Numerical study on the calculation of hydrodynamic coefficient and judgment of motion stability of submersibles
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摘要: 圆碟形潜水器具备零转弯直径、精准着陆和良好的悬停能力，对提高海底观测系统效率意义重大。然而，关于圆碟形潜水器水动力性能的研究并不多。本研究提出将平面运动机构（PMM）数值模拟实验与Routh判据相结合的创新方法，用于判断圆碟形潜水器的运动稳定性。一方面，推导了潜水器运动控制方程和运动稳定性判断标准，另一方面，建立了数值模拟模型并设计了PMM数值模拟来计算水动力系数。研究首次以圆碟形潜水器为例，结合Routh准则，对比分析了HG1和HG3两种船体的水动力性能，得到HG1和HG3水平运动及垂直运动的稳定系数，结果表明HG3在运动稳定性方面表现更优。此结论在缩尺模型水池实验中也得到了验证。该数值研究方法可扩展用于各类作业型潜水器的运动稳定性研究，避免实际实验高昂的成本，进一步提升潜水器在海洋工程中的作业性能。Abstract: The disk-shaped submersible exhibits exceptional maneuverability, including zero-radius turning, precise landing, and stable hovering capabilities, which hold significant importance for enhancing the operational efficiency of seabed observation systems. However, research focusing on the hydrodynamic performance of disk-shaped submersibles remains limited. This study innovatively proposes a methodology integrating Planar Motion Mechanism (PMM) numerical experiments with the Routh criterion to evaluate the motion stability of disk-shaped submersibles. Firstly, the governing equations for submersible motion and motion stability criteria were derived. Subsequently, a numerical simulation model was established, and PMM-based numerical experiments were designed to calculate hydrodynamic coefficients. This research represents the first systematic comparison of hydrodynamic performance between HG1 and HG3 hull configurations in disk-shaped submersibles using the Routh criterion. The derived stability coefficients for horizontal and vertical motions demonstrate superior motion stability in the HG3 configuration. These findings were further validated through scale model basin experiments. The proposed numerical approach can be extended to motion stability analysis of various operational submersibles, effectively reducing the substantial costs associated with physical experiments while enhancing submersible performance in marine engineering applications.
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图 2 HG1和HG3的模型及其参数
a. 潜水器在湖中进行实验，b. HG1示意图，c. HG3示意图，d. HG1和HG3的剖面和几何参数
Fig. 2 Models and parameters of HG1 and HG3
a. Submersible conducting experiments in the lake, b. original disk shape (HG1), c. modified anterior-posterior asymmetric shape of the tail (HG3), d. profiles and geometrical parameters of HG1 and HG3
图 6 MATLAB拟合的简谐横荡仿真结果
Fig. 6 Simulation results of MATLAB fitted simple harmonic transverse oscillation
图 7 MATLAB拟合的简谐横摇仿真结果
Fig. 7 Simulation results of MATLAB fitted simple harmonic transverse rocking
表 1 原始圆碟形潜水器（HG1）基本参数
Tab. 1 Basic parameters of the current disk-shaped submersible (HG1)
直径/mm 高度/mm 重量/kg 巡航速度/(m·s−1) 值 1557 801 ≤300 ≤2 
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表 2 不同网格数量下HG3潜水器在1 m/s速度下的水动力
Tab. 2 The hydrodynamic performance of the HG3 submersible under different grid numbers, 1 m/s
物理量 网格数量(百万) 3.7 6 10 14 阻力Fx/N 17.12 18.81 19.18 19.22 变化率 / 9.87% 1.97% 0.2%
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表 3 HG1的无量纲水动力系数
Tab. 3 Dimensionless hydrodynamic coefficients for HG1
无量纲系数 无量纲公式 值 无量纲系数 无量纲公式 值 $ X_{\dot{u}}^{'} $ $ \dfrac{{X}_{\dot{u}}}{1/2\rho {L}^{3}} $ −0.117 $ X_{u}^{{'}} $ $ \dfrac{{X}_{u}}{1/2\rho {L}^{2}U} $ 0 $ X_{u|u|}^{'} $ $ \dfrac{{X}_{u|u|}}{1/2\rho {L}^{2}} $ −0.050 $ X_{0}^{{'}} $ $ \dfrac{{X}_{0}}{1/2\rho {L}^{2}{U}^{2}} $ 0.027 $ Y_{\dot{v}}^{'} $ $ \dfrac{{Y}_{\dot{v}}}{1/2\rho {L}^{3}} $ −0.161 $ Y_{v}^{\mathrm{'}} $ $ \dfrac{{Y}_{v}}{1/2\rho {L}^{2}U} $ −0.068 $ Y_{v|v|}^{'} $ $ \dfrac{{Y}_{v|v|}}{1/2\rho {L}^{2}} $ 0.017 $ K_{\dot{v}}^{\mathrm{'}} $ $ \dfrac{{K}_{\dot{v}}}{1/2\rho {L}^{4}} $ −0.002 $ K_{v}^{'} $ $ \dfrac{{K}_{v}}{1/2\rho {L}^{3}U} $ 0.008 $ K_{\dot{v}|v|}^{\mathrm{'}} $ $ \dfrac{{K}_{v|v|}}{1/2\rho {L}^{3}} $ −0.001 $ Z_{\dot{w}}^{'} $ $ \dfrac{{Z}_{\dot{w}}}{1/2\rho {L}^{3}} $ −0.658 $ Z_{w}^{\mathrm{'}} $ $ \dfrac{{Z}_{w}}{1/2\rho {L}^{2}U} $ −1.540 $ Z_{w|w|}^{'} $ $ Z_{w|w|}^{1/2\rho {L}^{2}} $ 0.288 $ M_{\dot{w}}^{\mathrm{'}} $ $ \dfrac{{M}_{\dot{w}}}{1/2\rho {L}^{4}} $ 0.034 $ M_{w}^{'} $ $ \dfrac{{M}_{w}}{1/2\rho {L}^{3}U} $ −0.155 $ M_{w|w|}^{\mathrm{'}} $ $ \dfrac{{M}_{w|w|}}{1/2\rho {L}^{3}} $ −0.056 $ Z_{\dot{q}}^{'} $ $ \dfrac{{Z}_{\dot{q}}}{1/2\rho {L}^{4}} $ 0.003 $ Z_{q}^{\mathrm{'}} $ $ \dfrac{{Z}_{q}}{1/2\rho {L}^{3}U} $ 0.855 $ Z_{q|q|}^{'} $ $ \dfrac{{Z}_{q|q|}}{1/2\rho {L}^{4}} $ −0.141 $ M_{\dot{q}}^{\mathrm{'}} $ $ \dfrac{{M}_{\dot{q}}}{1/2\rho {L}^{5}} $ −0.041 $ M_{q}^{'} $ $ \dfrac{{M}_{q}}{1/2\rho {L}^{4}U} $ −0.094 $ M_{q|q|}^{\mathrm{'}} $ $ \dfrac{{M}_{q|q|}}{1/2\rho {L}^{5}} $ 0.027 $ N_{\dot{r}}^{'} $ $ \dfrac{{N}_{\dot{r}}}{1/2\rho {L}^{5}} $ −0.001 $ N_{r}^{\mathrm{'}} $ $ \dfrac{{N}_{r}}{1/2\rho {L}^{4}U} $ −0.003 $ N_{r|r|}^{'} $ $ \dfrac{{N}_{r\mid r}}{1/2\rho {L}^{5}} $ 0.001 $ Y_{\dot{p}}^{\mathrm{'}} $ $ \dfrac{{Y}_{\dot{p}}}{1/2\rho {L}^{4}} $ 0 $ Y_{p}^{'} $ $ \dfrac{{Y}_{p}}{1/2\rho {L}^{3}U} $ 0 $ Y_{p|p|}^{\mathrm{'}} $ $ \dfrac{{Y}_{p|p|}}{1/2\rho {L}^{4}} $ 0 $ K_{\dot{p}}^{'} $ $ \dfrac{{K}_{\dot{p}}}{1/2\rho {L}^{5}} $ −0.022 $ K_{p}^{\mathrm{'}} $ $ \dfrac{{K}_{p}}{1/2\rho {L}^{4}U} $ −0.019 $ K_{p|p|}^{'} $ $ \dfrac{{K}_{p|p|}}{1/2\rho {L}^{5}} $ 0
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表 4 HG3的无量纲水动力系数
Tab. 4 Dimensionless hydrodynamic coefficients for HG3
无量纲系数 无量纲公式 值 无量纲系数 无量纲公式 值 $ X_{\dot{u}}^{'} $ $ \dfrac{{X}_{\dot{u}}}{1/2\rho {L}^{3}} $ −0.135 $ X_{u}^{\mathrm{'}} $ $ \dfrac{{X}_{u}}{1/2\rho {L}^{2}U} $ −0.019 $ X_{u|u|}^{'} $ $ \dfrac{{X}_{u|u|}}{1/2\rho {L}^{2}} $ −0.033 $ X_{0}^{\mathrm{'}} $ $ \dfrac{{X}_{0}}{1/2\rho {L}^{2}{U}^{2}} $ 0.043 $ Y_{\dot{v}}^{'} $ $ \dfrac{{Y}_{\dot{v}}}{1/2\rho {L}^{3}} $ −0.173 $ Y_{v}^{\mathrm{'}} $ $ \dfrac{{Y}_{v}}{1/2\rho {L}^{2}U} $ −0.096 $ Y_{v|v|}^{'} $ $ \dfrac{{Y}_{v|v|}}{1/2\rho {L}^{2}} $ 0.117 $ K_{\dot{v}}^{\mathrm{'}} $ $ \dfrac{{K}_{\dot{v}}}{1/2\rho {L}^{4}} $ 0.056 $ K_{v}^{'} $ $ \dfrac{{K}_{v}}{1/2\rho {L}^{3}U} $ 0.031 $ K_{\dot{v}|v|}^{\mathrm{'}} $ $ \dfrac{{K}_{v|v|}}{1/2\rho {L}^{3}} $ −0.070 $ Z_{\dot{w}}^{'} $ $ \dfrac{{Z}_{\dot{w}}}{1/2\rho {L}^{3}} $ 0.555 $ Z_{w}^{\mathrm{'}} $ $ \dfrac{{Z}_{w}}{1/2\rho {L}^{2}U} $ −0.499 $ Z_{w|w|}^{'} $ $ Z_{w|w|}^{1/2\rho {L}^{2}} $ 0.096 $ M_{\dot{w}}^{\mathrm{'}} $ $ \dfrac{{M}_{\dot{w}}}{1/2\rho {L}^{4}} $ −0.045 $ M_{w}^{'} $ $ \dfrac{{M}_{w}}{1/2\rho {L}^{3}U} $ −0.778 $ M_{w|w|}^{\mathrm{'}} $ $ \dfrac{{M}_{w|w|}}{1/2\rho {L}^{3}} $ 0.103 $ Z_{\dot{q}}^{'} $ $ \dfrac{{Z}_{\dot{q}}}{1/2\rho {L}^{4}} $ −0.021 $ Z_{q}^{\mathrm{'}} $ $ \dfrac{{Z}_{q}}{1/2\rho {L}^{3}U} $ 0.572 $ Z_{q|q|}^{'} $ $ \dfrac{{Z}_{q|q|}}{1/2\rho {L}^{4}} $ −0.154 $ M_{\dot{q}}^{\mathrm{'}} $ $ \dfrac{{M}_{\dot{q}}}{1/2\rho {L}^{5}} $ −0.068 $ M_{q}^{'} $ $ \dfrac{{M}_{q}}{1/2\rho {L}^{4}U} $ −0.024 $ M_{q|q|}^{\mathrm{'}} $ $ \dfrac{{M}_{q|q|}}{1/2\rho {L}^{5}} $ 0.046 $ N_{\dot{r}}^{'} $ $ \dfrac{{N}_{\dot{r}}}{1/2\rho {L}^{5}} $ 0 $ N_{r}^{\mathrm{'}} $ $ \dfrac{{N}_{r}}{1/2\rho {L}^{4}U} $ −0.004 $ N_{r|r|}^{'} $ $ \dfrac{{N}_{r\mid r}}{1/2\rho {L}^{5}} $ 0 $ Y_{\dot{p}}^{\mathrm{'}} $ $ \dfrac{{Y}_{\dot{p}}}{1/2\rho {L}^{4}} $ 0.034 $ Y_{p}^{'} $ $ \dfrac{{Y}_{p}}{1/2\rho {L}^{3}U} $ 0.016 $ Y_{p|p|}^{\mathrm{'}} $ $ \dfrac{{Y}_{p|p|}}{1/2\rho {L}^{4}} $ − 0.0068 $ K_{\dot{p}}^{'} $ $ \dfrac{{K}_{\dot{p}}}{1/2\rho {L}^{5}} $ −0.046 $ K_{p}^{\mathrm{'}} $ $ \dfrac{{K}_{p}}{1/2\rho {L}^{4}U} $ −0.021 $ K_{p|p|}^{'} $ $ \dfrac{{K}_{p|p|}}{1/2\rho {L}^{5}} $ 0.005 $ {m}^{\mathrm{'}} $ $ \dfrac{m}{1/2\rho {L}^{3}} $ 0.178
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表 A1 符号定义
Tab. A1 Parameter Definitions
参数 定义 AUV 自主水下潜水器 AUG 自主水下滑翔机 AUH 自主圆碟形潜水器 ROV 遥控潜水器 HOV 载人潜水器 HG1 原始圆碟形潜水器 HG3 改进外形后的圆碟形潜水器 PMM 平面运动机构 VG 潜水器重心的速度矢量 LG 潜水器重心的角动量矢量 MG 作用于潜水器重心的合力矩 W, B AUH的重力与浮力 V AUH初始速度 L AUH长度 u, v, w 纵荡速度，横荡速度，垂荡速度 ub 水池实验背景流速 $ {\dot{u}} $ 纵荡速度的导数 p, q, r 横摇速度，纵摇速度，艏摇速度 My 倾覆力矩 ω 正弦运动频率 η 横荡运动振幅 o0x0y0z0 空间固定坐标系 oxyz 运动坐标系 xG, yG, zG 潜水器重心在动系中的矢量位置 ϕ, θ, ψ 横摇角，俯仰角，偏航角 Ixx, Iyy, Izz 潜水器对ox、oy、oz 3轴的转动惯量 ${ I_{xy}^{G},\, I_{yz}^{G},\, I_{zx}^{G}} $ 潜水器对ox、oy、oz 3轴的惯性积 X, Y, Z x，y，z方向的所有外力 K, M, N x，y，z方向的所有外力矩 τRB 外力和外力矩 g(η) 静力和静力矩 τH 流体动力和力矩 τE 环境力量和力矩 τ 推进控制力和力矩 ρ 液体密度 m AUH质量 GH, GV 水平和垂直平面动态稳定指数 ε 耗散率 Δy 边界层第一层厚度 δ 边界层厚度 $ {{X}_{\dot{u}}} $ X对$ {\dot{u}} $的偏导数，其他同理 ${ {X}_{u\left| u\right| }} $ X对u|u|的偏导数，其他同理 Xu X对u的偏导数，其他同理 FP 螺旋桨的推力 R 螺旋桨的安装半径 Fx, Fy, Fz x，y，z方向的水动力
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