ =60°)

3 结　语

1）舱内液体的运动对垂向速度影响较小，对旋转速度和水平速度影响较大。垂向速度主要由楔形液舱的重力和垂直方向的水动力决定，垂向水动力对于液舱内液体的流动并不敏感，舱内流体质点的运动对垂向运动影响较小。而对于旋转速度和水平速度，随着液舱内流体的运动，不同的流体质点会以不同的加速度运动，这会导致附加质量不同于相同重量的冰，此时液舱更容易加速和减速。除此之外，舱内液体的流动会使楔形液舱的质心更快的移动，液舱将获得更大的合力或力矩。所以当液舱装有水时，旋转速度和水平速度的幅值更大。

2）楔形液舱内半角的增大会使垂向速度、旋转速度和水平速度随时间减小的更快且峰值更小。内半角增大会导致两侧的底升角减小，从而导致相同时刻下物面湿面积更大、逆物体旋转方向的力矩更大以及楔形体两侧压力差更大。因此速度峰值更早的到来，液舱的垂向速度、旋转速度和水平速度减小的更快。

 [1] HOWISON S D, OCKENDON J R, WILSON S K. Incompressible water-entry problems at small deadrise angles[J]. Journal of Fluid Mechanics, 1991, 222: 215-230. DOI:10.1017/S0022112091001076 [2] KOROBKIN A A. Second-order wagner theory of wave impact[J]. Journal of Engineering Mathematics, 2007, 58: 121-139. DOI:10.1007/s10665-006-9105-7 [3] WU G X, SUN H, HE Y S. Numerical simulation and experimental study of water entry of a wedge in free fall motion[J]. Journal of Fluids and Structures, 2004, 19: 277-289. DOI:10.1016/j.jfluidstructs.2004.01.001 [4] DOBROVOL'S KAYA Z N. On some problems of similarity flow of fluid with a free surface[J]. Journal of Fluid Mechanics, 1969, 36: 805-829. DOI:10.1017/S0022112069001996 [5] ZHAO R, FALTINSEN O M. Water entry of two-dimensional bodies[J]. Journal of Fluid Mechanics, 1993, 246: 593-612. DOI:10.1017/S002211209300028X [6] XU G D, DUAN W Y, WU GX. Numerical simulation of oblique water entry of asymmetrical wedge[J]. Ocean Engineering, 2008, 35: 1597-1603. DOI:10.1016/j.oceaneng.2008.08.002 [7] WU G X. Numerical simulation of water entry of twin wedges[J]. Journal of Fluids and Structures, 2006, 22: 99-108. DOI:10.1016/j.jfluidstructs.2005.08.013 [8] XU G D, DUAN W Y, WU G X. Simulation of water entry of a wedge through free fall in three degrees of freedom[J]. Proceeding of the Royal SocietyA:Mathematical,, 2010, 466: 2219-2239. DOI:10.1098/rspa.2009.0614 [9] SUN S Y, SUN S L, WU G X. Oblique water entry of a wedge into waves with gravity effect [J]. Journal of Fluids and Structures, 2015, 52: 49-64. [10] 张书谊, 段文洋. 矩形液舱横荡流体载荷的Fluent数值模拟[J]. 中国舰船研究, 2011, 6(5): 73-77. [11] 朱仁庆, 侯玲. LNG船液舱晃荡数值模拟[J]. 江苏科技大学学报(自然科学版), 2010, 24(1): 1-6. [12] 甄长文, 吴文锋, 朱柯壁, 等. 共振频率下油船液舱晃荡数值模拟研究[J]. 中国修船, 2019, 32(1): 40-43. [13] 李裕龙, 朱仁传, 缪国平, 等. 基于OpenFOAM的船舶与液舱流体晃荡在波浪中时域耦合运动的数值模拟[J]. 船舶力学, 2012, 016(7): 750-758. DOI:10.3969/j.issn.1007-7294.2012.07.004 [14] 骆阳, 朱仁庆, 刘永涛. FPSO与运输船旁靠时液舱晃荡与船舶运动耦合效应分析[J]. 江苏科技大学学报: 自然科学版, 2015, 29(4): 307-316. [15] 朱仁庆, 李辰, 顾思琪. 弹性液舱内液体晃荡研究[J]. 江苏科技大学学报: 自然科学版, 2013, 27(3): 214-218. [16] 龚少军, 姚震球. 基于粒子法的液舱共振晃荡现象研究[J]. 江苏科技大学学报: 自然科学版, 2010, 24(6): 534-538. [17] LU C H, HE Y S, WU G X. Coupled analysis of nonlinear interaction between fluid and structure during impact [J]. J. Journal of Fluids & Structures, 2ooo,14(1):127-146 [18] WU G. Hydrodynamic force on a rigid body during impact with liquid [J], J. Journal of Fluids & Structures,1998,12(5):549-559