﻿ 圆球上浮运动特性数值仿真及试验研究
 舰船科学技术  2024, Vol. 46 Issue (3): 28-33    DOI: 10.3404/j.issn.1672-7649.2024.03.005 PDF

1. 武汉第二船舶设计研究所，湖北 武汉 430064;
2. 中国船舶集团有限公司第七〇五研究所昆明分部，云南 昆明 650106;
3. 华中科技大学 船舶与海洋工程学院，湖北 武汉 430074

Numerical simulation and experimental research on floating motion characteristics of round spheres
LIANG Lai-yu1, CHEN Lin2, SONG Lei3
1. Wuhan Second Ship Design and Research Institute, Wuhan 430064, China;
2. Kunming Branch of the 705 Research Institute of CSSC, Kunming 650106, China;
3. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Techonology, Wuhan 430074, China
Abstract: Taking the spherical floating body as the research object, the influence of scale and density ratio on its floating motion performance was studied. Numerical simulation method for the floating motion of a ball in hydrostatic water was established, and the reliability of the numerical method was verified by comparing the test results through model tests. Then, the floating motion of multiple spheres of different scales was simulated and analyzed. It is found that the drag coefficient of the sphere in the stable floating stage is roughly constant 1, which does not change greatly with the change of Reynolds number. If two spheres meet similar conditions and the ratio of scales is λ, the ratio of their stable velocities to the total time to float is approximately λ0.5. In this study, through the calculation and research analysis of the floating motion of the sphere, the correlation between the floating motion of the sphere of different scales is sought, which can provide an important reference for the study and analysis of the scale correlation of the floating motion performance of other structures.
Key words: free floating     model testing     scale effects
0 引　言

1 建模与试验验证 1.1 研究对象

 图 1 坐标系定义 Fig. 1 Definition of coordinate system
1.2 数值模拟方法 1.2.1 控制方程

 $\frac{{\partial \rho }}{{\partial t}} + \frac{{\partial (\rho {u_i})}}{{\partial {x_i}}}{\text{ = }}0 ，$ (1)

 $\begin{split} & \frac{{\partial \left( {\rho {u_i}} \right)}}{{\partial t}} + \frac{{\partial \left( {\rho {u_i}{u_j}} \right)}}{{\partial {x_j}}} = - \frac{{\partial \rho }}{{\partial {x_i}}} + \\ & \left. {\frac{\partial }{{\partial {x_i}}}\left[ {\mu (\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {x_i}}}) - } \right.\frac{2}{3}\mu \frac{{\partial {u_l}}}{{\partial {x_l}}}{\delta _{ij}}} \right] + \frac{\partial }{{\partial {x_j}}}\left( { - p\overline {u_i^{'} u_j^{'}} } \right) + \rho {f_i}。\\[-1pt] \end{split}$ (2)

1.2.2 计算方案

1.3 网格无关性验证

 图 2 计算域及网格 Fig. 2 Computational domain and grid

2 结果与讨论 2.1 试验验证

 图 3 模型试验方案 Fig. 3 Model test scheme

 图 4 圆球上浮运动加速度时历对比曲线 Fig. 4 Comparison curve of the acceleration of the floating motion of the sphere

 图 5 圆球上浮运动速度时历对比曲线 Fig. 5 Comparison curve of the floating velocity of the sphere

 图 6 圆球上浮运动水深时历对比曲线 Fig. 6 Comparison curve of water depth in the floating motion of the sphere
2.2 不同尺度圆球上浮稳定速度分析

 图 7 圆球大小对比 Fig. 7 Comparison of sphere sizes

 图 8 R=0.04 m、0.06 m、0.08 m圆球上浮速度时历曲线 Fig. 8 Calendar curves of the floating velocity of R=0.04 m, 0.06 m, and 0.08 m spheres

 图 9 R=0. 1m、0.12 m、0.14 m圆球上浮速度时历曲线 Fig. 9 Calendar curves of the floating velocity of R=0.1 m, 0.12 m, and 0.14 m spheres

 图 10 R=0.16 m、0.18 m、0.2 m圆球上浮速度时历曲线 Fig. 10 Calendar curves of the floating velocity of the R=0.16 m, 0.18 m, and 0.2 m spheres

 图 11 不同直径圆球上浮稳定速度 Fig. 11 Stable floating velocity of balls with different diameters

 $F = \frac{4}{3}{\text{π}} {R^3}\rho (1 - {m^*})g。$ (3)

 $\dfrac{F}{{\dfrac{1}{2}\rho {v^2}S}} = \frac{{\dfrac{4}{3}{\text{π}} {R^3}\rho (1 - {m^*})g}}{{\dfrac{1}{2}\rho {v^2}{\text{π}} {R^2}}} = {C_d} = 1。$ (4)

 $v = \sqrt {\frac{{32g}}{{15}}R}。$ (5)

2.3 不同密度圆球上浮稳定速度分析

2.4 不同尺度圆球上浮总时间分析

 图 12 不同直径圆球上浮总时间 Fig. 12 The total floating time of balls with different diameters

 $\frac{{{t_{a1}}}}{{{t_{a2}}}} = {\raise0.7ex\hbox{${\frac{{{l_1}}}{{{v_1}}}}$} \mathord{\left/ {\vphantom {{\frac{{{l_1}}}{{{v_1}}}} {\frac{{{l_2}}}{{{v_2}}}}}}\right.} \lower0.7ex\hbox{${\frac{{{l_2}}}{{{v_2}}}}$}} = \frac{{{l_1}}}{{{l_2}}}\frac{{{v_2}}}{{{v_1}}} = \lambda \frac{1}{{\sqrt \lambda }} = \sqrt \lambda。$ (6)

3 结　语

1）圆球在稳定上浮阶段的阻力系数大致为常数1，不随雷诺数或圆球密度比的变化而发生变化；

2）当满足相似条件时，2个圆球在上浮时的稳定速度满足如下关系：若2个圆球的尺度之比为$\lambda$，则其上浮时的稳定速度之比大致为$\sqrt \lambda$

3）当满足相似条件时，2个圆球的上浮总时间满足如下关系：若2个圆球的尺度之比为$\lambda$，则其上浮总时间之比大致为$\sqrt \lambda$

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