﻿ 水下滑翔机附加质量数值计算
 舰船科学技术  2016, Vol. 38 Issue (12): 116-120,134 PDF

Numerical method for added mass of an underwater glider
YANG Lei, CAO Jun-jun, YAO Bao-heng, ZENG Zheng, LIAN Lian
Naval Architecture and Ocean Engineering National Laboratory, Shanghai Jiaotong University, Shanghai 200240, China
Abstract: Added mass of an underwater glider is quite important for the motions of glider. In this paper, the added mass of an arbitrary three-dimensional body is obtained through Hess-Smith method. Then an underwater glider which was designed by our laboratory is meshed by Gambit software in order to obtain its added mass. Besides, the Planar Motion Mechanism (PMM) tests of the glider are simulated by using CFD software, dynamic mesh technique and UDF. By comparing with the Hess-Smith results, the characters and advantages of Hess-Smith method and PMM are analyzed.
Key words: added Mass     underwater glider     Hess-Smith     PMM
0 引言

1 基于面元法的附加质量计算 1.1 基本方程和边界条件

 $\Phi =x{{V}_{\infty x}}+y{{V}_{\infty y}}+z{{V}_{\infty z}}+\varphi ,$ (1)
 ${{\nabla }^{2}}\varphi =0,$ (2)
 $\frac{\partial \varphi }{\partial n}=-V\cdot n,$ (3)
 $\varphi =0,$ (4)

1.2 速度势离散

 $\varphi (p)=\iint\limits_{s}{\frac{\sigma (q)}{{{r}_{pq}}}}\text{d}{{s}_{q}},$ (5)

 $2\pi \sigma (p)+\iint\limits_{s-\varepsilon }{\sigma }\left( q \right)\frac{\partial }{\partial {{n}_{p}}}\left( \frac{1}{{{r}_{pq}}} \right)\text{d}{{s}_{q}}=-V\cdot n,$ (6)

 $\iint\limits_{s}{\sigma }\left( q \right)\frac{\partial }{\partial {{n}_{p}}}\left( \frac{1}{{{r}_{pq}}} \right)\text{d}{{s}_{q}}\approx \sum\limits_{j=1}^{N}{{{\sigma }_{j}}\iint\limits_{\Delta {{Q}_{j}}}{\frac{\partial }{\partial {{n}_{p}}}}\left( \frac{1}{{{r}_{pq}}} \right)}\text{d}{{s}_{q}},$ (7)

 $2\pi \sigma (p)+\sum\limits_{j=1}^{N}{{{\sigma }_{j}}\iint\limits_{\Delta {{Q}_{j}}}{\frac{\partial }{\partial {{n}_{p}}}}\left( \frac{1}{{{r}_{pq}}} \right)}\text{d}{{s}_{q}}=-V\cdot n$ (8)

 图 1 水下滑翔机三角形网格划分 Fig. 1 The generation of triangular mesh for an underwater glider
1.3 附加质量求解

 ${{\lambda }_{ij}}={{\lambda }_{ji}}$ (9)

 $\lambda = \left[{\begin{array}{*{20}{c}} {{\lambda _{11}}}&0&0&0&0&0\\[8pt] 0&{{\lambda _{22}}}&0&{{\lambda _{24}}}&0&{{\lambda _{26}}}\\[8pt] 0&0&{{\lambda _{33}}}&0&{{\lambda _{35}}}&0\\[8pt] 0&{{\lambda _{42}}}&0&{{\lambda _{44}}}&0&{{\lambda _{46}}}\\[8pt] 0&0&{{\lambda _{53}}}&0&{{\lambda _{55}}}&0\\[8pt] 0&{{\lambda _{62}}}&0&{{\lambda _{64}}}&0&{{\lambda _{66}}} \end{array}} \right] = \left[{\begin{array}{*{20}{c}} {\frac{1}{2}\rho {L^3}{{X'}_{\dot u}}}&0&0&0&0&0\\[8pt] 0&{\frac{1}{2}\rho {L^3}{{Y'}_{\dot v}}}&0&{\frac{1}{2}\rho {L^4}{{K'}_{\dot v}}}&0&{\frac{1}{2}\rho {L^4}{{N'}_{\dot v}}}\\[8pt] 0&0&{\frac{1}{2}\rho {L^3}{{Z'}_{\dot w}}}&0&{\frac{1}{2}\rho {L^4}{{M'}_{\dot w}}}&0\\[8pt] 0&{\frac{1}{2}\rho {L^4}{{Y'}_{\dot p}}}&0&{\frac{1}{2}\rho {L^5}{{K'}_{\dot p}}}&0&{\frac{1}{2}\rho {L^5}{{N'}_{\dot p}}}\\[8pt] 0&0&{\frac{1}{2}\rho {L^4}{{Z'}_{\dot q}}}&0&{\frac{1}{2}\rho {L^5}{{M'}_{\dot q}}}&0\\[8pt] 0&{\frac{1}{2}\rho {L^4}{{Y'}_{\dot r}}}&0&{\frac{1}{2}\rho {L^5}{{K'}_{\dot r}}}&0&{\frac{1}{2}\rho {L^5}{{N'}_{\dot r}}} \end{array}} \right]{\text {。}}\!\!$ (10)

 ${{\lambda }_{ij}}=-\rho \text{ }\iint\limits_{s}{{{\varphi }_{i}}}\frac{\partial {{\varphi }_{j}}}{\partial n}\text{d}s=-\rho \iint\limits_{s}{{{\varphi }_{i}}}{{n}_{j}}\text{d}s$ (11)

2 基于 CFD 软件的附加质量计算

Fluent 软件可提供动网格技术，动网格模型可以用来模拟流程形状由于边界运动而随时间改变的问题，利用 UDF 可以定义边界的运动方式。本文则是对水下滑翔机进行 PMM 试验模拟，从而得到附加质量。

2.1 计算方程

 $\left\{ \begin{array}{*{35}{l}} \zeta =a\sin (\omega t), \\ \theta =\dot{\theta }=0, \\ w=\dot{\zeta }=aw\cos (\omega t), \\ \dot{w}=\ddot{\zeta }=-a{{w}^{2}}\sin (\omega t). \\ \end{array} \right.$ (12)

 图 2 水下滑翔机纯升沉运动示意图 Fig. 2 The heaving motion of an underwater glider

 $\left\{ \begin{array}{*{35}{l}} Z={{Z}_{{\dot{w}}}}\dot{w}+{{Z}_{w}}w+{{Z}_{0}}= \\ -a{{\omega }^{2}}{{Z}_{{\dot{w}}}}\sin (\omega t)+a\omega {{Z}_{w}}\cos (\omega t)+{{Z}_{0}}, \\ M={{M}_{{\dot{w}}}}\dot{w}+{{M}_{w}}w+{{M}_{0}}= \\ -a{{\omega }^{2}}{{M}_{{\dot{w}}}}\sin (\omega t)+a\omega {{M}_{w}}\cos (\omega t)+{{M}_{0}}. \\ \end{array} \right.$ (13)

 $\left\{ \begin{array}{*{35}{l}} \zeta =a\sin (\omega t), \\ w=\dot{\zeta }=aw\cos (\omega t), \\ \dot{w}=\ddot{\zeta }=-a{{w}^{2}}\sin (\omega t), \\ \theta ={{\theta }_{0}}\cos (\omega t), \\ q=\dot{\theta }=-\omega {{\theta }_{0}}\sin (\omega t), \\ \dot{q}=\ddot{\theta }=-{{\omega }^{2}}{{\theta }_{0}}\cos (\omega t). \\ \end{array} \right.$ (14)

 $\left\{ \begin{array}{*{35}{l}} \theta \approx \tan \theta =\frac{w}{U}=\frac{a\omega }{U}\cos (\omega t)={{\theta }_{0}}\cos (\omega t), \\ {{\theta }_{0}}=\frac{a\omega }{U}. \\ \end{array} \right.$ (15)
 图 3 水下滑翔机纯俯仰运动示意图 Fig. 3 The pitching motion of an underwater glider

 $\left\{ \begin{array}{*{35}{l}} Z={{Z}_{{\dot{q}}}}\dot{q}+{{Z}_{q}}q+{{Z}_{0}}= \\ -{{\theta }_{0}}{{\omega }^{2}}{{Z}_{{\dot{q}}}}\sin (\omega t)+{{\theta }_{0}}\omega {{Z}_{q}}\cos (\omega t)+{{Z}_{0}}, \\ M={{M}_{{\dot{q}}}}\dot{q}+{{M}_{q}}q+{{M}_{0}}= \\ -{{\theta }_{0}}{{\omega }^{2}}{{M}_{{\dot{q}}}}\sin (\omega t)+{{\theta }_{0}}\omega {{M}_{q}}\cos (\omega t)+{{M}_{0}}. \\ \end{array} \right.$ (16)
2.2 CFD 计算准备

 图 4 计算域内域和外域网格划分 Fig. 4 The mesh of internal and external flow domain

2.3 结果处理与分析

 图 5 纯升沉运动 f = 0.4 时垂向力 Z 的变化 Fig. 5 The changes of force in Z-direction when the frequency of heaving motion is 0.4

 图 6 纯升沉运动 f = 0.4 时绕 Z 轴力矩的变化 Fig. 6 The changes of moment around Z-direction when the frequency of heaving motion is 0.4

Origin 软件具有自定义拟合函数的功能，利用式（13）编写出相应的拟合函数，从而得到高精度的拟合结果。

f = 0.4 时，得到拟合力和力矩的曲线函数为：

 $\left\{ \begin{array}{*{35}{l}} Z={{Z}_{{\dot{w}}}}\dot{w}+{{Z}_{w}}w+{{Z}_{0}}= \\ 26.037\sin (\omega t)-16.627\cos (\omega t)-0.116, \\ M={{M}_{{\dot{w}}}}\dot{w}+{{M}_{w}}w+{{M}_{0}}= \\ 0.313\sin (\omega t)-2.065\cos (\omega t)-0.002. \\ \end{array} \right.$

 $\left\{ \begin{array}{*{35}{l}} {{Z}_{a}}=-\frac{a{{\omega }^{2}}L}{{{V}^{2}}}{{{{Z}'}}_{w}}, \\ {{M}_{a}}=-\frac{a{{\omega }^{2}}L}{{{V}^{2}}}{{{{M}'}}_{w}}. \\ \end{array} \right.$ (17)

 图 7 利用最小二乘法计算加速度系数 Fig. 7 The acceleration coefficients applying the least square method

3 结 语

1）对于水下滑翔机，Hess-Smith 面元法程序和 Fluent 仿真得到的附加质量结果均比较精确，满足工程需求。

2）相比于 Fluent 软件仿真需要花费较多时间多次建模与计算，Hess-Smith 方法仅需要重新划分物体的面网格，节省大量时间，但编写程序较为复杂。

3）本文提供了计算水下滑翔机附加质量的 2 种方法，介绍了两者原理与优劣，同时得到精确的附加质量，验证了 2 种方法的可行性和准确性，对水下滑翔机的水动力设计和运动仿真具有重要指导意义和参考价值。

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