﻿ 基于SPH和FVM的穿浪双体船跌落仿真研究
 舰船科学技术  2022, Vol. 44 Issue (16): 33-38    DOI: 10.3404/j.issn.1672-7649.2022.16.007 PDF

Research on the simulation of wave-piercing catamaran drop based on SPH and FVM
LIU Di, SUN Ya-jun
Weihai Ocean Vocational College, Weihai 264300, China
Abstract: The methods of the research on slamming of wave-piercing catamaran's wave piercing bow include theory calculation, numerical forecast and experiment. The model of catamaran drop is applied in numerical forecast as a simplified model. SPH and FVM are common methods in meshless and mesh CFD. To analyze the catamaran drop, the change of acceleration and velocity during catamaran drop is calculated on SPH and FVM respectively. The experimental results are also taken into comparative analysis. Based on the calculation image and data, the application of two methods on the simulation of catamaran drop has been researched. The results indicate the feasibility and characteristics of SPH and FVM in the simulation.
Key words: SPH     FVM     wave-piercing catamaran     drop
0 引　言

1 SPH应用理论基础 1.1 粒子点的生成

1.2 光滑函数的选取

 $W\left( {R,h} \right) = \left\{ {\begin{array}{*{20}{l}} \dfrac{{105}}{{16{\text{π}} {h^2}}}\left( {1 + 3R} \right){{\left( {1 - R} \right)}^2}，R \leqslant 1 ，\\ 0，R > 1 。\end{array}} \right.$ (1)

1.3 物理特性设置

 $\dfrac{{{\rm{d}}{\rho _i}}}{{{\rm{d}}t}} = \sum\limits_{j = 1}^N {{m_j}\nu _{ij}^\beta \dfrac{{\partial {W_{ij}}}}{{\partial x_i^\beta }}},$ (2)

 $\dfrac{{{\rm{d}}\nu _i^a}}{{{\rm{d}}t}} = \sum\limits_{j = 1}^N {{m_j}\dfrac{{\sigma _i^{\alpha \beta } + \sigma _j^{\alpha \beta }}}{{{\rho _i}{\rho _j}}}\dfrac{{\partial {W_{ij}}}}{{\partial x_i^\beta }}},$ (3)

 $\dfrac{{{\rm{d}}{e_i}}}{{{\rm{d}}t}} = \dfrac{1}{2}\sum\limits_{j = 1}^N {{m_j}\dfrac{{{p_i} + {p_j}}}{{{\rho _i}{\rho _j}}}\nu _{ij}^\beta \dfrac{{\partial {W_{ij}}}}{{\partial x_i^\beta }}} + \frac{{{\mu _i}}}{{2{\rho _i}}}\varepsilon _i^{\alpha \beta }\varepsilon _i^{\alpha \beta }。$ (4)

 $p = B\left( {{{\left( {\frac{\rho }{{{\rho _0}}}} \right)}^r} - 1} \right)。$ (5)

1.4 动边界

 $pe = \frac{{{h_i} + {h_j}}}{{2{r_{ij}}}} \geqslant 1，$ (6)

 $P = \left\{ {\begin{array}{*{20}{l}} \dfrac{{{{10}^5}}}{{{r_{ij}}}}\left( {p{e^6} - p{e^4}} \right)，pe \geqslant 1 ，\\ 0，pe < 1 。\end{array}} \right.$ (7)

2 有限体积法应用基础 2.1 有限体积法理论基础

 图 1 离散原理图 Fig. 1 The theory of discretion

 ${a_p}{\varphi _p} = \sum {{a_{nb}}{\varphi _{nb}} + b}。$ (8)

1）求解整个流体区域内控制单元上的积分方程；

2）运用相应的离散格式把控制体上的积分方程转化为代数方程求解，其中包括对方程对流项，扩散项，源项的离散；

3）使用设当的迭代方法求解代数方程。

2.2 计算过程 2.2.1 湍流模型的确立

 $\dfrac{\partial }{{\partial t}}(\rho k) + \dfrac{\partial }{{\partial {x_i}}}(\rho k{u_i}) = \dfrac{\partial }{{\partial {x_j}}}\left({\Gamma _k}\dfrac{{\partial k}}{{\partial {x_j}}}\right) + {\tilde G_k} - {Y_k} + {S_k}，$ (9)
 $\dfrac{\partial }{{\partial t}}(\rho \omega ) + \dfrac{\partial }{{\partial {x_i}}}(\rho \omega {u_i}) = \dfrac{\partial }{{\partial {x_j}}}\left({\Gamma _\omega }\dfrac{{\partial \omega }}{{\partial {x_j}}}\right) + {\tilde G_\omega } - {Y_\omega } + {S_\omega }。$ (10)

 $\begin{gathered} {\alpha _1} = 0.31, {\beta _{t,1}} = 0.075, {\beta _{t,2}} = 0.0828，\\ {\sigma _{k,1}} = 1.176,{\sigma _{\omega ,1}} = 2.0,{\sigma _{\omega ,2}} = 1.168。\\ \end{gathered}$
2.2.2 边界条件

 图 2 边界条件示意图 Fig. 2 The boundary conditions
2.2.3 重叠网格

 图 3 控制体单元分布形式 Fig. 3 The distribution form of control units

Overset meshes在设置流体区域时，需要一个能够用于完整计算的大的流体域以及一个包含物体表面的小的流体域，如图4所示。在网格划分时，首先在整个流域中设置一套大尺寸网格，再在物体周围区域设置一套尺寸较小的网格，使其包括在流域中，在网格重叠部分需要设置过渡区，目的是使物体在下落过程中，重叠区域附近的非活动单元能够有效的转为活动单元。

 图 4 重叠网格图 Fig. 4 The overlapping grid
3 计算分析

SPH方法和FVM计算结果与试验值[20]的对比如图5图6所示。本文的试验数据来源于Michael R. Davis和James R. Whelan发表的数据。由于本文主要研究穿浪双体船跌落所受的砰击，故试验值给出的是跌落加速度的变化以表征其所受的砰击载荷，数值计算也侧重跌落速度和加速度的变化以研究穿浪双体船的跌落过程。

 图 5 跌落速度曲线 Fig. 5 The curves of dropping velocity

 图 6 跌落加速度数值计算与试验对比曲线 Fig. 6 The comparison of numerical calculation and experiment on dropping acceleration

 图 7 跌落图像对比 Fig. 7 The comparison of drop images
4 结　语

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