Numerical solution and sensitivity analysis of hydrodynamic force derivatives on maneuverability prediction
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摘要:
目的 为了兼顾船舶操纵运动预报的成本与精度，基于数值计算方法，结合水动力导数敏感度分析，提出一种船舶操纵运动预报方法。 方法 首先，求解RANS方程，应用流体体积（VOF）法捕捉自由液面，采用动态网格方法对DTMB 5415船型进行约束运动的数值计算，并将回归得到的线性水动力导数与试验值进行对比，验证数值方案的有效性；然后，基于MMG分离建模方法建立DTMB 5415船模的操纵数学模型，并利用龙格－库塔算法进行求解，对船舶回转和Z形操纵运动进行仿真；最后，分析水动力导数对这2种操纵运动的敏感度。 结果 结果显示，采用所提方法得到的操纵轨迹和衡准参数仿真结果与试验结果一致，回转运动参数的平均误差为5.1%，Z形操纵运动参数的平均误差为11.7%，较文献使用CFD进行自航船模模拟得到运动参数的精度与计算成本均有所改善；水动力敏感度分析结果也验证了部分非线性水动力导数对操纵性衡准的影响较小，可以使用经验公式进行估算。 结论 研究表明采用所提方法进行船舶操纵性预报方法可行，可在满足工程应用精度的同时大大减少计算成本，尤其适用于船舶设计阶段的操纵性预报及优化。 Abstract:Objectives Aiming at balancing the cost and accuracy of ship maneuvering motion prediction, a numerical calculation based prediction approach is presented, combined with the sensitivity analysis of hydrodynamic derivatives. Methods First, the numerical calculation is carried out by solving the RANS equations, employing the method of volume-of-fluid (VOF) to capture the free-water surface and putting constraints on the motion of DTMB 5415 model, additional comparison of the linear hydrodynamic derivatives obtained from the regression are conducted with the experimental data so as to verify the validity of the proposed numerical scheme. Furthermore, a ship maneuvering mathematical model of DTMB 5415 is established on the basis of the maneuvering mathematical model group (MMG) method, and the Runge-Kutta algorithm is utilized to solve the equations and the model's turning and zigzag maneuvering motions are simulated. Finally, the sensitivity of the hydrodynamic derivatives of the two maneuvering motions are analyzed. Results The results show that the modelling results of ship motion trajectory and parameter for criteria obtained by the proposed methods are agree well with the experimental data, among which the average errors of the parameters of turning and zigzag maneuvering motion are 5.1% and 11.7% respectively. Compared with the results of the self-propelled ship model simulation using CFD, both the accuracy and cost are improved. The sensitivity analysis also verify that some nonlinear hydrodynamic derivatives have little influence on the maneuverability criterion, and can be estimated using empirical formulas. Conclusions The proposed method is feasible for ship maneuverability motions prediction, which can meet the engineering application precision and reduce the calculation cost greatly, especially suitable for the maneuverability motions prediction and optimization in ship design stage. -
图 2 DTMB 5415船模在流域中的位置及边界条件示意图
Figure 2. Computational domain and boundary conditions of DTMB 5415 ship model
图 4 CFD阻力计算值、试验值及拟合曲线
Figure 4. The calculated results, experimental data and fitting curve of CFD resistance
图 5 纯横荡运动横向力、回转力矩的计算值与试验值比较
Figure 5. Comparison of the calculated results and experimental data of sway force and yaw moment in pure sway motion
图 6 纯艏摇运动横向力、回转力矩的计算值与试验值比较
Figure 6. Comparison of the calculated and experimental data of sway force and yaw moment of pure yaw motion
图 9 回转运动中仿真值与试验值的对比
Figure 9. Comparison of the simulated results and experimental data for turning motion
图 10 Z形运动仿真值与试验值的对比
Figure 10. Comparison of the simulated results and experimental data for zigzag motion
表 1 DTMB 5415船模主要参数
Table 1. Main parameters of DTMB 5415 ship model
主尺度 全尺度模型 MARIN模型 缩尺比λ 1 35.48 垂线间长Lpp /m 142 4.002 型宽B/m 19.06 0.538 吃水T/m 6.15 0.173 $排水体积 \nabla/{\mathrm{m} }^{3}$ 8 424.4 0.189 湿表面面积S/m2 2 972.6 2.361 螺旋桨直径DP /m 6.15 0.173 舵面积AR /m2 15.4 0.012 2 
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表 2 网格收敛性分析
Table 2. Grid convergence analysis
网格类型 网格数量 总阻力Rf /N 误差/% 计算值 试验值 细网格 2.46×106 18.416 18.41 0.03 中等网格 1.10×106 18.534 18.41 0.67 粗网格 0.56×106 18.821 18.41 2.23 ${R_{\rm{G}}}$ 0.41 − −
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表 3 DTMB 5415船模的总阻力CFD计算值与试验值比较
Table 3. Comparison of the CFD calculated results and experimental data of total resistance for DTMB 5415 ship model
$F_r= U/\sqrt{gL}$ U/(m·s−1) 总阻力Rf /N 误差/% 计算值 试验值 0.138 0.864 4.286 4.32 −0.79 0.210 1.315 10.15 − − 0.280 1.751 18.534 18.41 0.67 0.330 2.067 27.12 − − 0.410 2.567 55.05 55.46 −0.73
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表 4 纯横荡计算工况
Table 4. Calculation conditions of pure sway motion
Fr u/(m·s−1) f /(r·min−1) a/m $ v'_{\text{max}} $ 0.28 1.755 7 0.083 6 0.03 0.28 1.755 7 0.167 0 0.07 0.28 1.755 7 0.415 8 0.17
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表 5 纯艏摇计算工况
Table 5. Calculation conditions of pure yaw motion
Fr u/(m·s−1) f /(r·min−1) $ {\psi }_{\mathrm{m}\mathrm{a}\mathrm{x}} $/rad $ r'_{\text{max}} $ 0.28 1.755 7 0.029 91 0.05 0.28 1.755 7 0.179 5 0.30 0.28 1.755 7 0.269 2 0.45 0.28 1.755 9 0.279 2 0.60 0.28 1.755 9 0.349 0 0.75
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表 6 二次拟合的水动力导数
Table 6. Hydrodynamic derivatives obtained by quadratic fitting
水动力导数 计算值 试验值 误差/% $ {Y}_{\nu } $ −160.7 −190.2 −15.51 $ {Y}_{\dot{\nu }} $ −151.6 −161.5 −6.13 $ {N}_{\nu } $ −431.7 −426.3 1.27 $ {N}_{\dot{\nu }} $ −85.45 −86.26 −0.94 $ {Y}_{r} $ −181 −172.5 4.93 $ {Y}_{\dot{r}} $ −67.27 −63.78 5.47 $ {N}_{r} $ −551.9 −547.6 0.79 $ {N}_{\dot{r}} $ −203.4 −203 0.20
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表 7 经验公式计算的水动力导数
Table 7. Hydrodynamic derivatives calculated by empirical formulas
水动力导数 数值 水动力导数 数值 $ X_{vv}' $ −0.086 $ Y_{vrr}' $ −0.940 $ X_{rr}' $ 0.007 $ N_{vv}' $ 0.086 $ X_{vr}' $ −0.052 $ N_{rr}' $ −0.056 $ Y_{vv}' $ −4.162 $ N_{vvr}' $ −0.614 $ Y_{rr}' $ −0.015 $ N_{vrr}' $ −0.016 $ Y_{vvr}' $ −0.407 $ X\dot{u} $ −0.020
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表 8 回转运动参数仿真值与试验值的对比
Table 8. Comparison of the simulated results and experimental data for turning motion parameters
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表 9 Z形操纵运动参数仿真值与试验值的对比
Table 9. Comparison of simulated results and experiment data for zigzag motion parameters
参数 仿真值 试验值 SB[18] CFDB[5] 本文方法与
试验值的误差/%$ {\psi }_{\mathrm{m}\mathrm{a}\mathrm{x}1}/ $(°) 25.82 24.61 24.15 26.38 4.9 $ {\psi }_{\mathrm{m}\mathrm{a}\mathrm{x}2}/ $(°) 25.84 24.96 24.69 27.02 3.5 $ {\psi }_{\mathrm{O}\mathrm{V}1}/ $(°) 5.82 4.61 4.15 6.38 26.2 $ {\psi }_{\mathrm{O}\mathrm{V}2}/ $(°) 5.84 4.96 4.69 7.02 17.7 $ {t}_{\alpha }/ $s 21.8 25.1 − − −13.1 $ {t}_{\mathrm{l}}/ $s 3.556 3.4 − − 4.6 预报平均误差/% 4.6 23.8 11.7
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表 10 水动力导数对回转运动参数的敏感度
Table 10. Sensitivity of hydrodynamic derivatives to turning motion parameters
水动力导数 敏感度/% 敏感度
平均值/%Ad Tr DT Do rs Us Yv 1.75 7.59 4.53 4.15 −1.76 2.78 3.76 $ {Y}_{\dot{\nu }} $ −0.27 −2.63 −3.12 −3.73 −2.61 −6.38 3.13 Yr −1.73 −7.47 −4.36 −4.06 1.77 −2.65 3.67 Nv −31.01 −36.09 −35.72 −35.50 32.60 0.40 28.55 Nr 30.32 34.97 34.76 34.69 −36.94 −0.65 28.72 $ {N}_{\dot{r}} $ 2.80 −0.31 −0.23 0.00 0.00 0.00 0.56 Xvr 0.84 1.85 1.67 1.77 1.22 3.04 1.73 Xvv −0.39 −0.63 −0.80 −0.86 −0.61 −0.27 0.59 Xrr 0.40 1.08 0.76 0.79 0.55 1.37 0.82 Yvv 2.14 12.28 7.47 7.19 −2.96 4.89 6.15 Yrr −0.11 −0.59 −0.43 −0.41 0.18 −0.27 0.33 Yvrr 0.95 6.48 3.99 4.08 −1.72 2.79 3.34 Yvvr −0.11 −0.67 −0.52 −0.52 0.23 −0.35 0.40 Nvv 1.89 2.41 2.40 2.41 −2.44 −0.03 1.93 Nrr 14.18 18.25 18.46 18.60 −20.13 −0.32 14.99 Nvrr −0.56 −0.71 −0.79 −0.81 0.82 0.01 0.62 Nvvr 6.52 8.80 8.99 9.08 −9.23 −0.12 7.12
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表 11 水动力导数对Z形操纵运动参数的敏感度
Table 11. Sensitivity of hydrodynamic derivatives to zigzag motion parameters
水动力导数 敏感度/% 敏感度
平均值/%$ {\psi }_{\mathrm{O}\mathrm{V}1} $ $ {\psi }_{\mathrm{O}\mathrm{V}2} $ $ {t}_{\alpha } $ $ {t}_{\mathrm{l}} $ Yv −3.45 −3.22 4.36 −33.75 11.19 $ {Y}_{\dot{\nu }} $ 0.19 0.16 4.13 16.87 5.34 Yr 2.63 2.62 −4.36 22.63 8.06 Nv 16.55 16.03 −32.80 88.59 38.49 Nr −17.20 −16.09 29.36 −66.09 32.18 $ {N}_{\dot{r}} $ 4.43 4.27 6.88 50.62 16.55 Xvr 0.27 0.38 0.00 1.41 0.51 Xvv −0.14 −0.47 0.00 0.00 0.15 Xrr 0.12 0.36 0.00 0.00 0.12 Yvv −4.53 −3.97 3.90 −39.37 12.94 Yrr 0.42 0.06 0.00 1.41 0.47 Yvrr −1.23 −0.96 1.38 −11.25 3.71 Yvvr 0.39 0.07 0.00 1.41 0.47 Nvv −0.53 −0.77 1.61 −4.22 1.78 Nrr −4.60 −5.08 8.94 −16.87 8.87 Nvrr 0.42 0.45 0.00 0.00 0.22 Nvvr −2.30 −1.80 2.75 −5.62 3.12
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